\DocumentMetadata{lang=en}
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%% $Id: pst-kepler-doc.tex 1224 2026-01-01 07:30:39Z herbert $
\documentclass[fontsize=11pt,english,BCOR=10mm,DIV=12,bibliography=totoc,parskip=false,
   headings=small, headinclude=false,footinclude=false,twoside,ngerman,english]{pst-doc}
\usepackage{pst-eucl,pst-plot}
%\usepackage{xintexpr,polexpr,xinttrig}
%\usepackage{xstring}
\usepackage{pst-tools,minted-code,pst-kepler}
\let\PSTfileversion\fileversion
\let\PSTfiledate\filedate

\renewcommand\bgImage{\includegraphics[height=8cm]{images/kepler_model}}

\addbibresource{\jobname.bib}

\usepackage{enumitem,xltabular,hvindex}
\setlist{nosep}

\usepackage{pstricks-add}
\usepackage{wrapfig}
\usepackage{pifont}

\NewExpandableDocumentCommand{\XcalcR}{sO{8}m}{%  calculates #3 and rounds to #2 decimals; non formating: for formated output use \Xcalc not \XcalcR
\PolDecToString{\xintREZ{\xinttheiexpr[#2]\xintfloatexpr\empty#3\relax\relax}}%
}%                                        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^


\begin{document}

\title{\texttt{pst-kepler}}
\subtitle{The earth model of Johannes Kepler; v.\PSTfileversion}
\author{Jürgen Gilg$\dagger$}
\docauthor{Jürgen Gilg$\dagger$\\Jean-François Burnol\\Manuel Luque\\ 
Thomas Ossig\\Herbert Voß}
\date{\today}
\settitle

\tableofcontents

\vfill
\begin{center}
\fbox{\parbox{0.9\linewidth}{This package is dedicated to Jürgen Gilg 
(08.02.1966 -- 06.05.2022). His work for PSTricks will never be forgotten.}}
\end{center}

\vfill

\begin{abstract}
Jean-François Burnol's tribute to Jürgen Gilg ($\dagger$ 2022):

»It is with deep sadness that I learned of Jürgen's completely 
unexpected passing. How I regret not having visited him in Germany! 
I received his first email on January 2, 2018, some time after 
I began interacting with Thomas Söll. It was titled ›Polynomials 
with XINT‹ and is the direct inspiration for my \TeX/\LaTeX\ 
package \LPack{polexpr}. Since then, Jürgen, Thomas, and I have had 
many exchanges about \LPack{xint}, \LPack{polexpr}, and also PS-Tricks, of which 
Jürgen was an expert, and where he was my teacher in everything. 
Jürgen, along with Thomas Söll and others, did extraordinary things 
with PS-Tricks. As the months and years went by, we talked more and 
more about other subjects, about events in our respective lives, and 
we shared painful experiences. I think Jürgen was incredibly 
kind-hearted; I feel like I've known him for ten or twenty years, 
but it's only been four \ldots«

\hfill Jean-François Burnol, May 8, 2022
\end{abstract}


\clearpage



\section{Curriculum Vit{\ae} of Johannes Kepler (December 27, 1571 -- November 15, 1630)}

%\begin{wrapfigure}[lineheight]{position}{width}
%  ...
%\end{wrapfigure}
%The position parameter has eight possible values:
%
%r	R	right side of the text
%l	L	left side of the text
%i	I	inside edge–near the binding (in a twoside document)
%o	O	outside edge–far from the binding
%The uppercase version allows the figure to float. The lowercase version means exactly here.
%
\begin{wrapfigure}{r}{0.45\textwidth}
  \centering
    \includegraphics[width=0.9\linewidth]{images/kepler_picture}
  \caption{Johannes Kepler}
%
%   \includegraphics[width=0.9\linewidth]{images/Brahe}
%  \caption{Tycho Brahe}
\end{wrapfigure}
Johannes Kepler was born on December 27, 1571 in Weil (today: \Index{Weil der Stadt}) 
as son of Katharina and Heinrich Kepler.
He lived in the epoch of Renaissance till the beginning of the »Thirty Years' 
War«. The big change in spiritual subjects, which arised to that time, 
manifested itself in acrimonious religious wars during the reformation 
and intellectually for famous explorers to compete for a new true knowledge 
about the world.

\NameIndex{Nikolaus Kopernikus} (1473-1543) described in his astronomical opus 
»\textit{Von den Umdrehungen der Hemisph\"{a}ren}«, which he released 
short time before he died, that not the Earth, but the Sun is the center 
of the planet system, hence the Earth would be only a planet among other planets.
Duke Christoph donated a scholarship for talented children 
for the University of \Index{Tübingen}. In 1589 Kepler, who received such a 
scholarship, started to study in the subjects of Theology, Mathematics 
and Astronomy. In this connection he came to know of Kopernikus' 
revoluting thesis.  After attending two convent schools Kepler came 
into the Tübinger monastery. Due to the fact of his critical intellect 
not fitting with all the dogmas of the post-lutherian Orthodoxy 
(amongst others Lord's Supper,  geocentric world view), he didn't 
receive any employment in \Index{Württemberg} anymore. So he had to go 
in a foreign country. In \Index{Graz} he worked from 1594 to  1600 as a 
mathematician of the province. There he came into his own which was 
the astronomy. In 1597 he married \NameIndex{Barbara Müller}.

\begin{wrapfigure}{r}{0.45\textwidth}
  \centering
    \includegraphics[width=0.9\linewidth]{images/Brahe}
  \caption{Tycho Brahe}
\end{wrapfigure}
With 24, Kepler published his first opus »\textit{Mysterium cosmographicum}« 
(»\textit{The Mysteries of the Universe}«).~\parencite{kepler_mystery} 
In 1600 he moved with his wife to \Index{Prague}, 
to become a collaborator of \NameIndex{Tycho Brahe}. 
After Brahe's death in the following year, Kepler became his successor as an astronomer 
under Emperor \Index{Rudolf II.} and as an imperial mathematician respectively and continued 
Brahe's astronomical work that he has left. In 1605 he realized, that the orbit of 
Mars is not a \Index{circle}, but an \Index{ellipse}. This finding is the base for the Kepler Laws. 

The first two were released in 1609 in 
»\textit{Astronomia nova}« (»\textit{New Astronomy}«). 
His Third Law Kepler posted in 1619 in the opus »\textsc{Harmonices mundi}« 
(»\textit{Harmony of the World}«).

\begin{figure}[!htb]
\frame{\includegraphics[width=0.3\linewidth]{images/astronomia_nova}}
\qquad
\frame{\includegraphics[width=0.3\linewidth]{images/harmony}}
\quad
\frame{\includegraphics[width=0.3\linewidth]{images/dioptrice}}
%  \caption{Tycho Brahe}
\end{figure}

%\begin{center}
%\includegraphics[scale=0.1]{images/Brahe_kepler}
%\end{center}

\begin{wrapfigure}{r}{0.45\textwidth}
  \centering
    \includegraphics[width=0.9\linewidth]{images/rudolphine}
%  \caption{Tycho Brahe}
\end{wrapfigure}
Apart from the astronomy, Kepler contributed significant things in the subject of Optics. 
In 1611 »\textit{Dioptrice} (»\textit{Dioptre}«) was published, where he progressed 
in the theory of lenses of a \Index{telescope} (with two convex lenses).
After his wife died in 1611 and later his benefactor \Index{Rudolf II.} as well, 
he became mathematician of the province in \Index{Linz}. There he married 
\NameIndex{Susanne Reutinger}. 


He published "\textit{Abriss der kopernikanischen Astronomie}" 
(7 Volumes, 1618-22) and released the »\textit{Tabulae Rudolphinae}« 
(»\textit{Rudolphine Tables}«) in 1627. They include till then the 
most exact tables of the planet movements and for this reason with the 
appropriate planet positions.

From 1628 he was in attendance on 
\Index{Wallenstein} in \Index{Ulm} and \Index{Sagan}. He mainly 
was asking fir some exact horoscopes. During that time the astronomy and 
astrology were strongly connected. At a later date it came out of its 
shadowy existence.  An anecdote tells, that Kepler really prognosticates 
Wallenstein's date of death. He was well known for his precise horoscopes, 
although nowadays one can be doubtfully about that. For sure the astrology j
ust was a means to an end for earning some money for his astronomical research.

In 1630 Kepler went on a journey to Linz. He fell ill in \Index{Regensburg} and died 
on November 15, shotly after his arrival.


\section{The symbolic of Kepler's solar system}
With his debut feature, he likes to resolve, why \Index{God} created exactly six 
and not any more planets. His first idea is to inscribe plain geometrical 
figures one within one to explain the number of planets. This failed, 
cause there are too many regular plain geometrical figures, but only five were needed.

\begin{figure}
\includegraphics[width=0.19\linewidth,page=2]{images/tetra.png}
\hfill
\includegraphics[width=0.19\linewidth]{images/hexa.png}
\hfill
\includegraphics[width=0.19\linewidth]{images/octa.png}
\hfill
\includegraphics[width=0.19\linewidth]{images/dode.png}
\hfill
\includegraphics[width=0.19\linewidth]{images/ikosa.png}
%Tetrahedron\qquad  Hexahedron\qquad  Octahedron\qquad Dodecahedron \qquad Icosahedron
\caption{The five regular Platonic Solids}
\end{figure}


Since the system of planets is three dimensional, Kepler exploits finally 
the \textbf{five regular Platonic Solids} for his thought experiment: 
the \textit{\Index{Tetrahedron}}, the \textit{\Index{Hexahedron}} (\Index{Cube}), 
the \textit{\Index{Octahedron}}, the \textit{\Index{Dodecahedron}} and the 
\textit{\Index{Icosahedron}}. These regular solids are in--- respectively 
circumscried by spheres. The ratio of the radiuses between inner and 
outer sphere expresses the ratio of the distances to the sun of adjacent 
planets (see figure~\vref{solarsystem}). 
For taking into account the orbit \Index{excentricity}, Kepler emanates 
from a certain thickness of the spheres. The copernican data of distances 
suggest the following order:


%\newpage
\begin{xltabular}{\linewidth}{@{}l l m{3cm} X @{}}
\symbol{"2644} & Sphere of Saturn. & 
  \vspace{0pt}\includegraphics[trim=0 1cm 0 0,clip,width=3cm]{images/saturn} &
\Index{Hexahedron} (\Index{Cube}, the first of the regular bodies in geometry, showing the distance 
between the orbits of \Index{Saturn} and \Index{Jupiter}.\\

\symbol{"2643} & Sphere of Jupiter. & 
  \vspace{0pt}\includegraphics[trim=0 1cm 0 0,clip,width=3cm]{images/jupiter}&
\Index{Tetrahedron}, or \Index{pyramid}, touching the sphere of \Index{Jupiter} 
on the outside and the sphere of \Index{Mars} on the inside, and responsible 
for the greatest distance between the two planets.\\

%\symbol{"2644} 
& Sphere of Mars. & 
  \vspace{0pt}\includegraphics[trim=0 1cm 0 0,clip,width=3cm]{images/mars}&
\Index{Dodecahedron}, the third body, representing the 
distance extending from the sphere of \Index{Mars} to the 
great orbit that encompasses the Earth and the Moon.\\

\symbol{"2641} & \shortstack{Great orbit\\ (of the Earth).} & 
  \vspace{0pt}\includegraphics[trim=0 1cm 0 0,clip,width=3cm]{images/earth}&
\Index{Icosahedron}, which indicates the true distance between the great orbit and the sphere of Venus.
\Index{Octahedron}, which shows the distance between the sphere of 
\Index{Venus} and that of \Index{Mercury}.\\

\symbol{"263F} & Sphere of Mercury. & 
  \vspace{0pt}\includegraphics[trim=0 1cm 0 0,clip,width=3cm]{images/mercury}&
\Index{Sun}, the center of the stationary center of the Universe.\\
\end{xltabular}


\begin{figure}
\centering
\includegraphics[scale=0.4]{images/Keplerienne3}
\caption{Kepler's solar system.}\label{solarsystem}
\end{figure}

The greek philosopher Plato (about 300 B. C.) described detailed the solids and 
assigns them to the elements of the \textit{\Index{Platonic World View}}. 
They were assigned as schown in figure~\vref{platonic}.

\begin{figure}
\centering
\includegraphics[scale=0.6]{images/universe}
\caption{Platon's World View}\label{platonic}
\end{figure}

In his dialog \textit{Timaios}, \Index{Plato} illustrates a model of the universe, 
wherein he combines the regular solids with the four elements \Index{Earth}
 (\Index{Hexahedron}/\Index{Cube}), 
\Index{Water} (\Index{Icosahedron}), \Index{Fire} (\Index{Tetrahedron}) and \Index{Air} 
(\Index{Octahedron}).~\parencite{jorgenson2020plato,2016timaios}
Plato associates the \Index{Dodecahedron} with the \textit{quinta essentia}, 
the Heavenly \Index{Ether}. Every of the twelve side faces correspond to one of the 
twelve constellations.


\clearpage

\section{Kepler's laws}
\subsection{Kepler's 1st Law}
During his work in \Index{Prague}, Kepler manifested his »First Law«. He phrased the 
following:

\begin{quotation}
»The cause is like so: The planet's orbit is not a circle; it comes in little 
by little on both sides and then back 
till the circumference of the circle in the perigree [the point of the moon's 
orbit that is closest to the Earth]. A such shaped orbit is called oval.«
\end{quotation}

Nowadays it will be expressed:

\textit{All planets orbit the \Index{sun} in \Index{elliptical orbit}s 
with the sun at one focus.}


\subsection{Kepler's 2nd Law}
His »Second Law« reads like so:

\begin{quotation}
\begin{otherlanguage}{ngerman}
»Unvollkommenes, jedoch f\"{u}r die Sonnen- oder Erdbahn ausreichendes 
Verfahren zur Berechnung der Gleichungen auf Grund der physikalischen 
Hypothese. Da ich mir bewu{\ss}t war, dass es unendlich viele Punkte 
auf dem Exzenter [au{\ss}erhalb des Mittelpunktes liegende angebrachte 
Steuerungsscheibe] und entsprechend unendlich viele Abst\"{a}nde gibt, 
kam mir der Gedanke, dass in der Fl\"{a}che des Exzenters alle diese 
Abst\"{a}nde enthalten seien.«
\end{otherlanguage}
\end{quotation}

Nowadays it will be phrased like:
\begin{center}
\begin{pspicture}(-4,-2.75)(4,3.2)%\psgrid

\pscustom[fillstyle=solid,fillcolor=lightgray]{%
 \psline(-1.658,0)(-2.298,1.607)
 \psellipticarc(3,2.5){140}{220}
 \psline(-1.658,0)(-2.298,-1.607)
}
\pscustom[fillstyle=solid,fillcolor=lightgray]{%
 \psline(-1.658,0)(2.954,-0.434)
 \psellipticarc(3,2.5){350}{10}
 \psline(2.954,0.434)(-1.658,0)
}
\parametricplot[algebraic,plotpoints=200,linestyle=dashed]{0}{TwoPi}{3*cos(t)|2.5*sin(t)}
\rput(-1.658,0){\textcolor{yellow}{\LARGE$\bullet$}}
\rput(0,2.5){\textcolor{blue}{\LARGE$\bullet$}}
\uput[90](0,2.5){Planet}
\psline[linecolor=blue]{->}(3,0)(3,1)
\psline[linecolor=blue]{->}(-3,0)(-3,-2)

\uput[0](3,1){$\vec{v}_1$}
\uput[180](-3,-2){$\vec{v}_2$}
\rput(1.25,0){$A_1$}
\rput(-2.25,0){$A_2$}

\rput(3,0){$\bullet$}
\rput(-3,0){$\bullet$}
\uput[180](-3,0){Perihelion}
\uput[0](3,0){Aphelion}
\rput[l](-1.658,-0.4){Sun}
\end{pspicture}
\end{center}

\textit{\Index{Perihelion}} is the point on its orbit when a planet is closest to the sun, 
\textit{\Index{Aphelion}} is the point on its orbit when a planet is farthest from the sun. 
The sun is in one focus of the elliptical orbit.


\textit{The line connecting a planet to the sun sweeps out equal areas in equal 
amounts of time. A planet moves faster nearer to the sun. Thus: $A_1=A_2$.}


\subsection{Kepler's 3rd Law}
Further researches leaded Kepler to his famous third Law of planet's movements, 
in 1619 published in »\textit{Harmonices Mundi Libri V}«.~\parencite{kepler1619joannis}

\begin{otherlanguage}{ngerman}
»Allein es ist ganz sicher und stimmt vollkommen, dass die Proportion, die 
zwischen den Umlaufzeiten irgendzweier Planenten besteht, genau das Anderthalbe 
der Proportion der mittleren Abst\"{a}nde, d.\,h. der Bahnen selber, ist.«
\end{otherlanguage}

In other words:

\textit{The squares of the periods $T$ of the planets are proportional t
o the cubes of their semi-major axes $a$.}

Brought into a mathematical formula:
\begin{equation*}
  \dfrac{T^2}{a^3}=\text{const.}
\end{equation*}
The Third Law describes the complex relation between period and the distance to the sun.

The following table shows the relation:
\begin{itemize}
\item $T$ measured as a multiple of an Earth year.
\item $a$ measured as a multiple of the distance Earth--Sun.
\end{itemize}
\begin{tabular}{lrrrr}
\Index{Planet}  & $T$            & $T^2$     & $a$        & $a^3$\\\hline
\Index{Mercury}  & 0,241          & 0,058     & 0,387      &  0,058\\
\Index{Venus}   & 0,615          & 0,378     & 0,723      &  0,378\\
\Index{Earth}    & 1,000          & 1,000     & 1,000      &  1,000\\
\Index{Mars}    & 1,881          & 3,538     & 1,524      &  3,540\\
\Index{Jupiter} & 11,860         & 140,660   & 5,203      &  140,852\\
\Index{Saturn}  & 29,460         & 867,892   & 9,539      &  867,978\\
\end{tabular}




\begin{center}
%\includegraphics[scale=0.75]{loglog_engl}
\end{center}


\section{Background informations}
\subsection{John Banville}
In his book »Kepler« (1981), \NameIndex{John Banville} imagines how the idea came to Kepler, pages
42 and 43 of the French edition, of which a German translation must exist.~\parencite{banville}

Perhaps he was mistaken, perhaps the universe was not an ordered construction 
governed by immutable laws? Who knows if God, like his creatures, does not ultimately prefer the temporal to the eternal, the near-perfect to the perfect, children's trumpets and the frenzied bleating of disorder to the music of the spheres? But, no, no, despite these doubts, no: his God was above all a God of order. The universe obeys geometric laws because geometry embodies the earthly paradigm of divine thought.
He worked very late into the night, and went through the days swaying, in an ecstatic state. Summer came. He had been working non-stop for six months and had just reached the conclusion—if one could even call it a conclusion—that it was not so much the planets, their position and their speed, that he should concern himself with first, but the intervals between their orbits. If, for these distances, he possessed only Copernicus's estimates, which were scarcely more reliable than Ptolemy's, he nevertheless had to assume, for his own peace of mind, that they were precise enough to serve his purposes. Time and time again, he combined and recombined them in the hope of uncovering the relationship they concealed. Why were there only six planets? That was a question. But it was more pertinent to inquire into the distances that separated them. Why these distances and not others? He waited patiently for the rustling of wings.

On that July morning, though like so many other mornings, the angel appeared who brought him the answer. Johannes was in class. It was a hot and radiant day. A fly buzzed against the high window, a luminous diamond stood out on the floor at his feet. With glazed eyes, his students, petrified with boredom, stared at him without seeing him. Busy
demonstrating a theorem of \Index{Euclid}—though he tried afterward to remember which one, he
could not—he had drawn an equilateral triangle on the board. He seized the heavy wooden compass and, immediately, as always, that monstrosity pricked him. He put his injured thumb to his mouth, resumed his work, and began to draw two circles; the first, inscribed
inside the triangle, touched its three sides, while the second, circumscribed, encompassed it
and passed through its three vertices. He stepped back to the lozenge of dusty light, blinked, and
suddenly something, perhaps his heart, leaped, like an athlete performing a
feat on a trampoline; for no reason, he said to himself, ecstatically: I will live forever.
The proportion between the two circles was identical to that which existed between the orbits
of \Index{Saturn} and \Index{Jupiter}, the most distant planets, and behold, between the circles that
delimited this proportion was inscribed that basic figure which was the equilateral triangle
in geometry. And if we placed a square between the orbits of Jupiter and \Index{Mars}, a pentagon
between Mars and Earth, and between Earth and \Index{Venus} a \ldots\  Yes. Oh, yes. 
The figure, the painting,
even the walls of the room melted into a shimmering liquid, and the students of young
Professor Kepler had the chance, a rare and gratifying spectacle, to see their master dab
his eyes and hear him blow his nose noisily into a handkerchief of dubious whiteness.



\subsection{Jacques Blamont}
In his book »The Number and the Dream«, Jacques Blamont~\parencite{blamont}
describes his “discovery” as follows:

Now, \Index{Euclid} showed that in three-dimensional space, only five solids (called Pythagorean) 
exist whose faces are identical: 
the tetrahedron (made up of four triangles), 
the cube (four squares), the octahedron (eight triangles), 
the dodecahedron (twelve pentagons), and the icosahedron (twenty triangles). 
Their symmetry allows them to 
be inscribed in a sphere (that is, their vertices are placed on it) 
or to be circumscribed about a sphere (that is, their sides are tangent to it). 
Five solids, five intervals between the planets! 
This coincidence provided the immediate solution to the enigma of the number of planets 
and the mystery of their distances; The radii of the spheres calculated by Copernicus 
allowed the five solids to be placed between the spheres in the following order:

\medskip
Saturn-cube-Jupiter-tetrahedron-Mars-dodecahedron-Earth-icosahedron-Venus-octahedron-Mercury

\medskip
The numbers had to be stretched a bit, but overall the agreement was good, except for
Jupiter, »but no one will be surprised by that, given the great distance.« That's enough to make you forget
many boils. The twenty-three-year-old leaped onto his pen and wrote a book
which, despite its central error, was nonetheless a masterpiece, the Mysterium
cosmographicum, whose genesis we know thanks to its preface:



\begin{quotation}
»I did not yet clearly see in what order the perfect solids should be arranged, and
nevertheless I succeeded [\ldots] in arranging them so happily that later, when I checked these
arrangements, I had nothing to change. I no longer regretted the lost time; I was no longer
weary of my work; I shrank from no calculation, however difficult. Day and night,
I performed my calculations to see if the proposition I had just formulated agreed with the
orbits of Copernicus or if my joy would be carried away by the wind [\ldots]. 
In a few days,
everything was in place. I saw the symmetrical solids fit one after the other with such
precision between the appropriate orbits [\ldots] that if a peasant He asked, “On what hooks 
are the heavens fixed so as not to fall? It would be easy to answer.
\end{quotation}

\clearpage

\section{The macros}
\subsection{World model}

\begin{codeblockA}[title=Show single Platonic solids]
\psKeplerModel[Options](x,y)
\end{codeblockA}

The coordinates for the center are also optional, because they are preset to $(0,0)$.

\begin{exampleA}{Kepler's world model}
\begin{pspicture}[showgrid](-3,-3)(3,3)
\psKeplerModel[Decran=25]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{Tetrahedron Model}
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psKeplerModel[type=tetrahedron,Decran=25]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{Hexahedron Model}
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psKeplerModel[type=hexahedron,Decran=25]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{Octahedron Model}
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psKeplerModel[type=octahedron,
  viewpoint=50 -20 10 rtp2xyz, Decran=25]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{Dodekahedron Model}
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psKeplerModel[type=dodekahedron,Decran=25]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{Icosahedron Model}
\begin{pspicture}[showgrid](-2,-2)(2,2)
\psKeplerModel[type=icosahedron,Decran=25]
\end{pspicture}
\end{exampleA}

\clearpage


\begin{codeblockA}[title=Show solids without a sphere]
\psKeplerShow[Options](x,y)
\end{codeblockA}


The coordinates for the center are also optional, because they are preset to $(0,0)$.
With this macro one can draw the five platonic solids. The scaling factor can
be changed by the optional arguments \Lkeyword{scaleTetra}, \Lkeyword{scaleHexa},
\Lkeyword{scaleOcta}, \ldots. The values are preset to 2, 4, 6, \ldots\ 
All boolean options for the solids are preset to true.
All five solids are shown if no optional argument is used.

\begin{exampleA}{title=Tetrahedron and Hexahedron}
\begin{pspicture}(-4,-4)(4,4)
\psKeplerShow[unit=0.8,
              octahedron=false,
              dodecahedron=false,
              icosahedron=false]
\end{pspicture}
\end{exampleA}

\begin{exampleA}{title={Tetrahedron, Hexahedron, and Octahedron}}
\begin{pspicture}(-5,-5)(5,5)
\psKeplerShow[unit=0.5,
              dodecahedron=false,
              icosahedron=false,
              scaleHexa=3.5,   % was preset to 4
              scaleOcta=5.5]  % was preset to 6
\end{pspicture}
\end{exampleA}


\clearpage

\subsection{2nd Law}

The area computed by numerical integration\footnote{Simpson method with evaluations
points separated by about $\num{1}$ degrees} is evaluated to be $\Delta
A_{1/2} = \frac{1}{2}\int\limits_{\varphi_1}^{\varphi_2} (r(\varphi))^2
\mathrm{d}\varphi \approx \num{\xintDecToString{\xintfloateval{\AreaStart}}}$.
The exact value in this example is actually 
$\xintTeXsignedFrac{\xinttheexpr reduce(\AreaFractionOfTotal/360)\relax}$ 
of the total ellipse area, i.\,e. 
$\xintTeXsignedFrac{\xinttheexpr reduce(\AreaFractionOfTotal/360)\relax} 
\mathcal{A} \approx
\num{\xintDecToString{\xintfloateval{Pi*a_el*b_el/(360/\AreaFractionOfTotal)}}}$.

\FloatEdef\AreaStart{
    %  spaces do not matter here...
    AreaEllipseS(
    \EKeplerphi,
    \EKeplerIphi,
    0.05
    )
}%

The starting angle was $\varphi_1 = \num{\xintfloateval{\EKeplerphi}}^{\circ}$
and the ending angle $\varphi_2 = \num{\xintfloateval{\EKeplerIphi}}^{\circ}$ and
7 evaluation points were used. With evaluation points separated by a bit less
than $0.05^{\circ}$ ($115$ evaluation points) we gain almost 6 digits of
precision: $\Delta A_{1/2} \approx
\num{\xintDecToString{\xintfloateval{\AreaStart}}}$.%
\FloatEdef\EKepler{EKepler(0)}
\FloatEdef\EKeplerI{EKepler(\AreaFractionOfTotal)}
\FloatEdef\EKeplerphi{E_to_phi_Pol(\EKepler)}
\FloatEdef\EKeplerIphi{E_to_phi_Pol(\EKeplerI)}
\FloatEdef\AreaLater{
    %  spaces do not matter here...
    AreaEllipseS(
    \EKeplerphi,
    \EKeplerIphi,
    1
    )
}%
Let's use now starting angle $\varphi_1 =
\num{\xintfloateval{\EKeplerphi}}^{\circ}$ and ending angle $\varphi_2 =
\num{\xintfloateval{\EKeplerIphi}}^{\circ}$ (itself computed approximately).
We obtain (using $135$ evaluations points)
$\frac{1}{2}\int\limits_{\varphi_1}^{\varphi_2} (r(\varphi))^2
\mathrm{d}\varphi \approx \num{\xintDecToString{\xintfloateval{\AreaLater}}}$.
This is, not surprisingly, less precise that our previous evaluation on a
shorter interval with comparable number of evaluation points.

The macro \Lcs{psKeplerAnimate} creates an inline \Index{PDF-animation} which can only
be viewed with the \Index{AdobeReader}. This document also comes with a \Index{GIF-animation}
which can be used as standalone image or as part of a webpage.


\begin{exampleM}{Kepler's 2nd law as animation}
\psKeplerAnimate % needs AdobeReader
\end{exampleM}

\printbibliography

\printindex
\end{document}

