The three regular cross sections of Platonic dodecahedron that are in red: one equilateral triangle and two hexagons of the three‑dimensional space, are not distorted by the projection onto the drawing plane, because these three cross sections are parallel to the drawing plane. On a same drawing of dodecahedron, 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 star pentagons, each on a face of the solid. The twenty hexagonal cross sections, of which the edge length is 2Φ+1{\displaystyle 2\,\Phi +1\,} within the greatest drawing of dodecahedron, have  20×6=12×10=120 edges{\displaystyle 20\times 6=12\times 10=120\,{\text{ edges}}\,} in total: the edges of twelve regular star decagons, each on a pentagonal face. The Schäfli symbol of these twelve congruent star decagons is {10/3}. We can reveal properties of these nonconvex regular polygons through a tiling of convex regular pentagon by golden triangles: a golden triangle is an isosceles triangle of which the greatest length ratio of an edge to another is the golden ratio: Φ=2cos⁡36∘=12cos⁡72∘=5+12.{\displaystyle {\text{golden ratio:  }}\,\Phi =2\,\cos 36\,^{\circ }={\frac {1}{\,2\,\cos 72\,^{\circ }}}={\frac {\,{\sqrt {5}}+1\,}{2}}.\,} In its two possible forms, a golden triangle has one or two  36∘ angles,{\displaystyle 36^{\,\circ }{\text{ angles,}}\,} its other angle or angles measuring either  3×36∘=108∘,{\displaystyle 3\times 36^{\,\circ }=108^{\,\circ },\,} or 2×36∘=72∘.{\displaystyle {\text{or }}\,2\times 36^{\,\circ }=72^{\,\circ }.}
Own work
Arthur Baelde