\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}
%
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	% 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);
	\coordinate(4) at (-\escala*1.61803, -\escala*0.525731, \escala*0.);
	\coordinate(5) at (-\escala*1.61803, \escala*0.525731, \escala*0.);
	\coordinate(6) at (-\escala*1.44721, \escala*0., -\escala*1.89443);
	\coordinate(7) at (-\escala*1.44721, \escala*0., -\escala*0.894427);
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	\coordinate(9) at (-\escala*1.17082, -\escala*0.850651, \escala*1.89443);
	\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.);
	\coordinate(13) at (-\escala*1., \escala*1.37638, \escala*0.);
	\coordinate(14) at (-\escala*0.894427, \escala*0., \escala*0.447214);
	\coordinate(15) at (-\escala*0.894427, \escala*0., \escala*1.44721);
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	\coordinate(17) at (-\escala*0.723607, -\escala*0.525731, -\escala*1.44721);
	\coordinate(18) at (-\escala*0.723607, -\escala*0.525731, -\escala*0.447214);
	\coordinate(19) at (-\escala*0.723607, \escala*0.525731, -\escala*1.44721);
	\coordinate(20) at (-\escala*0.723607, \escala*0.525731, -\escala*0.447214);
	%
	\coordinate(21) at (-\escala*0.723607, \escala*2.22703, -\escala*0.447214);
	\coordinate(22) at (-\escala*0.447214, -\escala*1.37638, -\escala*1.89443);
	\coordinate(23) at (-\escala*0.447214, -\escala*1.37638, -\escala*0.894427);
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	\coordinate(25) at (-\escala*0.447214, \escala*1.37638, -\escala*0.894427);
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	%
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	\coordinate(32) at (\escala*0., \escala*0., \escala*1.);
	\coordinate(33) at (\escala*0., \escala*1.7013, \escala*0.);
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	\coordinate(36) at (\escala*0.276393, \escala*0.850651, -\escala*1.44721);
	\coordinate(37) at (\escala*0.276393, \escala*0.850651, -\escala*0.447214);
	\coordinate(38) at (\escala*0.447214, -\escala*1.37638, \escala*0.894427);
	\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);
	\coordinate(42) at (\escala*0.723607, -\escala*2.22703, \escala*0.447214);
	\coordinate(43) at (\escala*0.723607, -\escala*0.525731, \escala*0.447214);
	\coordinate(44) at (\escala*0.723607, -\escala*0.525731, \escala*1.44721);
	\coordinate(45) at (\escala*0.723607, \escala*0.525731, \escala*0.447214);
	\coordinate(46) at (\escala*0.723607, \escala*0.525731, \escala*1.44721);
	\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);
	\coordinate(56) at (\escala*1.44721, \escala*0., \escala*0.894427);
	\coordinate(57) at (\escala*1.44721, \escala*0., \escala*1.89443);
	\coordinate(58) at (\escala*1.61803, -\escala*0.525731, \escala*0.);
	\coordinate(59) at (\escala*1.61803, \escala*0.525731, \escala*0.);
	\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]  (28) -- (33) -- (21) -- (13) -- cycle;
	%
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (19) -- (24) -- (36) -- (31) -- cycle;
	%
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (20) -- (19) -- (6) -- (7) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (31) -- (17) -- (6) -- (19) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (7) -- (6) -- (17) -- (18) -- cycle;
	%
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (33) -- (47) -- (51) -- (37) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (45) -- (51) -- (47) -- (40) -- cycle;
	% 
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (15) -- (11) -- (10) -- (14) -- cycle;
	% 
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (29) -- (41) -- (40) -- (28) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (32) -- (46) -- (41) -- (29) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (45) -- (40) -- (41) -- (46) -- cycle;
	%
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (14) -- (8) -- (9) -- (15) -- cycle;
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (45) -- (59) -- (61) -- (51) -- cycle;
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (18) -- (17) -- (22) -- (23) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (23) -- (22) -- (34) -- (35) -- cycle;
	% 
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (48) -- (52) -- (34) -- (31) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (53) -- (52) -- (48) -- (49) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (35) -- (34) -- (52) -- (53) -- cycle;
	% 
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (18) -- (23) -- (16) -- (12) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (12) -- (16) -- (30) -- (26) -- cycle;
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (38) -- (42) -- (50) -- (43) -- cycle;
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (59) -- (62) -- (58) -- (49) -- cycle;
	% 
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (58) -- (60) -- (53) -- (49) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (53) -- (60) -- (50) -- (35) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (43) -- (50) -- (60) -- (58) -- cycle;	
	% 
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (46) -- (57) -- (56) -- (45) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (32) -- (44) -- (57) -- (46) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (43) -- (56) -- (57) -- (44) -- cycle;
	% 
	\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;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (44) -- (39) -- (38) -- (43) -- cycle;
	\draw[red,thick,fill=cyan!35,opacity=0.75]  (26) -- (38) -- (39) -- (27) -- cycle;	
	%
	\end{tikzpicture}
	%
\end{document}



