langfassung

14 files changed, 1199 insertions, 406 deletions

diff --git a/langfassung/docs/2_einleitung.aux b/langfassung/docs/2_einleitung.aux
deleted file mode 100644
index 44243d1..0000000
--- a/langfassung/docs/2_einleitung.aux
+++ /dev/null
@@ -1,30 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Themen}{3}{subsection.1.1}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Motivation}{3}{subsection.1.2}}
-\@setckpt{docs/2_einleitung}{
-\setcounter{page}{4}
-\setcounter{equation}{0}
-\setcounter{enumi}{0}
-\setcounter{enumii}{0}
-\setcounter{enumiii}{0}
-\setcounter{enumiv}{0}
-\setcounter{footnote}{2}
-\setcounter{mpfootnote}{0}
-\setcounter{part}{0}
-\setcounter{section}{1}
-\setcounter{subsection}{2}
-\setcounter{subsubsection}{0}
-\setcounter{paragraph}{0}
-\setcounter{subparagraph}{0}
-\setcounter{figure}{1}
-\setcounter{table}{0}
-\setcounter{parentequation}{0}
-\setcounter{Item}{0}
-\setcounter{Hfootnote}{2}
-\setcounter{bookmark@seq@number}{3}
-\setcounter{ContinuedFloat}{0}
-\setcounter{lstnumber}{1}
-\setcounter{section@level}{2}
-\setcounter{lstlisting}{0}
-}
diff --git a/langfassung/docs/3_hauptteil.aux b/langfassung/docs/3_hauptteil.aux
deleted file mode 100644
index f598ad5..0000000
--- a/langfassung/docs/3_hauptteil.aux
+++ /dev/null
@@ -1,67 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{5}{subsection.2.1}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{5}{subsubsection.2.1.1}}
-\newlabel{eq:NFW_profile}{{1}{5}{Das Navarro-Frenk-White Profil}{equation.2.1}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{5}{subsubsection.2.1.2}}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }}{6}{figure.caption.3}}
-\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
-\newlabel{fig:galaxy}{{2}{6}{Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }{figure.caption.3}{}}
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen.   Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie   Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }}{6}{figure.caption.4}}
-\newlabel{fig:rho}{{3}{6}{Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen. \\ Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie \\ Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }{figure.caption.4}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{7}{subsubsection.2.1.3}}
-\newlabel{subsec:lookup}{{2.1.3}{7}{Lookup Tabellen}{subsubsection.2.1.3}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{7}{subsection.2.2}}
-\newlabel{eq:dark_matter}{{3}{7}{Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{equation.2.3}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{7}{subsection.2.3}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{8}{subsection.2.4}}
-\newlabel{subsec:big_o}{{2.4}{8}{Rechenaufwand}{subsection.2.4}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{8}{subsection.2.5}}
-\newlabel{subsec:speeding_things_up}{{2.5}{8}{Beschleunigung der Generation}{subsection.2.5}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{8}{subsubsection.2.5.1}}
-\newlabel{subsec:lookuptable}{{2.5.1}{8}{Lookuptable}{subsubsection.2.5.1}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{9}{subsubsection.2.5.2}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Amazon Web Services}{9}{section*.5}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{9}{subsubsection.2.5.3}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{9}{subsection.2.6}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{9}{subsubsection.2.6.1}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{10}{section*.6}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{11}{subsection.2.7}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{11}{subsubsection.2.7.1}}
-\newlabel{eq:n-body-2nd}{{10}{11}{Das n-Körper Problem}{equation.2.10}{}}
-\newlabel{eq:hamilton}{{11}{11}{Das n-Körper Problem}{equation.2.11}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{12}{subsubsection.2.7.2}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{12}{subsubsection.2.7.3}}
-\newlabel{eq:gravitation_law}{{16}{12}{Berechnung der wirkenden Kräfte}{equation.2.16}{}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Masse der Sterne}{12}{section*.7}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Abstand der Sterne}{12}{section*.8}}
-\newlabel{eq:pytagoras}{{17}{12}{Abstand der Sterne}{equation.2.17}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.8}Weiteres}{12}{subsection.2.8}}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }}{13}{figure.caption.9}}
-\newlabel{fig:spiralgalaxy}{{4}{13}{Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }{figure.caption.9}{}}
-\@setckpt{docs/3_hauptteil}{
-\setcounter{page}{14}
-\setcounter{equation}{17}
-\setcounter{enumi}{0}
-\setcounter{enumii}{0}
-\setcounter{enumiii}{0}
-\setcounter{enumiv}{0}
-\setcounter{footnote}{3}
-\setcounter{mpfootnote}{0}
-\setcounter{part}{0}
-\setcounter{section}{2}
-\setcounter{subsection}{8}
-\setcounter{subsubsection}{0}
-\setcounter{paragraph}{0}
-\setcounter{subparagraph}{0}
-\setcounter{figure}{4}
-\setcounter{table}{0}
-\setcounter{parentequation}{0}
-\setcounter{Item}{0}
-\setcounter{Hfootnote}{3}
-\setcounter{bookmark@seq@number}{22}
-\setcounter{ContinuedFloat}{0}
-\setcounter{lstnumber}{15}
-\setcounter{section@level}{2}
-\setcounter{lstlisting}{0}
-}
diff --git a/langfassung/docs/4_ergebnisse.aux b/langfassung/docs/4_ergebnisse.aux
deleted file mode 100644
index ec4d7a3..0000000
--- a/langfassung/docs/4_ergebnisse.aux
+++ /dev/null
@@ -1,31 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{15}{subsection.3.1}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{15}{subsection.3.2}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Fazit}{15}{subsection.3.3}}
-\@setckpt{docs/4_ergebnisse}{
-\setcounter{page}{16}
-\setcounter{equation}{17}
-\setcounter{enumi}{0}
-\setcounter{enumii}{0}
-\setcounter{enumiii}{0}
-\setcounter{enumiv}{0}
-\setcounter{footnote}{3}
-\setcounter{mpfootnote}{0}
-\setcounter{part}{0}
-\setcounter{section}{3}
-\setcounter{subsection}{3}
-\setcounter{subsubsection}{0}
-\setcounter{paragraph}{0}
-\setcounter{subparagraph}{0}
-\setcounter{figure}{4}
-\setcounter{table}{0}
-\setcounter{parentequation}{0}
-\setcounter{Item}{0}
-\setcounter{Hfootnote}{3}
-\setcounter{bookmark@seq@number}{26}
-\setcounter{ContinuedFloat}{0}
-\setcounter{lstnumber}{15}
-\setcounter{section@level}{2}
-\setcounter{lstlisting}{0}
-}
diff --git a/langfassung/docs/5_quellen.aux b/langfassung/docs/5_quellen.aux
deleted file mode 100644
index f7e47e9..0000000
--- a/langfassung/docs/5_quellen.aux
+++ /dev/null
@@ -1,32 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{17}{section*.11}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tim Tugendhat}{17}{section*.12}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{17}{section*.13}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{17}{section*.14}}
-\@setckpt{docs/5_quellen}{
-\setcounter{page}{18}
-\setcounter{equation}{17}
-\setcounter{enumi}{0}
-\setcounter{enumii}{0}
-\setcounter{enumiii}{0}
-\setcounter{enumiv}{0}
-\setcounter{footnote}{3}
-\setcounter{mpfootnote}{0}
-\setcounter{part}{0}
-\setcounter{section}{4}
-\setcounter{subsection}{0}
-\setcounter{subsubsection}{0}
-\setcounter{paragraph}{0}
-\setcounter{subparagraph}{0}
-\setcounter{figure}{4}
-\setcounter{table}{0}
-\setcounter{parentequation}{0}
-\setcounter{Item}{0}
-\setcounter{Hfootnote}{3}
-\setcounter{bookmark@seq@number}{27}
-\setcounter{ContinuedFloat}{0}
-\setcounter{lstnumber}{15}
-\setcounter{section@level}{4}
-\setcounter{lstlisting}{0}
-}
diff --git a/langfassung/docs/6_abgabe.tex b/langfassung/docs/6_abgabe.tex
index 22fe221..f8112d9 100755
--- a/langfassung/docs/6_abgabe.tex
+++ b/langfassung/docs/6_abgabe.tex
@@ -59,7 +59,7 @@
 %     \end{pmatrix}
 % \end{equation}
 
-\subsection{Spiralgalaxien}
+\subsection{Spiralgalaxies}
 
 \subsection{Using Object Oriented Programming (OOP) techniques}
 
@@ -271,3 +271,535 @@ Not to be considered:
   \item any kind of resistance
   \item acceleration
 \end{itemize}
+
+\subsection{Notes}
+
+\begin{itemize}
+  \item Don't search for spheres very far away!
+\end{itemize}
+
+\begin{equation}
+  % \sum_{lower}^{upper} + \sum_{lower}^{upper} - \sum_{lower}^{upper}
+  \sum A_{fi} + \sum B_{fi} - \sum AB_{fij}
+\end{equation}
+
+\begin{itemize}
+  \item USE dictionaries to store which stars are in wich spheres
+\end{itemize}
+
+\subsection{exec.py}
+
+The exec.py file is used to execute the galaxytools defined in galaxytools.py.
+
+\subsubsection{Importing the galaxytools}
+
+\begin{lstlisting}
+  import galaxytools as galaxytools
+\end{lstlisting}
+
+The complete prgramm is compressed into one object. This Object has to be
+imported in order to be used.
+
+\subsubsection{Generate a new galaxy}
+
+\begin{lstlisting}
+  galaxy = galaxytools.new_galaxy(100)
+\end{lstlisting}
+
+Using the previously imported library, one can start building a galaxy by
+calling the function new\_galaxy(...). The parameter inside the braces defines
+the size of the galaxy.
+
+\subsubsection{Generate new stars in the galaxy}
+
+\begin{lstlisting}
+  galaxy.gen_new_stars(100)
+\end{lstlisting}
+
+The function new\_stars(...) is used to generate in given amount of new stars.
+
+\subsubsection{Print the coordinates of every star in the galaxy relative to
+the origin}
+
+\begin{lstlisting}
+  galaxy.print_stars()
+\end{lstlisting}
+
+Printing the coordinates of every star in the galaxy is useful for debugging:
+It is clearly visible if something is going wrong on the first look. The
+range of the galaxy might be wrong or the whole galaxy might be completely
+wrong scaled.
+
+\subsubsection{Calculate the forces acting inbetween all the stars in the
+galaxy}
+
+\begin{lstlisting}
+  galaxy.calc_all_forces()
+\end{lstlisting}
+
+the function calc\_all\_forces() if used to calculate all the forces acting
+in the selected galaxy. The O notation for this can be calculated using the
+following equation: \( O(n) = n^2 \).
+
+\subsubsection{Print the individual forces acting on the stars}
+
+\begin{lstlisting}
+  galaxy.print_individual_forces()
+\end{lstlisting}
+
+The individual forces (x, y, z) acting on the star can be printed out too!
+Just use the function print\_individual\_forces() and you will recieve the
+individual forces nicley formatted.
+
+\subsubsection{Generate the coordinates of the positions for the spheres}
+
+\begin{lstlisting}
+  galaxy.gen_sphere_positions(2)
+\end{lstlisting}
+
+To generate the sphere positions subdividing the galaxy, the
+gen\_sphere\_positions(...) function is utilized. The Parameter defines how
+many spheres are generated on one axis of the galaxy, so a higher value equals
+more spheres and so a longer time to compute. An infinite high value can be used
+if the value between each star should be calculated (use at own risk!).
+
+\subsubsection{Calculate the forces after 1 time step}
+
+\begin{lstlisting}
+  galaxy.gen_forces_after_t(1)
+\end{lstlisting}
+
+Calculating the new position after one timestep makes it possible to animate
+the galaxy and so visualizing it in an exciting way making people think you've
+done something awesome! This can be acchieved by using the
+gen\_forces\_after\_t(...) function. It uses a timestep as an argument and
+uses it to calculate the new coordinates of the star.
+
+\subsection{galaxytools.py}
+
+Inside this file, pretty much everything for building a galaxy is defined.
+
+\subsubsection{Importing important libraries}
+
+\begin{lstlisting}
+# Import libraries
+import math as math  # general math
+import numpy as np  # advanced math
+# import matplotlib.pyplot as plt  # plotting things
+\end{lstlisting}
+
+This part of the code is used to import libraries which are then used to do
+e.g. advanced math.
+
+\subsubsection{Generating the new\_galaxy class}
+
+\begin{lstlisting}
+# class used to create galaxies
+class new_galaxy(object):
+\end{lstlisting}
+
+The class definition defines the galaxytools classname as new\_galaxy
+
+\subsubsection{Initialisation}
+
+\begin{lstlisting}
+
+    # Initialisation
+    def __init__(self, galaxy_range):
+        print(
+            """>>> Initialising the list storing coordinates, forces and other
+            values"""
+        )
+
+        # list used for storing the coordinates os the stars
+        self.list_coords = []
+
+        # list storing the overall force acting on one star
+        self.list_force_star = []
+
+        # list storing the coordinates of the midpoints of the spheres dividing
+        # the galaxy into equaly big sized cells
+        self.list_sphere_coords = []
+
+        # self.list_sphere_stars = np.array(3, )
+
+        print("\tDone\n")
+        print(">>> Initialising variables and constants")
+
+        # variable storing the number of stars generated
+        self.num_of_stars = 0
+
+        self.galaxy_range = int(galaxy_range)
+
+        # define the universal gravitational constant
+        self.G = 6.67408 * 10e11
+
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Generating new stars}
+
+\begin{lstlisting}
+
+    # generate n new stars and store the coordinates in list_coords
+    # n = number of stars to be generated
+    # galaxy_range = size of the galaxy
+    def gen_new_stars(self, n):
+        print(">>> Generating Stars...")
+
+        # for a given number of stars
+        for i in range(0, n):
+
+            # generate a temporary random coordinate inside a given range using
+            # numpy
+            self.temp_coord = np.random.uniform(
+                low=0, high=self.galaxy_range, size=(4, ))
+
+            # append the random coordinate to the list storing the coordinates
+            self.list_coords.append(self.temp_coord)
+
+        # increment the generated star counter
+        self.num_of_stars += n
+        print("\tDone")
+        print("\tGenerated " + str(n) + " Stars\n")
+
+\end{lstlisting}
+
+\subsubsection{Print out all the star coordinates}
+
+\begin{lstlisting}
+
+    # print out all the coordinates in list_coords
+    def print_stars(self):
+        print(">>> Listing the coordinates of all stars:")
+        # print the coordinates of every star
+        for value in self.list_coords:
+            print(value)
+
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Calculate the forces acting inbetween two stars}
+
+\begin{lstlisting}
+
+    # calculate the forces acting between two stars on a specified axis
+    # star1 = coordinates of the first star
+    # star2 = coordinates of the second star
+    # axes = "x", "y" or "z" (CASE SENSITIVE!)
+    def calc_forces(self, star1, star2, axes):
+        if axes == "x":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[0] - star2[0], 2))
+        elif axes == "y":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[1] - star2[1], 2))
+        elif axes == "z":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[2] - star2[2], 2))
+
+        # stop division by zero
+        if distance == 0:
+            pass
+        else:
+            # return the acting force
+            return self.G * mass / math.pow(distance, 2)
+
+\end{lstlisting}
+
+\subsubsection{Calculate all the forces acting in the galaxy}
+
+\begin{lstlisting}
+
+    # calculate all the forces acting in the current galaxy
+    def calc_all_forces(self):
+        print(">>> Calculating all the forces acting inbetween the stars:")
+
+        if (self.num_of_stars <= 5):
+            # print some information above the columns
+            print(">>> Printing the forces acting inbetween every star")
+            print("{:-<60}".format(""))
+            print("\t| {:<3}| {:<3}| ".format("a", "b"))
+            print("\t+{:-<4}+{:-<4}+{:-<60} ".format("", "", ""))
+
+        else:
+            print("\t[W] Too many stars to print out!")
+            print("{:-<60}".format(""))
+
+        # for every star
+        for i in range(0, self.num_of_stars):
+
+            # initialize
+            self.force = 0
+
+            # every other star:
+            for j in range(0, self.num_of_stars):
+
+                # don't calculate the force between a star and and itself
+                if i != j and i < j:
+                    self.arr_force = np.array((0, 0, 0))
+
+                    # calculate the force between the two stars
+                    force_x = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "x")
+                    force_y = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "y")
+                    force_z = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "z")
+
+                    # print("overall force: ", end="")
+                    self.arr_force = np.array((force_x, force_y, force_z))
+
+                    if (self.num_of_stars <= 5):
+                        print("\t| {:<3}| {:<3}| {:<60}".format(
+                            str(i), str(j), str(self.arr_force)))
+                    """
+                    force_x = 42
+                    force_y = 36
+                    force_z = 24
+
+                    (0, 0, 0) --> (42, 36, 24)
+                    """
+
+            # append the variable to the list storing all the forces
+            self.list_force_star.append(self.arr_force)
+
+        print("{:-<60}".format(""))
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Print the individual forces acting on one star}
+
+\begin{lstlisting}
+
+    # print the individual forces acting on a star
+    def print_individual_forces(self, n=None, print_confirm=False):
+        print(">>> Printing the individual forces acting on every star")
+
+        if self.num_of_stars > 10:
+            print("\t[W] Too many stars to print out!")
+            print("{:-<60}".format(""))
+
+            for i in range(0, 3):
+                print("\t" + str(i) + " " + str(self.list_force_star[i]))
+
+            print("\n\t...\n")
+
+            for i in range(
+                    int(len(self.list_force_star) - 3),
+                    len(self.list_force_star)):
+                print("\t" + str(i) + " " + str(self.list_force_star[i]))
+            print("{:-<60}".format(""))
+
+        else:
+            print("{:-<60}".format(""))
+            if n is None:
+                # for value in self.list_force_star:
+                for i in range(0, len(self.list_force_star)):
+                    print("\t" + str(i) + " " + str(self.list_force_star[i]))
+            else:
+                print(self.list_force_star[n])
+
+            print("{:-<60}".format(""))
+            print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Find out if a star is inside one sphere}
+
+\begin{lstlisting}
+
+    # star      [x, y, z, mass]
+    # sphere    [x, y, z, radius]
+    def is_star_in_sphere(self, star, sphere):
+
+        # define the sphere values
+        self.sphere_x = sphere[0]
+        self.sphere_y = sphere[1]
+        self.sphere_z = sphere[2]
+        self.sphere_r = sphere[3]
+
+        # define the star coordinates
+        self.star_x = star[0]
+        self.star_y = star[1]
+        self.star_z = star[2]
+
+        # find out the distance between the point and the center of the sphere
+        # if the distance is bigger than the radius of the sphere, the point is
+        # not inside the sphere. Elsewise, the point is inside the sphere
+
+        x = math.pow(self.sphere_x - self.star_x, 2)
+        y = math.pow(self.sphere_y - self.star_y, 2)
+        z = math.pow(self.sphere_z - self.star_z, 2)
+        r = math.sqrt(x + y + z)
+
+        if r > self.sphere_r:
+            return False
+        else:
+            return True
+
+        # self.sphere_x_neg = self.sphere_x - self.sphere_r
+        # self.sphere_x_pos = self.sphere_x + self.sphere_r
+        #
+        # self.sphere_y_neg = self.sphere_y - self.sphere_r
+        # self.sphere_y_pos = self.sphere_y + self.sphere_r
+        #
+        # self.sphere_z_neg = self.sphere_z - self.sphere_r
+        # self.sphere_z_pos = self.sphere_z + self.sphere_r
+        #
+        # # find out if the star is inside the sphere
+        # if self.sphere_x_neg < self.star_x < self.sphere_x_pos:
+        #     if self.sphere_y_neg < self.star_y < self.sphere_y_pos:
+        #         if self.sphere_z_neg < self.star_z < self.sphere_z_pos:
+        #             return True
+        #         else:
+        #             return False
+        #     else:
+        #         return False
+        # else:
+        #     return False
+
+\end{lstlisting}
+
+\subsubsection{Find out which star in in which spheres}
+
+\begin{lstlisting}
+
+    # find out which stars in in which spheres
+    def is_star_in_sphere_all(self):
+
+        # print(self.sphrer_rs)
+
+        print(">>> is_star_in_sphere_all")
+
+        # initialize a temporary counter in order to index the spheres
+        tmp_counter = 0
+
+        # cycle through all the stars
+        for sphere in self.sphere_coords:
+            # print("sphere: " + str(sphere))
+
+            tmp_list = []
+            for star in self.list_coords:
+                # parse the needed values from the sphere list
+
+                # if the star is inside the sphere
+                if (self.is_star_in_sphere(star, sphere) is True):
+                    # print("\nstar: " + str(star))
+
+                    star_x = []
+
+                    for value in star:
+                        # print(value, end=" ")
+                        # print("")
+                        star_x.append(value)
+
+                    # print("star_x :" + str(star_x))
+
+                    tmp_list.append(star_x)
+
+            # print("")
+            # print("tmp_list: " + str(tmp_list))
+            # print("END")
+
+            self.sphere_coords[sphere] = tmp_list
+
+        print("")
+        # print(self.sphere_coords)
+
+        # cycle through the dictionary storing which star is in which cell
+        for value in self.sphere_coords:
+            stars_in_sphere = self.sphere_coords[value]
+
+            # calculate the individual forces in the sphere
+            self.calc_forces_sphere(stars_in_sphere)
+
+            # for value in stars_in_sphere:
+                # print(value)
+
+\end{lstlisting}
+
+\subsubsection{Generate the sphere positions}
+
+\begin{lstlisting}
+
+    # function generating the positions of the sphere cells
+    def gen_sphere_positions(self, sampling_rate):
+
+        print(">>> Generating the sphere positions")
+
+        # initialize a dictionary linking the sphere coordinates to the
+        # coordiantes of the stars in the sphere
+        self.sphere_coords = {}
+
+        # calculate the distance between the midpoints of the spheres
+        sphere_distance = int(round(self.galaxy_range / sampling_rate, 0))
+
+        # define the sphere_radius
+        tmp_var = math.pow(sphere_distance, 2)
+        sphere_radius = math.sqrt(tmp_var + tmp_var + tmp_var)
+
+        # define a sphere counter for "labeling" the spheres
+        tmp_counter = 0
+
+        # cycle through all potential points
+        for i in range(-self.galaxy_range, self.galaxy_range, sphere_distance):
+            for j in range(-self.galaxy_range, self.galaxy_range,
+                           sphere_distance):
+                for k in range(-self.galaxy_range, self.galaxy_range,
+                               sphere_distance):
+
+                    # generate a temporary array combining all values
+                    # temp_arr = np.array((i, j, k, sphere_radius, tmp_counter))
+                    tmp_arr = (i, j, k, sphere_radius, tmp_counter)
+
+                    # append the array to the list storing the sphere infos
+                    # self.list_sphere_coords.append(temp_arr)
+
+                    # print("temp_arr: " + str(temp_arr))
+                    self.sphere_coords[tmp_arr[0:4]] = []
+
+                    # increment the sphere counter
+                    tmp_counter += 1
+
+        # print(self.sphere_coords)
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Calculate the forces acting inside the sphere}
+
+\begin{lstlisting}
+
+    def calc_forces_sphere(self, stars_in_sphere):
+        print("stars_in_sphere: ", end="")
+        print(stars_in_sphere)
+        # for value in stars_in_sphere:
+        #     self.calc_all_forces(stars_in_sphere)
+        #     print(value)
+
+\end{lstlisting}
+
+\subsubsection{calculate the forces acting in every sphere}
+
+\begin{lstlisting}
+
+    def calc_forces_sphere_all():
+        # for i in range(0, len(num_of_spheres)):
+        #     for star in sphere[i]:
+        #         for star_2 in len(0, num_of_stars_in_sphere[i])
+        #             a = calc_force(star, star[j])
+
+        pass
+
+    def gen_print_forces_after_t(t):
+        pass
+
+    # def all_stars_in_sphere(self, star, se)
+
+\end{lstlisting}
+
+\subsection{GAN}
diff --git a/langfassung/master.aux b/langfassung/master.aux
index 0a6be6c..7ee3365 100755..100644
--- a/langfassung/master.aux
+++ b/langfassung/master.aux
@@ -50,16 +50,16 @@
 \setcounter{section@level}{0}
 \setcounter{lstlisting}{0}
 }
-\@writefile{toc}{\contentsline {section}{\numberline {1}Einleitung}{2}{section.1}}
-\newlabel{Einleitung}{{1}{2}{Einleitung}{section.1}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Einleitung}{3}{section.1}}
+\newlabel{Einleitung}{{1}{3}{Einleitung}{section.1}{}}
 \@auxtag{docs/2_einleitung}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Themen}{2}{subsection.1.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Themen}{3}{subsection.1.1}}
 }
 \@auxtag{docs/2_einleitung}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Motivation}{2}{subsection.1.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Motivation}{3}{subsection.1.2}}
 }
 \@setckpt{docs/2_einleitung}{%
-\setcounter{page}{2}
+\setcounter{page}{3}
 \setcounter{equation}{0}
 \setcounter{enumi}{0}
 \setcounter{enumii}{0}
@@ -85,95 +85,95 @@
 \setcounter{section@level}{2}
 \setcounter{lstlisting}{0}
 }
-\@writefile{toc}{\contentsline {section}{\numberline {2}Hauptteil}{3}{section.2}}
-\newlabel{Hauptteil}{{2}{3}{Hauptteil}{section.2}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}Hauptteil}{4}{section.2}}
+\newlabel{Hauptteil}{{2}{4}{Hauptteil}{section.2}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{3}{subsection.2.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{4}{subsection.2.1}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{3}{subsubsection.2.1.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{4}{subsubsection.2.1.1}}
 }
-\newlabel{eq:NFW_profile}{{1}{3}{Das Navarro-Frenk-White Profil}{equation.2.1}{}}
+\newlabel{eq:NFW_profile}{{1}{4}{Das Navarro-Frenk-White Profil}{equation.2.1}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{3}{subsubsection.2.1.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{4}{subsubsection.2.1.2}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }}{4}{figure.caption.3}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }}{5}{figure.caption.3}}
 }
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
-\newlabel{fig:galaxy}{{2}{4}{Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }{figure.caption.3}{}}
+\newlabel{fig:galaxy}{{2}{5}{Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }{figure.caption.3}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen.   Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie   Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }}{4}{figure.caption.4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen.   Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie   Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }}{5}{figure.caption.4}}
 }
-\newlabel{fig:rho}{{3}{4}{Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen. \\ Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie \\ Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }{figure.caption.4}{}}
+\newlabel{fig:rho}{{3}{5}{Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen. \\ Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie \\ Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }{figure.caption.4}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{5}{subsubsection.2.1.3}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{6}{subsubsection.2.1.3}}
 }
-\newlabel{subsec:lookup}{{2.1.3}{5}{Lookup Tabellen}{subsubsection.2.1.3}{}}
+\newlabel{subsec:lookup}{{2.1.3}{6}{Lookup Tabellen}{subsubsection.2.1.3}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{5}{subsection.2.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{6}{subsection.2.2}}
 }
-\newlabel{eq:dark_matter}{{3}{5}{Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{equation.2.3}{}}
+\newlabel{eq:dark_matter}{{3}{6}{Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{equation.2.3}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{5}{subsection.2.3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{6}{subsection.2.3}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{6}{subsection.2.4}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{7}{subsection.2.4}}
 }
-\newlabel{subsec:big_o}{{2.4}{6}{Rechenaufwand}{subsection.2.4}{}}
+\newlabel{subsec:big_o}{{2.4}{7}{Rechenaufwand}{subsection.2.4}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{6}{subsection.2.5}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{7}{subsection.2.5}}
 }
-\newlabel{subsec:speeding_things_up}{{2.5}{6}{Beschleunigung der Generation}{subsection.2.5}{}}
+\newlabel{subsec:speeding_things_up}{{2.5}{7}{Beschleunigung der Generation}{subsection.2.5}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{6}{subsubsection.2.5.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{7}{subsubsection.2.5.1}}
 }
-\newlabel{subsec:lookuptable}{{2.5.1}{6}{Lookuptable}{subsubsection.2.5.1}{}}
+\newlabel{subsec:lookuptable}{{2.5.1}{7}{Lookuptable}{subsubsection.2.5.1}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{7}{subsubsection.2.5.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{8}{subsubsection.2.5.2}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Amazon Web Services}{7}{section*.5}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Amazon Web Services}{8}{section*.5}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{7}{subsubsection.2.5.3}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{8}{subsubsection.2.5.3}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{7}{subsection.2.6}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{8}{subsection.2.6}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{7}{subsubsection.2.6.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{8}{subsubsection.2.6.1}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{8}{section*.6}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{9}{section*.6}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{9}{subsection.2.7}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{10}{subsection.2.7}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{9}{subsubsection.2.7.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{10}{subsubsection.2.7.1}}
 }
-\newlabel{eq:n-body-2nd}{{10}{9}{Das n-Körper Problem}{equation.2.10}{}}
-\newlabel{eq:hamilton}{{11}{9}{Das n-Körper Problem}{equation.2.11}{}}
+\newlabel{eq:n-body-2nd}{{10}{10}{Das n-Körper Problem}{equation.2.10}{}}
+\newlabel{eq:hamilton}{{11}{10}{Das n-Körper Problem}{equation.2.11}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{10}{subsubsection.2.7.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{11}{subsubsection.2.7.2}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{10}{subsubsection.2.7.3}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{11}{subsubsection.2.7.3}}
 }
-\newlabel{eq:gravitation_law}{{16}{10}{Berechnung der wirkenden Kräfte}{equation.2.16}{}}
+\newlabel{eq:gravitation_law}{{16}{11}{Berechnung der wirkenden Kräfte}{equation.2.16}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Masse der Sterne}{10}{section*.7}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Masse der Sterne}{11}{section*.7}}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Abstand der Sterne}{10}{section*.8}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Abstand der Sterne}{11}{section*.8}}
 }
-\newlabel{eq:pytagoras}{{17}{10}{Abstand der Sterne}{equation.2.17}{}}
+\newlabel{eq:pytagoras}{{17}{11}{Abstand der Sterne}{equation.2.17}{}}
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.8}Weiteres}{10}{subsection.2.8}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.8}Weiteres}{11}{subsection.2.8}}
 }
 \@setckpt{docs/3_hauptteil}{%
-\setcounter{page}{10}
+\setcounter{page}{11}
 \setcounter{equation}{17}
 \setcounter{enumi}{0}
 \setcounter{enumii}{0}
@@ -200,22 +200,22 @@
 \setcounter{lstlisting}{0}
 }
 \@auxtag{docs/3_hauptteil}{%
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }}{11}{figure.caption.9}}
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }}{12}{figure.caption.9}}
 }
-\newlabel{fig:spiralgalaxy}{{4}{11}{Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }{figure.caption.9}{}}
-\@writefile{toc}{\contentsline {section}{\numberline {3}Ergebnisse}{11}{section.3}}
-\newlabel{ergebnisse}{{3}{11}{Ergebnisse}{section.3}{}}
+\newlabel{fig:spiralgalaxy}{{4}{12}{Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }{figure.caption.9}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {3}Ergebnisse}{12}{section.3}}
+\newlabel{ergebnisse}{{3}{12}{Ergebnisse}{section.3}{}}
 \@auxtag{docs/4_ergebnisse}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{11}{subsection.3.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{12}{subsection.3.1}}
 }
 \@auxtag{docs/4_ergebnisse}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{11}{subsection.3.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{12}{subsection.3.2}}
 }
 \@auxtag{docs/4_ergebnisse}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Fazit}{11}{subsection.3.3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Fazit}{12}{subsection.3.3}}
 }
 \@setckpt{docs/4_ergebnisse}{%
-\setcounter{page}{12}
+\setcounter{page}{13}
 \setcounter{equation}{17}
 \setcounter{enumi}{0}
 \setcounter{enumii}{0}
@@ -241,22 +241,22 @@
 \setcounter{section@level}{2}
 \setcounter{lstlisting}{0}
 }
-\@writefile{toc}{\contentsline {section}{\numberline {4}Quellen und Hilfen}{13}{section.4}}
-\newlabel{quellen}{{4}{13}{Quellen und Hilfen}{section.4}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {4}Quellen und Hilfen}{14}{section.4}}
+\newlabel{quellen}{{4}{14}{Quellen und Hilfen}{section.4}{}}
 \@auxtag{docs/5_quellen}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{13}{section*.11}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{14}{section*.11}}
 }
 \@auxtag{docs/5_quellen}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tim Tugendhat}{13}{section*.12}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tim Tugendhat}{14}{section*.12}}
 }
 \@auxtag{docs/5_quellen}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{13}{section*.13}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{14}{section*.13}}
 }
 \@auxtag{docs/5_quellen}{%
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{13}{section*.14}}
+\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{14}{section*.14}}
 }
 \@setckpt{docs/5_quellen}{%
-\setcounter{page}{13}
+\setcounter{page}{14}
 \setcounter{equation}{17}
 \setcounter{enumi}{0}
 \setcounter{enumii}{0}
@@ -282,68 +282,143 @@
 \setcounter{section@level}{4}
 \setcounter{lstlisting}{0}
 }
-\@writefile{toc}{\contentsline {section}{\numberline {5}Nach der Abgabe...}{14}{section.5}}
-\newlabel{nach_der_abgabe}{{5}{14}{Nach der Abgabe..}{section.5}{}}
+\@writefile{toc}{\contentsline {section}{\numberline {5}Nach der Abgabe...}{15}{section.5}}
+\newlabel{nach_der_abgabe}{{5}{15}{Nach der Abgabe..}{section.5}{}}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Spiralgalaxies}{15}{subsection.5.1}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{15}{subsection.5.2}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{15}{subsubsection.5.2.1}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Generation of new stars}{15}{subsection.5.3}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{15}{subsection.5.4}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{15}{subsection.5.5}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{16}{subsection.5.6}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{16}{subsection.5.7}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.8}Spherical cells}{16}{subsection.5.8}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{16}{subsubsection.5.8.1}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole galaxy}{16}{subsubsection.5.8.2}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{16}{subsubsection.5.8.3}}
+}
+\newlabel{sphere_distance}{{20}{16}{Generate the position of the spheres}{equation.5.20}{}}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The Alignment of the spheres  Left: The perfect alignment covering the complete space  Right: A previously generated alignment using small spheres to cover the missing space \relax }}{17}{figure.caption.15}}
+}
+\newlabel{sphere_alignment}{{5}{17}{The Alignment of the spheres\\ Left: The perfect alignment covering the complete space\\ Right: A previously generated alignment using small spheres to cover the missing space \relax }{figure.caption.15}{}}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{17}{subsubsection.5.8.4}}
+}
+\newlabel{sphere_radius}{{21}{17}{The radius of the spheres}{equation.5.21}{}}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{17}{subsubsection.5.8.5}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{18}{subsubsection.5.8.6}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{18}{subsubsection.5.8.7}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{18}{subsection.5.9}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.10}Notes}{18}{subsection.5.10}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.11}exec.py}{19}{subsection.5.11}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.1}Importing the galaxytools}{19}{subsubsection.5.11.1}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.2}Generate a new galaxy}{19}{subsubsection.5.11.2}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.3}Generate new stars in the galaxy}{19}{subsubsection.5.11.3}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.4}Print the coordinates of every star in the galaxy relative to the origin}{19}{subsubsection.5.11.4}}
+}
+\@auxtag{docs/6_abgabe}{%
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.5}Calculate the forces acting inbetween all the stars in the galaxy}{19}{subsubsection.5.11.5}}
+}
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Spiralgalaxien}{14}{subsection.5.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.6}Print the individual forces acting on the stars}{19}{subsubsection.5.11.6}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{14}{subsection.5.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.7}Generate the coordinates of the positions for the spheres}{20}{subsubsection.5.11.7}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{14}{subsubsection.5.2.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.11.8}Calculate the forces after 1 time step}{20}{subsubsection.5.11.8}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Generation of new stars}{14}{subsection.5.3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.12}galaxytools.py}{20}{subsection.5.12}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{14}{subsection.5.4}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.1}Importing important libraries}{20}{subsubsection.5.12.1}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{14}{subsection.5.5}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.2}Generating the new\_galaxy class}{20}{subsubsection.5.12.2}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{15}{subsection.5.6}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.3}Initialisation}{20}{subsubsection.5.12.3}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{15}{subsection.5.7}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.4}Generating new stars}{21}{subsubsection.5.12.4}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.8}Spherical cells}{15}{subsection.5.8}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.5}Print out all the star coordinates}{21}{subsubsection.5.12.5}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{15}{subsubsection.5.8.1}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.6}Calculate the forces acting inbetween two stars}{22}{subsubsection.5.12.6}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole galaxy}{15}{subsubsection.5.8.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.7}Calculate all the forces acting in the galaxy}{22}{subsubsection.5.12.7}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{15}{subsubsection.5.8.3}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.8}Print the individual forces acting on one star}{23}{subsubsection.5.12.8}}
 }
-\newlabel{sphere_distance}{{20}{15}{Generate the position of the spheres}{equation.5.20}{}}
 \@auxtag{docs/6_abgabe}{%
-\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The Alignment of the spheres  Left: The perfect alignment covering the complete space  Right: A previously generated alignment using small spheres to cover the missing space \relax }}{16}{figure.caption.15}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.9}Find out if a star is inside one sphere}{24}{subsubsection.5.12.9}}
 }
-\newlabel{sphere_alignment}{{5}{16}{The Alignment of the spheres\\ Left: The perfect alignment covering the complete space\\ Right: A previously generated alignment using small spheres to cover the missing space \relax }{figure.caption.15}{}}
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{16}{subsubsection.5.8.4}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.10}Find out which star in in which spheres}{25}{subsubsection.5.12.10}}
 }
-\newlabel{sphere_radius}{{21}{16}{The radius of the spheres}{equation.5.21}{}}
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{16}{subsubsection.5.8.5}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.11}Generate the sphere positions}{26}{subsubsection.5.12.11}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{17}{subsubsection.5.8.6}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.12}Calculate the forces acting inside the sphere}{26}{subsubsection.5.12.12}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{17}{subsubsection.5.8.7}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.13}calculate the forces acting in every sphere}{26}{subsubsection.5.12.13}}
 }
 \@auxtag{docs/6_abgabe}{%
-\@writefile{toc}{\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{17}{subsection.5.9}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {5.13}GAN}{27}{subsection.5.13}}
 }
 \@setckpt{docs/6_abgabe}{%
-\setcounter{page}{17}
-\setcounter{equation}{21}
+\setcounter{page}{27}
+\setcounter{equation}{22}
 \setcounter{enumi}{0}
 \setcounter{enumii}{0}
 \setcounter{enumiii}{0}
@@ -352,7 +427,7 @@
 \setcounter{mpfootnote}{0}
 \setcounter{part}{0}
 \setcounter{section}{5}
-\setcounter{subsection}{9}
+\setcounter{subsection}{13}
 \setcounter{subsubsection}{0}
 \setcounter{paragraph}{0}
 \setcounter{subparagraph}{0}
@@ -361,7 +436,7 @@
 \setcounter{parentequation}{0}
 \setcounter{Item}{0}
 \setcounter{Hfootnote}{3}
-\setcounter{bookmark@seq@number}{45}
+\setcounter{bookmark@seq@number}{70}
 \setcounter{IncludeDepth}{0}
 \setcounter{ContinuedFloat}{0}
 \setcounter{lstnumber}{15}
@@ -369,5 +444,5 @@
 \setcounter{lstlisting}{0}
 }
 \global\@namedef{scr@dte@section@lastmaxnumwidth}{9.89993pt}
-\global\@namedef{scr@dte@subsection@lastmaxnumwidth}{16.7777pt}
-\global\@namedef{scr@dte@subsubsection@lastmaxnumwidth}{24.55545pt}
+\global\@namedef{scr@dte@subsection@lastmaxnumwidth}{21.7777pt}
+\global\@namedef{scr@dte@subsubsection@lastmaxnumwidth}{34.55545pt}
diff --git a/langfassung/master.fdb_latexmk b/langfassung/master.fdb_latexmk
index 4204fe5..ce25814 100755..100644
--- a/langfassung/master.fdb_latexmk
+++ b/langfassung/master.fdb_latexmk
@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1518176572 "master.tex" "master.pdf" "master" 1518176574
+["pdflatex"] 1519571828 "master.tex" "master.pdf" "master" 1519571831
   "/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc" 1510784331 3486 c7eadf5dcc57b3b2d11736679f6636ba ""
   "/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc" 1510784331 2405 5dcf2c1b967ee25cc46c58cd52244aed ""
   "/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc" 1510784331 2840 216e6e45ad352e2456e1149f28885bee ""
@@ -26,6 +26,7 @@
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmr6.tfm" 1510784343 11836 e3b6ce3e601aec94f64a536e7f4224d5 ""
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmr7.tfm" 1510784343 11852 5a9022f105fd1ee2797df861e79ae9a0 ""
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmr8.tfm" 1510784343 11864 309fd7f43e4a0ba39f6f7644d76e8edf ""
+  "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmri10.tfm" 1510784343 16968 1b588d0f410bd67a83eaecf8c64b2a43 ""
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmss10.tfm" 1510784343 10976 59d4d2acacc56ba9abc92645023932ec ""
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmssbx10.tfm" 1510784343 10956 dad25672db463289a56f0700c0488a86 ""
   "/usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmtt10.tfm" 1510784343 1340 adaa94f9add991a099a6fc98bbb5f02c ""
@@ -43,6 +44,7 @@
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmr6.pfb" 1510784332 123394 d390152bb30feeb496aaaa93299ee9ba ""
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmr7.pfb" 1510784332 121145 68312a933e2c689ed40ec0aba373e279 ""
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmr8.pfb" 1510784332 122174 a7a08406857c9530a0320a2517f60370 ""
+  "/usr/share/texmf-dist/fonts/type1/public/lm/lmri10.pfb" 1510784332 112593 fda2373ba4420af33949610de4c28fe8 ""
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmssbx10.pfb" 1510784332 119663 e82fa1a58f98ccd89bdbd77311ac9cf1 ""
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmsy10.pfb" 1510784332 27863 09ce3735688ffde955e72da27c95b61a ""
   "/usr/share/texmf-dist/fonts/type1/public/lm/lmsy7.pfb" 1510784332 27941 d1f5d03f61a46c3fcc3a2ba904ddda52 ""
@@ -158,6 +160,8 @@
   "/usr/share/texmf-dist/tex/latex/listings/lstmisc.sty" 1510784350 77028 c3eb00afb55a32bc13ca8da7f5234377 ""
   "/usr/share/texmf-dist/tex/latex/lm/lmodern.sty" 1510784350 1606 c17281c7cff2bbd7ff0173e1433487ec ""
   "/usr/share/texmf-dist/tex/latex/lm/omllmm.fd" 1510784350 888 44447a3a3af84a22454ef89500942d93 ""
+  "/usr/share/texmf-dist/tex/latex/lm/omllmr.fd" 1510784350 1014 b44f996e1fb8d1548178b60325f9547d ""
+  "/usr/share/texmf-dist/tex/latex/lm/omslmr.fd" 1510784350 1013 f8133048bddeed1bf1041c9d830a3723 ""
   "/usr/share/texmf-dist/tex/latex/lm/omslmsy.fd" 1510784350 805 af340a8260c447aa315cfc740ff0152f ""
   "/usr/share/texmf-dist/tex/latex/lm/omxlmex.fd" 1510784350 566 a94661f7b66063f191960bb7935b6ba2 ""
   "/usr/share/texmf-dist/tex/latex/lm/ot1lmr.fd" 1510784350 1880 bae7b659316f7344a86218ad38b01d91 ""
@@ -188,27 +192,22 @@
   "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1517504663 2404736 6ca9e6023b4459829efec74926ef1c58 ""
   "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1517504637 7736837 37c869a4b6afb01be7fdde576c304f64 ""
   "docs/1_kurzfassung.tex" 1517318438 994 74b88a132e4f651b58ad24ce5a45aa13 ""
-  "docs/2_einleitung.tex" 1517318438 2688 3128813c5bf12e1bf9b6abf3149bb698 ""
+  "docs/2_einleitung.tex" 1518211606 2688 3128813c5bf12e1bf9b6abf3149bb698 ""
   "docs/3_hauptteil.tex" 1517318438 27077 2c6e67804bf2bae299b1587798dc19b2 ""
   "docs/4_ergebnisse.tex" 1517318438 1532 aa09ddbc132532cf7821686a80cfaee0 ""
   "docs/5_quellen.tex" 1518128318 1372 ae0519f8932ec3157eb1ca10f5c6176f ""
-  "docs/6_abgabe.tex" 1518176570 8643 ecf661de0bb8cc8207825a23b87ea6b2 ""
+  "docs/6_abgabe.tex" 1519571822 26231 eb07adba5a6d9d3ab3550589e12b99e0 ""
   "figs/galaxy.png" 1517318437 158503 2e9f6025259ce041725bc1968c473272 ""
   "figs/lookup_table_rho_r_function_grid.png" 1517318437 20668 e982125a5ea76649c632f90e900f7eef ""
   "figs/sphere_alignment_cc.jpg" 1518123171 264400 13884e280e86d92f54f6ff87556db6bf ""
   "figs/spiralgalaxy.png" 1517318437 321656 70b5e0b43994c9c1d7a43cad1dee7910 ""
-  "master.aux" 1518176574 15985 3324682537796f6d154afd0680828ceb ""
-  "master.out" 1518176574 3677 b61fa051f86f22a18af030789f5c3d0e ""
-  "master.tex" 1518173850 2781 f3bf2ca1618ca79e70062c24267db86c ""
-  "master.toc" 1518176574 5055 a92c318f404f71ca12e2078a175d2902 ""
+  "master.aux" 1519571831 19906 778d1da799a2aa806f9a828e9b7c0544 ""
+  "master.out" 1519571831 6037 db8588ad2d77e7447a32fb30b6947fe9 ""
+  "master.tex" 1518627582 2648 bf6b8ac646f65e3eda1ded67e9ac41d6 ""
+  "master.toc" 1519571831 7823 7225d135c036278eb3dbbba651637b68 ""
   (generated)
-  "master.toc"
-  "master.pdf"
   "master.aux"
   "master.out"
-  "docs/3_hauptteil.aux"
-  "docs/5_quellen.aux"
-  "docs/4_ergebnisse.aux"
+  "master.toc"
   "master.log"
-  "docs/2_einleitung.aux"
-  "docs/6_abgabe.aux"
+  "master.pdf"
diff --git a/langfassung/master.fls b/langfassung/master.fls
index 0ff9eba..7a9c757 100644
--- a/langfassung/master.fls
+++ b/langfassung/master.fls
@@ -311,11 +311,16 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmtt8.tfm
 INPUT figs/sphere_alignment_cc.jpg
 INPUT ./figs/sphere_alignment_cc.jpg
 INPUT ./figs/sphere_alignment_cc.jpg
+INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/rm-lmri10.tfm
+INPUT /usr/share/texmf-dist/tex/latex/lm/omllmr.fd
+INPUT /usr/share/texmf-dist/tex/latex/lm/omllmr.fd
+INPUT /usr/share/texmf-dist/tex/latex/lm/omslmr.fd
+INPUT /usr/share/texmf-dist/tex/latex/lm/omslmr.fd
 INPUT master.aux
 INPUT ./master.out
 INPUT ./master.out
-INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rmtt.enc
 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc
+INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rmtt.enc
 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc
 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc
 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc
@@ -331,6 +336,7 @@ INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmr5.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmr6.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmr7.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmr8.pfb
+INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmri10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmssbx10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmsy10.pfb
 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmsy7.pfb
diff --git a/langfassung/master.log b/langfassung/master.log
index f9b7036..e66875b 100755..100644
--- a/langfassung/master.log
+++ b/langfassung/master.log
@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Arch Linux) (preloaded format=pdflatex 2018.2.1)  9 FEB 2018 12:42
+This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Arch Linux) (preloaded format=pdflatex 2018.2.1)  25 FEB 2018 16:17
 entering extended mode
  restricted \write18 enabled.
  file:line:error style messages enabled.
@@ -1201,7 +1201,7 @@ LaTeX Font Info:    Try loading font information for OT1+lmtt on input line 82.
  (/usr/share/texmf-dist/tex/latex/lm/ot1lmtt.fd
 File: ot1lmtt.fd 2009/10/30 v1.6 Font defs for Latin Modern
 )
-<figs/galaxy.png, id=187, 866.23625pt x 790.955pt>
+<figs/galaxy.png, id=288, 866.23625pt x 790.955pt>
 File: figs/galaxy.png Graphic file (type png)
 <use figs/galaxy.png>
 Package pdftex.def Info: figs/galaxy.png  used on input line 86. on input line 86.
@@ -1221,7 +1221,21 @@ Package babel Info: Redefining ngerman shorthand "~
 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} <./figs/galaxy.png>] (./master.toc
 LaTeX Font Info:    External font `lmex10' loaded for size
 (Font)              <5> on input line 3.
-)
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 64.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 64.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 64.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 64.
+pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored
+<to be read again> 
+                   \relax 
+l.64 ...ter 1 time step}{20}{subsubsection.5.11.8}
+                                                   [1
+
+])
 \tf@toc=\write5
 \openout5 = `master.toc'.
 
@@ -1233,13 +1247,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 104.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 104.
-pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored
-<to be read again> 
-                   \relax 
-l.104 \newpage
-               [1
-
-]
+ [2]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 106.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1263,18 +1271,18 @@ Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 38.
 )
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 109.
+(babel)             in language append on input line 108.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 109.
+(babel)             in language append on input line 108.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 109.
+(babel)             in language append on input line 108.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 109.
- [2]
+(babel)             in language append on input line 108.
+ [3]
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 111.
+(babel)             in language append on input line 110.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 111.
+(babel)             in language append on input line 110.
  (./docs/3_hauptteil.tex
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 1.
@@ -1288,7 +1296,7 @@ File: figs/galaxy.png Graphic file (type png)
 <use figs/galaxy.png>
 Package pdftex.def Info: figs/galaxy.png  used on input line 57. on input line 57.
 Package pdftex.def Info: Requested size: 341.43307pt x 311.75397pt on input line 57.
-<figs/lookup_table_rho_r_function_grid.png, id=265, 683.55376pt x 444.66125pt>
+<figs/lookup_table_rho_r_function_grid.png, id=395, 683.55376pt x 444.66125pt>
 File: figs/lookup_table_rho_r_function_grid.png Graphic file (type png)
 <use figs/lookup_table_rho_r_function_grid.png>
 Package pdftex.def Info: figs/lookup_table_rho_r_function_grid.png  used on input line 64. on input line 64.
@@ -1309,7 +1317,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 95.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 95.
- [3]
+ [4]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 95.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1318,7 +1326,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 95.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 95.
- [4 <./figs/lookup_table_rho_r_function_grid.png>]
+ [5 <./figs/lookup_table_rho_r_function_grid.png>]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 96.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1335,7 +1343,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 163.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 163.
- [5]
+ [6]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 177.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1360,7 +1368,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 236.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 236.
- [6]
+ [7]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 258.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1381,7 +1389,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 331.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 331.
- [7]
+ [8]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 502.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1394,7 +1402,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 511.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 511.
- [8]
+ [9]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 517.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1407,7 +1415,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 577.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 577.
- [9]
+ [10]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 585.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1420,25 +1428,25 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 628.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 628.
-<figs/spiralgalaxy.png, id=355, 1927.2pt x 1084.05pt>
+<figs/spiralgalaxy.png, id=485, 1927.2pt x 1084.05pt>
 File: figs/spiralgalaxy.png Graphic file (type png)
 <use figs/spiralgalaxy.png>
 Package pdftex.def Info: figs/spiralgalaxy.png  used on input line 635. on input line 635.
 Package pdftex.def Info: Requested size: 341.43307pt x 192.0439pt on input line 635.
 )
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 114.
+(babel)             in language append on input line 112.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 114.
+(babel)             in language append on input line 112.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 114.
+(babel)             in language append on input line 112.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 114.
- [10]
+(babel)             in language append on input line 112.
+ [11]
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 116.
+(babel)             in language append on input line 114.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 116.
+(babel)             in language append on input line 114.
  (./docs/4_ergebnisse.tex
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 1.
@@ -1454,41 +1462,41 @@ Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 32.
 )
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 119.
- [11 <./figs/spiralgalaxy.png>]
+(babel)             in language append on input line 116.
+ [12 <./figs/spiralgalaxy.png>]
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 119.
+(babel)             in language append on input line 116.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 119.
- [12]
+(babel)             in language append on input line 116.
+ [13]
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 121.
+(babel)             in language append on input line 118.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 121.
+(babel)             in language append on input line 118.
  (./docs/5_quellen.tex)
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 124.
+(babel)             in language append on input line 120.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 124.
+(babel)             in language append on input line 120.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 124.
+(babel)             in language append on input line 120.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 124.
- [13]
+(babel)             in language append on input line 120.
+ [14]
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 126.
+(babel)             in language append on input line 122.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 126.
+(babel)             in language append on input line 122.
  (./docs/6_abgabe.tex
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 62.
@@ -1526,7 +1534,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 122.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 122.
- [14]
+ [15]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 125.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1555,12 +1563,12 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 185.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 185.
- [15]
+ [16]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 190.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 190.
-<figs/sphere_alignment_cc.jpg, id=401, 963.6pt x 507.39563pt>
+<figs/sphere_alignment_cc.jpg, id=531, 963.6pt x 507.39563pt>
 File: figs/sphere_alignment_cc.jpg Graphic file (type jpg)
 <use figs/sphere_alignment_cc.jpg>
 Package pdftex.def Info: figs/sphere_alignment_cc.jpg  used on input line 207. on input line 207.
@@ -1581,7 +1589,7 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 231.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 231.
- [16 <./figs/sphere_alignment_cc.jpg>]
+ [17 <./figs/sphere_alignment_cc.jpg>]
 Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 231.
 Package babel Info: Redefining ngerman shorthand "~
@@ -1590,39 +1598,327 @@ Package babel Info: Redefining ngerman shorthand "|
 (babel)             in language append on input line 266.
 Package babel Info: Redefining ngerman shorthand "~
 (babel)             in language append on input line 266.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 275.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 275.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 288.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 288.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 288.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 288.
+ [18]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 290.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 290.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 294.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 294.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 303.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 303.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 313.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 313.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 322.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 322.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 334.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 334.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 344.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 344.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 354.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 354.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 356.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 356.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 356.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 356.
+ [19]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 366.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 366.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 378.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 378.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 382.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 382.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 394.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 394.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 403.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 403.
+LaTeX Font Info:    Try loading font information for OML+lmr on input line 410.
+ (/usr/share/texmf-dist/tex/latex/lm/omllmr.fd
+File: omllmr.fd 2009/10/30 v1.6 Font defs for Latin Modern
+)
+LaTeX Font Info:    Font shape `OML/lmr/m/it' in size <10> not available
+(Font)              Font shape `OML/lmm/m/it' tried instead on input line 410.
+
+Overfull \hbox (12.76065pt too wide) in paragraph at lines 410--411
+[][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 420.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 420.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 420.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 420.
+[20]
+Overfull \hbox (18.76071pt too wide) in paragraph at lines 420--421
+[][][][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+LaTeX Font Info:    Try loading font information for OMS+lmr on input line 426.
+(/usr/share/texmf-dist/tex/latex/lm/omslmr.fd
+File: omslmr.fd 2009/10/30 v1.6 Font defs for Latin Modern
+)
+LaTeX Font Info:    Font shape `OMS/lmr/m/n' in size <10> not available
+(Font)              Font shape `OMS/lmsy/m/n' tried instead on input line 426.
+LaTeX Font Info:    Font shape `OML/lmr/m/n' in size <10> not available
+(Font)              Font shape `OML/lmm/m/it' tried instead on input line 427.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 441.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 441.
+
+Overfull \hbox (18.76071pt too wide) in paragraph at lines 454--455
+[][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+
+Overfull \hbox (12.76065pt too wide) in paragraph at lines 459--460
+[][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 469.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 469.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 480.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 480.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 480.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 480.
+[21]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 484.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 484.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 512.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 512.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 542.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 542.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 542.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 542.
+ [22]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 574.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 574.
+
+Overfull \hbox (6.76059pt too wide) in paragraph at lines 602--603
+[][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 603.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 603.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 603.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 603.
+[23]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 611.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 611.
+
+Overfull \hbox (12.76065pt too wide) in paragraph at lines 630--631
+[][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+
+Overfull \hbox (18.76071pt too wide) in paragraph at lines 631--632
+[][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 665.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 665.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 665.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 665.
+[24]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 667.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 667.
+LaTeX Font Info:    Font shape `OMS/lmr/m/it' in size <10> not available
+(Font)              Font shape `OMS/lmsy/m/n' tried instead on input line 691.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 725.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 725.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 727.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 727.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 727.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 727.
+ [25]
+Overfull \hbox (18.76071pt too wide) in paragraph at lines 749--750
+[][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+
+Overfull \hbox (24.76077pt too wide) in paragraph at lines 756--757
+[][][][][][][][][][][][][][][][][][][][][][][][][][][][] 
+ []
+
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 773.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 773.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 786.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 786.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 790.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 790.
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 790.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 790.
+[26]
+Package babel Info: Redefining ngerman shorthand "|
+(babel)             in language append on input line 805.
+Package babel Info: Redefining ngerman shorthand "~
+(babel)             in language append on input line 805.
+
+
+./docs/6_abgabe.tex:810: Package pgf Error: No shape named z is known.
+
+See the pgf package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.810 	\node[left=of z]
+                        (i) {};
+This error message was generated by an \errmessage
+command, so I can't give any explicit help.
+Pretend that you're Hercule Poirot: Examine all clues,
+and deduce the truth by order and method.
+
+
+./docs/6_abgabe.tex:810: Package pgf Error: No shape named z is known.
+
+See the pgf package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.810 	\node[left=of z] (i) {};
+                               
+(That was another \errmessage.)
+
+
+./docs/6_abgabe.tex:811: Package pgf Error: No shape named z is known.
+
+See the pgf package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.811 ... -- node[above] {$p_\theta(\vec{z})$} (z)
+                                                  ;
+(That was another \errmessage.)
+
+
+./docs/6_abgabe.tex:811: Package pgf Error: No shape named z is known.
+
+See the pgf package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.811 ... -- node[above] {$p_\theta(\vec{z})$} (z)
+                                                  ;
+(That was another \errmessage.)
+
 )
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 130.
+(babel)             in language append on input line 125.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 130.
+(babel)             in language append on input line 125.
 Package babel Info: Redefining ngerman shorthand "|
-(babel)             in language append on input line 130.
+(babel)             in language append on input line 125.
 Package babel Info: Redefining ngerman shorthand "~
-(babel)             in language append on input line 130.
- [17]
-Package atveryend Info: Empty hook `BeforeClearDocument' on input line 130.
-Package atveryend Info: Empty hook `AfterLastShipout' on input line 130.
+(babel)             in language append on input line 125.
+ [27]
+Package atveryend Info: Empty hook `BeforeClearDocument' on input line 125.
+Package atveryend Info: Empty hook `AfterLastShipout' on input line 125.
  (./master.aux)
-Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 130.
-Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 130.
+Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 125.
+Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 125.
 Package rerunfilecheck Info: File `master.out' has not changed.
-(rerunfilecheck)             Checksum: B61FA051F86F22A18AF030789F5C3D0E;3677.
-Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 130.
+(rerunfilecheck)             Checksum: DB8588AD2D77E7447A32FB30B6947FE9;6037.
+Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 125.
  ) 
 Here is how much of TeX's memory you used:
- 23098 strings out of 492985
- 415144 string characters out of 6138639
- 599311 words of memory out of 5000000
- 26128 multiletter control sequences out of 15000+600000
- 43589 words of font info for 44 fonts, out of 8000000 for 9000
+ 23726 strings out of 492985
+ 424820 string characters out of 6138639
+ 630284 words of memory out of 5000000
+ 26342 multiletter control sequences out of 15000+600000
+ 47807 words of font info for 45 fonts, out of 8000000 for 9000
  1141 hyphenation exceptions out of 8191
- 64i,14n,71p,8886b,1179s stack positions out of 5000i,500n,10000p,200000b,80000s
-{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rmtt.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-ts1.enc}</usr/share/texmf-dist/fonts/type1/public/lm/lmbx10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmex10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi5.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr12.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr5.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr6.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr8.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmssbx10.pfb></usr/sha
-re/texmf-dist/fonts/type1/public/lm/lmsy10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmsy7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmtt10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmtt12.pfb>
-Output written on master.pdf (18 pages, 878421 bytes).
+ 64i,14n,71p,8886b,1834s stack positions out of 5000i,500n,10000p,200000b,80000s
+{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rmtt.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathex.enc}{/usr/share/texmf-dist/fonts/enc/dvips/lm/lm-ts1.enc}</usr/share/texmf-dist/fonts/type1/public/lm/lmbx10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmex10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi5.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmmi7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr12.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr5.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr6.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmr8.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmri10.pfb></usr/share
+/texmf-dist/fonts/type1/public/lm/lmssbx10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmsy10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmsy7.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmtt10.pfb></usr/share/texmf-dist/fonts/type1/public/lm/lmtt12.pfb>
+Output written on master.pdf (28 pages, 952660 bytes).
 PDF statistics:
- 499 PDF objects out of 1000 (max. 8388607)
- 451 compressed objects within 5 object streams
- 117 named destinations out of 1000 (max. 500000)
- 399 words of extra memory for PDF output out of 10000 (max. 10000000)
+ 1130 PDF objects out of 1200 (max. 8388607)
+ 1065 compressed objects within 11 object streams
+ 524 named destinations out of 1000 (max. 500000)
+ 599 words of extra memory for PDF output out of 10000 (max. 10000000)
 
diff --git a/langfassung/master.out b/langfassung/master.out
index 12deeab..21b162f 100755..100644
--- a/langfassung/master.out
+++ b/langfassung/master.out
@@ -26,7 +26,7 @@
 \BOOKMARK [2][-]{subsection.3.3}{Fazit}{section.3}% 26
 \BOOKMARK [1][-]{section.4}{Quellen und Hilfen}{}% 27
 \BOOKMARK [1][-]{section.5}{Nach der Abgabe...}{}% 28
-\BOOKMARK [2][-]{subsection.5.1}{Spiralgalaxien}{section.5}% 29
+\BOOKMARK [2][-]{subsection.5.1}{Spiralgalaxies}{section.5}% 29
 \BOOKMARK [2][-]{subsection.5.2}{Using Object Oriented Programming \(OOP\) techniques}{section.5}% 30
 \BOOKMARK [3][-]{subsubsection.5.2.1}{Initialisation}{subsection.5.2}% 31
 \BOOKMARK [2][-]{subsection.5.3}{Generation of new stars}{section.5}% 32
@@ -43,3 +43,28 @@
 \BOOKMARK [3][-]{subsubsection.5.8.6}{Calculate the forces acting on all the spheres together}{subsection.5.8}% 43
 \BOOKMARK [3][-]{subsubsection.5.8.7}{Benchmarks}{subsection.5.8}% 44
 \BOOKMARK [2][-]{subsection.5.9}{Calculate the Position of a Star after a timestep}{section.5}% 45
+\BOOKMARK [2][-]{subsection.5.10}{Notes}{section.5}% 46
+\BOOKMARK [2][-]{subsection.5.11}{exec.py}{section.5}% 47
+\BOOKMARK [3][-]{subsubsection.5.11.1}{Importing the galaxytools}{subsection.5.11}% 48
+\BOOKMARK [3][-]{subsubsection.5.11.2}{Generate a new galaxy}{subsection.5.11}% 49
+\BOOKMARK [3][-]{subsubsection.5.11.3}{Generate new stars in the galaxy}{subsection.5.11}% 50
+\BOOKMARK [3][-]{subsubsection.5.11.4}{Print the coordinates of every star in the galaxy relative to the origin}{subsection.5.11}% 51
+\BOOKMARK [3][-]{subsubsection.5.11.5}{Calculate the forces acting inbetween all the stars in the galaxy}{subsection.5.11}% 52
+\BOOKMARK [3][-]{subsubsection.5.11.6}{Print the individual forces acting on the stars}{subsection.5.11}% 53
+\BOOKMARK [3][-]{subsubsection.5.11.7}{Generate the coordinates of the positions for the spheres}{subsection.5.11}% 54
+\BOOKMARK [3][-]{subsubsection.5.11.8}{Calculate the forces after 1 time step}{subsection.5.11}% 55
+\BOOKMARK [2][-]{subsection.5.12}{galaxytools.py}{section.5}% 56
+\BOOKMARK [3][-]{subsubsection.5.12.1}{Importing important libraries}{subsection.5.12}% 57
+\BOOKMARK [3][-]{subsubsection.5.12.2}{Generating the new\137galaxy class}{subsection.5.12}% 58
+\BOOKMARK [3][-]{subsubsection.5.12.3}{Initialisation}{subsection.5.12}% 59
+\BOOKMARK [3][-]{subsubsection.5.12.4}{Generating new stars}{subsection.5.12}% 60
+\BOOKMARK [3][-]{subsubsection.5.12.5}{Print out all the star coordinates}{subsection.5.12}% 61
+\BOOKMARK [3][-]{subsubsection.5.12.6}{Calculate the forces acting inbetween two stars}{subsection.5.12}% 62
+\BOOKMARK [3][-]{subsubsection.5.12.7}{Calculate all the forces acting in the galaxy}{subsection.5.12}% 63
+\BOOKMARK [3][-]{subsubsection.5.12.8}{Print the individual forces acting on one star}{subsection.5.12}% 64
+\BOOKMARK [3][-]{subsubsection.5.12.9}{Find out if a star is inside one sphere}{subsection.5.12}% 65
+\BOOKMARK [3][-]{subsubsection.5.12.10}{Find out which star in in which spheres}{subsection.5.12}% 66
+\BOOKMARK [3][-]{subsubsection.5.12.11}{Generate the sphere positions}{subsection.5.12}% 67
+\BOOKMARK [3][-]{subsubsection.5.12.12}{Calculate the forces acting inside the sphere}{subsection.5.12}% 68
+\BOOKMARK [3][-]{subsubsection.5.12.13}{calculate the forces acting in every sphere}{subsection.5.12}% 69
+\BOOKMARK [2][-]{subsection.5.13}{GAN}{section.5}% 70
diff --git a/langfassung/master.pdf b/langfassung/master.pdf
index 350b08d..d17c7a6 100755..100644
--- a/langfassung/master.pdf
+++ b/langfassung/master.pdf
Binary files differdiff --git a/langfassung/master.synctex.gz b/langfassung/master.synctex.gz
index 02b0ae0..19617b2 100644
--- a/langfassung/master.synctex.gz
+++ b/langfassung/master.synctex.gz
Binary files differdiff --git a/langfassung/master.tex b/langfassung/master.tex
index 75e232f..7717ab0 100755
--- a/langfassung/master.tex
+++ b/langfassung/master.tex
@@ -31,7 +31,7 @@
 % ???
 \usepackage{textcomp}
 % \usepackage[onehalfspacing]{setspace}
-% \linespread{0.95}
+\linespread{1}
 
 % Tikz for designing graphics
 \usepackage{tikz}
@@ -42,8 +42,8 @@
 \lstset{ %
   frame=single,	                   % adds a frame around the code
   language=Python,                 % the language of the code
-  numbers=left,                    % where to put the line-numbers; possible values are (none, left, right)
-  stepnumber=1,                    % the step between two line-numbers. If it's 1, each line will be numbered
+  % numbers=left,                    % where to put the line-numbers; possible values are (none, left, right)
+  % stepnumber=1,                    % the step between two line-numbers. If it's 1, each line will be numbered
 }
 
 % hyperref / hyperlink settings
@@ -104,27 +104,22 @@ urlcolor=black
 \newpage
 
 \section{Einleitung} \label{Einleitung}
-% \input{docs/2_einleitung}
 \include*{docs/2_einleitung}
 \newpage
 
 \section{Hauptteil} \label{Hauptteil}
-% \input{docs/3_hauptteil}
 \include*{docs/3_hauptteil}
 \newpage
 
 \section{Ergebnisse} \label{ergebnisse}
-% \input{docs/4_ergebnisse}
 \include*{docs/4_ergebnisse}
 \newpage
 
 \section{Quellen und Hilfen} \label{quellen}
-% \input{docs/5_quellen}
 \include*{docs/5_quellen}
 \newpage
 
 \section{Nach der Abgabe...} \label{nach_der_abgabe}
-% \input{docs/6_abgabe}
 \include*{docs/6_abgabe}
 
 \end{document}
diff --git a/langfassung/master.toc b/langfassung/master.toc
index 4a85a0f..cc8faf0 100755..100644
--- a/langfassung/master.toc
+++ b/langfassung/master.toc
@@ -1,54 +1,79 @@
 \select@language {ngerman}
-\contentsline {section}{\numberline {1}Einleitung}{2}{section.1}
-\contentsline {subsection}{\numberline {1.1}Themen}{2}{subsection.1.1}
-\contentsline {subsection}{\numberline {1.2}Motivation}{2}{subsection.1.2}
-\contentsline {section}{\numberline {2}Hauptteil}{3}{section.2}
-\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{3}{subsection.2.1}
-\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{3}{subsubsection.2.1.1}
-\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{3}{subsubsection.2.1.2}
-\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{5}{subsubsection.2.1.3}
-\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{5}{subsection.2.2}
-\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{5}{subsection.2.3}
-\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{6}{subsection.2.4}
-\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{6}{subsection.2.5}
-\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{6}{subsubsection.2.5.1}
-\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{7}{subsubsection.2.5.2}
-\contentsline {paragraph}{\nonumberline Amazon Web Services}{7}{section*.5}
-\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{7}{subsubsection.2.5.3}
-\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{7}{subsection.2.6}
-\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{7}{subsubsection.2.6.1}
-\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{8}{section*.6}
-\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{9}{subsection.2.7}
-\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{9}{subsubsection.2.7.1}
-\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{10}{subsubsection.2.7.2}
-\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{10}{subsubsection.2.7.3}
-\contentsline {paragraph}{\nonumberline Masse der Sterne}{10}{section*.7}
-\contentsline {paragraph}{\nonumberline Abstand der Sterne}{10}{section*.8}
-\contentsline {subsection}{\numberline {2.8}Weiteres}{10}{subsection.2.8}
-\contentsline {section}{\numberline {3}Ergebnisse}{11}{section.3}
-\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{11}{subsection.3.1}
-\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{11}{subsection.3.2}
-\contentsline {subsection}{\numberline {3.3}Fazit}{11}{subsection.3.3}
-\contentsline {section}{\numberline {4}Quellen und Hilfen}{13}{section.4}
-\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{13}{section*.11}
-\contentsline {paragraph}{\nonumberline Tim Tugendhat}{13}{section*.12}
-\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{13}{section*.13}
-\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{13}{section*.14}
-\contentsline {section}{\numberline {5}Nach der Abgabe...}{14}{section.5}
-\contentsline {subsection}{\numberline {5.1}Spiralgalaxien}{14}{subsection.5.1}
-\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{14}{subsection.5.2}
-\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{14}{subsubsection.5.2.1}
-\contentsline {subsection}{\numberline {5.3}Generation of new stars}{14}{subsection.5.3}
-\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{14}{subsection.5.4}
-\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{14}{subsection.5.5}
-\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{15}{subsection.5.6}
-\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{15}{subsection.5.7}
-\contentsline {subsection}{\numberline {5.8}Spherical cells}{15}{subsection.5.8}
-\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{15}{subsubsection.5.8.1}
-\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole galaxy}{15}{subsubsection.5.8.2}
-\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{15}{subsubsection.5.8.3}
-\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{16}{subsubsection.5.8.4}
-\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{16}{subsubsection.5.8.5}
-\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{17}{subsubsection.5.8.6}
-\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{17}{subsubsection.5.8.7}
-\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{17}{subsection.5.9}
+\contentsline {section}{\numberline {1}Einleitung}{3}{section.1}
+\contentsline {subsection}{\numberline {1.1}Themen}{3}{subsection.1.1}
+\contentsline {subsection}{\numberline {1.2}Motivation}{3}{subsection.1.2}
+\contentsline {section}{\numberline {2}Hauptteil}{4}{section.2}
+\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{4}{subsection.2.1}
+\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{4}{subsubsection.2.1.1}
+\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{4}{subsubsection.2.1.2}
+\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{6}{subsubsection.2.1.3}
+\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{6}{subsection.2.2}
+\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{6}{subsection.2.3}
+\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{7}{subsection.2.4}
+\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{7}{subsection.2.5}
+\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{7}{subsubsection.2.5.1}
+\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{8}{subsubsection.2.5.2}
+\contentsline {paragraph}{\nonumberline Amazon Web Services}{8}{section*.5}
+\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{8}{subsubsection.2.5.3}
+\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{8}{subsection.2.6}
+\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{8}{subsubsection.2.6.1}
+\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{9}{section*.6}
+\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{10}{subsection.2.7}
+\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{10}{subsubsection.2.7.1}
+\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{11}{subsubsection.2.7.2}
+\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{11}{subsubsection.2.7.3}
+\contentsline {paragraph}{\nonumberline Masse der Sterne}{11}{section*.7}
+\contentsline {paragraph}{\nonumberline Abstand der Sterne}{11}{section*.8}
+\contentsline {subsection}{\numberline {2.8}Weiteres}{11}{subsection.2.8}
+\contentsline {section}{\numberline {3}Ergebnisse}{12}{section.3}
+\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{12}{subsection.3.1}
+\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{12}{subsection.3.2}
+\contentsline {subsection}{\numberline {3.3}Fazit}{12}{subsection.3.3}
+\contentsline {section}{\numberline {4}Quellen und Hilfen}{14}{section.4}
+\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{14}{section*.11}
+\contentsline {paragraph}{\nonumberline Tim Tugendhat}{14}{section*.12}
+\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{14}{section*.13}
+\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{14}{section*.14}
+\contentsline {section}{\numberline {5}Nach der Abgabe...}{15}{section.5}
+\contentsline {subsection}{\numberline {5.1}Spiralgalaxies}{15}{subsection.5.1}
+\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{15}{subsection.5.2}
+\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{15}{subsubsection.5.2.1}
+\contentsline {subsection}{\numberline {5.3}Generation of new stars}{15}{subsection.5.3}
+\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{15}{subsection.5.4}
+\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{15}{subsection.5.5}
+\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{16}{subsection.5.6}
+\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{16}{subsection.5.7}
+\contentsline {subsection}{\numberline {5.8}Spherical cells}{16}{subsection.5.8}
+\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{16}{subsubsection.5.8.1}
+\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole galaxy}{16}{subsubsection.5.8.2}
+\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{16}{subsubsection.5.8.3}
+\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{17}{subsubsection.5.8.4}
+\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{17}{subsubsection.5.8.5}
+\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{18}{subsubsection.5.8.6}
+\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{18}{subsubsection.5.8.7}
+\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{18}{subsection.5.9}
+\contentsline {subsection}{\numberline {5.10}Notes}{18}{subsection.5.10}
+\contentsline {subsection}{\numberline {5.11}exec.py}{19}{subsection.5.11}
+\contentsline {subsubsection}{\numberline {5.11.1}Importing the galaxytools}{19}{subsubsection.5.11.1}
+\contentsline {subsubsection}{\numberline {5.11.2}Generate a new galaxy}{19}{subsubsection.5.11.2}
+\contentsline {subsubsection}{\numberline {5.11.3}Generate new stars in the galaxy}{19}{subsubsection.5.11.3}
+\contentsline {subsubsection}{\numberline {5.11.4}Print the coordinates of every star in the galaxy relative to the origin}{19}{subsubsection.5.11.4}
+\contentsline {subsubsection}{\numberline {5.11.5}Calculate the forces acting inbetween all the stars in the galaxy}{19}{subsubsection.5.11.5}
+\contentsline {subsubsection}{\numberline {5.11.6}Print the individual forces acting on the stars}{19}{subsubsection.5.11.6}
+\contentsline {subsubsection}{\numberline {5.11.7}Generate the coordinates of the positions for the spheres}{20}{subsubsection.5.11.7}
+\contentsline {subsubsection}{\numberline {5.11.8}Calculate the forces after 1 time step}{20}{subsubsection.5.11.8}
+\contentsline {subsection}{\numberline {5.12}galaxytools.py}{20}{subsection.5.12}
+\contentsline {subsubsection}{\numberline {5.12.1}Importing important libraries}{20}{subsubsection.5.12.1}
+\contentsline {subsubsection}{\numberline {5.12.2}Generating the new\_galaxy class}{20}{subsubsection.5.12.2}
+\contentsline {subsubsection}{\numberline {5.12.3}Initialisation}{20}{subsubsection.5.12.3}
+\contentsline {subsubsection}{\numberline {5.12.4}Generating new stars}{21}{subsubsection.5.12.4}
+\contentsline {subsubsection}{\numberline {5.12.5}Print out all the star coordinates}{21}{subsubsection.5.12.5}
+\contentsline {subsubsection}{\numberline {5.12.6}Calculate the forces acting inbetween two stars}{22}{subsubsection.5.12.6}
+\contentsline {subsubsection}{\numberline {5.12.7}Calculate all the forces acting in the galaxy}{22}{subsubsection.5.12.7}
+\contentsline {subsubsection}{\numberline {5.12.8}Print the individual forces acting on one star}{23}{subsubsection.5.12.8}
+\contentsline {subsubsection}{\numberline {5.12.9}Find out if a star is inside one sphere}{24}{subsubsection.5.12.9}
+\contentsline {subsubsection}{\numberline {5.12.10}Find out which star in in which spheres}{25}{subsubsection.5.12.10}
+\contentsline {subsubsection}{\numberline {5.12.11}Generate the sphere positions}{26}{subsubsection.5.12.11}
+\contentsline {subsubsection}{\numberline {5.12.12}Calculate the forces acting inside the sphere}{26}{subsubsection.5.12.12}
+\contentsline {subsubsection}{\numberline {5.12.13}calculate the forces acting in every sphere}{26}{subsubsection.5.12.13}
+\contentsline {subsection}{\numberline {5.13}GAN}{27}{subsection.5.13}