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{\Large\bf Fundamental D-V cells for $\mathbb{E}^{4}$ space groups on the $\mathbf{2D}$-screen }

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{\large\bf\noindent   László Ács}\\

Department of Mathematics, Széchenyi István University,\\

  9026 Győr, Hungary\\

\\

\texttt{ acs@mail.sze.hu } \\

\textit{MSC 2000:} 20H15, 51F15

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{\large As an introduction we describe two algorithms for determining fundamental domains 

( D-V cells ) of \textbf{XIX.27/01/01} and \textbf{XXII.31/05/02/002} [BBNWZ] space groups. With the first algorithm we obtain a 4-polytope as fundamental domain of \textbf{XIX.27/01/01} space group [AM01] belonging to the decagonal crystal family. The second algorithm determines a fundamental domain of a space group with broken translations [AM02].



Then we describe a method for ortogonal projection of $\mathbb{E}^{d}$-pictures on 

$\mathbf{2D}$-screen, and we illustrate the edge structure of D-V cells for 25 space groups to the 

decagonal and icosahedral families of $\mathbb{E}^{4}$.}



\begin{thebibliography}{9}

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\bibitem[BBNWZ]{1}  Brown,~H.~--~B{\"u}low,~R.~--~Neub{\"u}ser,~J.~--~%

  Wondratschek,~H.~--~Zassenhaus,~H.,  

\textit{Crystallographic groups of four-dimensional space},  A.~Wiley~--~Interscience, 

(1978).

%

\bibitem[AM01] {2}   Ács,~L.~--~Molnár,~E. 

\textit{Algorithm for D-V~cells and fundamental domains of $\mathbb{E}^d$

  space groups (to decagonal and icosahedral families in $\mathbb{E}^4$)}, 

Pure Mathematics and Application, Vol 13. No. 1-2, pp 1-20 (2002).

%

\bibitem[AM02] {3}   Ács,~L.~--~Molnár,~E. 

\textit{Algorithm for D-V~cells and fundamental domains, $\mathbb{E}^4$

  space groups with broken translations to icosahedral family}, 

Journal for Geometry and Graphics \textbf{6} No. 1, pp 1-16 (2002).  

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