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\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{0.125pt}
%
% File name: rhombic-hexecontahedron.tex
% Description:
% A geometric representation of the rhombic hexecontahedron is shown.
%
% Date of creation: June, 30th, 2021.
% Date of last modification: October, 9th, 2022.
% Author: Efraín Soto Apolinar.
% https://www.aprendematematicas.org.mx/author/efrain-soto-apolinar/instructing-courses/
% Source: page 227 of the
% Glosario Ilustrado de Matem\'aticas Escolares.
% https://tinyurl.com/5udm2ufy
%
% Terms of use:
% According to TikZ.net
% https://creativecommons.org/licenses/by-nc-sa/4.0/
% Your commitment to the terms of use is greatly appreciated.
%
\begin{document}
\tdplotsetmaincoords{80}{30}
%
\begin{tikzpicture}[tdplot_main_coords]
% Change the value of the number at {\escala}{##} to scale the figure up or down
\pgfmathsetmacro{\escala}{1.25}
\pgfmathsetmacro{\r}{\escala}
% Coordinates of the vertices
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\coordinate(2) at (-\escala*1.89443, -\escala*1.37638, \escala*0.447214);
\coordinate(3) at (-\escala*1.89443, \escala*1.37638, \escala*0.447214);
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\coordinate(10) at (-\escala*1.17082, \escala*0.850651, \escala*0.894427);
%
\coordinate(11) at (-\escala*1.17082, \escala*0.850651, \escala*1.89443);
\coordinate(12) at (-\escala*1., -\escala*1.37638, \escala*0.);
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%
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%
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\coordinate(32) at (\escala*0., \escala*0., \escala*1.);
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\coordinate(39) at (\escala*0.447214, -\escala*1.37638, \escala*1.89443);
\coordinate(40) at (\escala*0.447214, \escala*1.37638, \escala*0.894427);
%
\coordinate(41) at (\escala*0.447214, \escala*1.37638, \escala*1.89443);
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\coordinate(47) at (\escala*0.723607, \escala*2.22703, \escala*0.447214);
\coordinate(48) at (\escala*0.894427, \escala*0., -\escala*1.44721);
\coordinate(49) at (\escala*0.894427, \escala*0., -\escala*0.447214);
\coordinate(50) at (\escala*1., -\escala*1.37638, \escala*0.);
%
\coordinate(51) at (\escala*1., \escala*1.37638, \escala*0.);
\coordinate(52) at (\escala*1.17082, -\escala*0.850651, -\escala*1.89443);
\coordinate(53) at (\escala*1.17082, -\escala*0.850651, -\escala*0.894427);
\coordinate(54) at (\escala*1.17082, \escala*0.850651, -\escala*1.89443);
\coordinate(55) at (\escala*1.17082, \escala*0.850651, -\escala*0.894427);
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\coordinate(60) at (\escala*1.89443, -\escala*1.37638, -\escala*0.447214);
%
\coordinate(61) at (\escala*1.89443, \escala*1.37638, -\escala*0.447214);
\coordinate(62) at (\escala*2.34164, \escala*0., \escala*0.447214);
% faces of the polyhedron
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (25) -- (24) -- (19) -- (20) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (37) -- (36) -- (24) -- (25) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (20) -- (19) -- (6) -- (7) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (20) -- (7) -- (1) -- (5) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (18) -- (4) -- (1) -- (7) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (5) -- (1) -- (4) -- (14) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (13) -- (3) -- (10) -- (28) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (20) -- (5) -- (3) -- (13) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (10) -- (3) -- (5) -- (14) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (4) -- (2) -- (8) -- (14) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (18) -- (12) -- (2) -- (4) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (8) -- (2) -- (12) -- (26) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (28) -- (40) -- (47) -- (33) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (29) -- (11) -- (15) -- (32) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (28) -- (10) -- (11) -- (29) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (29) -- (41) -- (40) -- (28) -- cycle;
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%
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\draw[red,thick,fill=cyan!35,opacity=0.75] (27) -- (9) -- (8) -- (26) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (15) -- (9) -- (27) -- (32) -- cycle;
% + + + + + + + + + + + + + + + + + + + + +
% + + + + + + + + + + + + + + + + + + + + +
% + + + + + + + + + + + + + + + + + + + + +
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (51) -- (61) -- (55) -- (37) -- cycle;
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\draw[red,thick,fill=cyan!35,opacity=0.75] (55) -- (61) -- (59) -- (49) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (55) -- (54) -- (36) -- (37) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (36) -- (54) -- (48) -- (31) -- cycle;
\draw[red,thick,fill=cyan!35,opacity=0.75] (49) -- (48) -- (54) -- (55) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (31) -- (34) -- (22) -- (17) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (48) -- (52) -- (34) -- (31) -- cycle;
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%
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\draw[red,thick,fill=cyan!35,opacity=0.75] (35) -- (30) -- (16) -- (23) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (26) -- (30) -- (42) -- (38) -- cycle;
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\draw[red,thick,fill=cyan!35,opacity=0.75] (35) -- (50) -- (42) -- (30) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (45) -- (56) -- (62) -- (59) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (58) -- (60) -- (53) -- (49) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (46) -- (57) -- (56) -- (45) -- cycle;
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%
\draw[red,thick,fill=cyan!35,opacity=0.75] (43) -- (58) -- (62) -- (56) -- cycle;
%
\draw[red,thick,fill=cyan!35,opacity=0.75] (27) -- (39) -- (44) -- (32) -- cycle;
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\draw[red,thick,fill=cyan!35,opacity=0.75] (26) -- (38) -- (39) -- (27) -- cycle;
%
\end{tikzpicture}
%
\end{document}
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