File:Assorted golden tilings and regular polygons on Platonic dodecahedron.svg

Six faces of Platonic dodecahedron are six tilings, each consisting of ${\displaystyle \;5\times 7=35\;{\text{elements that are isosceles triangles,}}}$ divided into: 

${\displaystyle 5\times 4=20\;}$ acute-angled elements having each only one 36 ° angle,

${\displaystyle 5\times 3=15\;}$ obtuse-angled elements having each each two 36 ° angle.

Every element of any type is called golden triangle because of its length ratio of two edges:

either equal to ${\displaystyle {\text{the golden number φ}}\ =2\,\cos 36\,^{\circ }={\frac {1}{\,2\,\cos 72\,^{\circ }}}={\frac {\,{\sqrt {5}}+1\,}{2}},}$

or equal to its multiplicative ${\displaystyle {\text{inverse φ}}-1={\frac {1}{\text{φ}}},}$

if not equal to 1, of course.

Any similar elements of tiling are congruent. These tilings are contructed edge‑to‑edge, any boundary between two adjacent elements is a full edge of one and the other element, even if this common edge is on an edge of the dodecahedron, when the two adjacent elements are not coplanar.

These six tilings draw nine regular polygons in blue or red lines: two decagons and seven pentagons, including a star pentagon with interlaced edges, each laying on one face of the solid, also three regular cross sections of the solid: a yellow equilateral triangle and two congruent hexagons. These hexagons and the star pentagon have the same edge length. The green hexagon is entirely visible, not distorted by the projection. Each of the two hidden edges of the violet hexagon is parallel and congruent to one of its larger edges on the drawing. The projection shrinks every edge of the violet hexagon.