The three <a href="https://en.wikipedia.org/wiki/Regular_polygon" class="extiw" title="w:Regular polygon">regular</a> cross sections of <a href="https://en.wikipedia.org/wiki/Platonic_solid" class="extiw" title="w:Platonic solid">Platonic dodecahedron</a> that are in red: one <a href="https://en.wikipedia.org/wiki/Equilateral_triangle" class="extiw" title="w:Equilateral triangle">equilateral</a> triangle and two <a href="https://en.wikipedia.org/wiki/Hexagon" class="extiw" title="w:Hexagon">hexagons</a> of the <a href="https://en.wikipedia.org/wiki/Three-dimensional_space" class="extiw" title="w:Three-dimensional space">three‑dimensional space</a>, are not distorted by the <a href="https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections" class="extiw" title="w:Projection (linear algebra)">projection</a> onto the <a href="https://en.wikipedia.org/wiki/Two-dimensional_space" class="extiw" title="w:Two-dimensional space">drawing plane</a>, because these three cross sections are parallel to the drawing plane. On a same drawing <a href="https://en.wikipedia.org/wiki/Dodecahedron" class="extiw" title="w:Dodecahedron">of dodecahedron</a>, the cross section in green lines is congruent to the red one. Such a Platonic solid has twenty vertices, as many as triangular cross sections formed by three diagonals of faces. The sixty edges of these twenty triangles are the edges of twelve regular <a href="https://en.wikipedia.org/wiki/Star_polygon" class="extiw" title="w:Star polygon">star pentagons</a>, each on a face of the solid. The twenty hexagonal cross sections, of which the edge length is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\,\Phi +1\,}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>2</mn><mspace width="thinmathspace"></mspace><mi mathvariant="normal">Φ<!-- Φ --></mi><mo>+</mo><mn>1</mn><mspace width="thinmathspace"></mspace></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 2\,\Phi +1\,}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bef3b81005bb52cc4588b6ed42837c21e4e8264" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:7.618ex; height:2.343ex;" alt="{\displaystyle 2\,\Phi +1\,}"></span> within the greatest drawing of dodecahedron, have  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20\times 6=12\times 10=120\,{\text{ edges}}\,}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>20</mn><mo>×<!-- × --></mo><mn>6</mn><mo>=</mo><mn>12</mn><mo>×<!-- × --></mo><mn>10</mn><mo>=</mo><mn>120</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mtext> edges</mtext></mrow><mspace width="thinmathspace"></mspace></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 20\times 6=12\times 10=120\,{\text{ edges}}\,}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed87d2ec494d7f51e531662e6f50118f6c86024" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:30.293ex; height:2.509ex;" alt="{\displaystyle 20\times 6=12\times 10=120\,{\text{ edges}}\,}"></span> in total: the edges of twelve regular star decagons, each on a pentagonal face. The <a href="https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol" class="extiw" title="w:Schläfli symbol">Schäfli symbol</a> of these twelve congruent star decagons is {10/3}. We can reveal properties of these <a href="https://en.wikipedia.org/wiki/Convex_set" class="extiw" title="w:Convex set">nonconvex</a> regular polygons through a tiling of convex regular pentagon by <a href="https://en.wikipedia.org/wiki/Golden_triangle_(mathematics)" class="extiw" title="w:Golden triangle (mathematics)">golden triangles</a>: a golden triangle is an <a href="https://en.wikipedia.org/wiki/Isosceles_triangle" class="extiw" title="w:Isosceles triangle">isosceles triangle</a> of which the greatest length <a href="https://en.wikipedia.org/wiki/Ratio" class="extiw" title="w:Ratio">ratio</a> of an edge to another is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{golden ratio:  }}\,\Phi =2\,\cos 36\,^{\circ }={\frac {1}{\,2\,\cos 72\,^{\circ }}}={\frac {\,{\sqrt {5}}+1\,}{2}}.\,}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mtext>golden ratio: </mtext></mrow><mspace width="thinmathspace"></mspace><mi mathvariant="normal">Φ<!-- Φ --></mi><mo>=</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mi>cos</mi><mo>⁡<!-- ⁡ --></mo><mn>36</mn><msup><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mo>∘<!-- ∘ --></mo></mrow></msup><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mspace width="thinmathspace"></mspace><mi>cos</mi><mo>⁡<!-- ⁡ --></mo><mn>72</mn><msup><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mo>∘<!-- ∘ --></mo></mrow></msup></mrow></mfrac></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>5</mn></msqrt></mrow><mo>+</mo><mn>1</mn><mspace width="thinmathspace"></mspace></mrow><mn>2</mn></mfrac></mrow><mo>.</mo><mspace width="thinmathspace"></mspace></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\text{golden ratio:  }}\,\Phi =2\,\cos 36\,^{\circ }={\frac {1}{\,2\,\cos 72\,^{\circ }}}={\frac {\,{\sqrt {5}}+1\,}{2}}.\,}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31cd6a34758ed0956793e93addcf44f7e648bc9a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:53.917ex; height:6.009ex;" alt="{\displaystyle {\text{golden ratio:  }}\,\Phi =2\,\cos 36\,^{\circ }={\frac {1}{\,2\,\cos 72\,^{\circ }}}={\frac {\,{\sqrt {5}}+1\,}{2}}.\,}"></span> In its two possible forms, a golden triangle has one or two  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{\,\circ }{\text{ angles,}}\,}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mn>36</mn><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>∘<!-- ∘ --></mo></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mtext> angles,</mtext></mrow><mspace width="thinmathspace"></mspace></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 36^{\,\circ }{\text{ angles,}}\,}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3380a55352d7a4355a60b3f11ab3b4a6692f51a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:11.594ex; height:2.676ex;" alt="{\displaystyle 36^{\,\circ }{\text{ angles,}}\,}"></span> its other angle or angles measuring either  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 36^{\,\circ }=108^{\,\circ },\,}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>3</mn><mo>×<!-- × --></mo><msup><mn>36</mn><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>∘<!-- ∘ --></mo></mrow></msup><mo>=</mo><msup><mn>108</mn><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>∘<!-- ∘ --></mo></mrow></msup><mo>,</mo><mspace width="thinmathspace"></mspace></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 3\times 36^{\,\circ }=108^{\,\circ },\,}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe3077bea09974e84401e3598bd26d808d7368f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:16.83ex; height:2.676ex;" alt="{\displaystyle 3\times 36^{\,\circ }=108^{\,\circ },\,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{or }}\,2\times 36^{\,\circ }=72^{\,\circ }.}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mtext>or </mtext></mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mo>×<!-- × --></mo><msup><mn>36</mn><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>∘<!-- ∘ --></mo></mrow></msup><mo>=</mo><msup><mn>72</mn><mrow class="MJX-TeXAtom-ORD"><mspace width="thinmathspace"></mspace><mo>∘<!-- ∘ --></mo></mrow></msup><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\text{or }}\,2\times 36^{\,\circ }=72^{\,\circ }.}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a529f2eb58007eb1e3f3e7aa5d4c587694a6c8f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:18.322ex; height:2.343ex;" alt="{\displaystyle {\text{or }}\,2\times 36^{\,\circ }=72^{\,\circ }.}"></span><br> 
<a href="//commons.wikimedia.org/wiki/User:Arthur_Baelde" title="User:Arthur Baelde">Arthur Baelde</a>
<span class="int-own-work" lang="en">Own work</span>