File:Golden number and Platonic dodecahedron exhibited through golden tilings.svg

When a face of this Platonic dodecahedron is entirely tiled,  like the regular pentagon on this image,  the face contains ${\displaystyle \;5\times 18=90\;{\text{ isosceles triangles,}}\,}$ divided into:

${\displaystyle 5\times 11=55\;}$ acute-angled elements with only one 36 ° angle each,

${\displaystyle 5\times 7=35\;}$ 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 ${\displaystyle {\text{the golden number φ}}\ =2\,\cos 36\,^{\circ },}$

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 tilings are congruent.  All 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.

A first projection of this Platonic dodecahedron distorts neither the face of solid at the center of the drawing nor all concentric regular decagons that it bears.  One of them is a star decagon with interlaces edges.  This first projection gives the solid a regular outline: a regular decagon again, but not oriented like the previous convex decagons.  The ten edges of the outline are ten images of edges of the solid reduced to the same scale by the projection.

On the right, a small drawing of the same solid represents its hidden edges in dashed lines.  We assume it is the same dodecahedron since we recognize certain parts of tilings that remain,  notably at the center of the drawing where a regular pyramid has three faces tiled each by two golden triangles.  This second projection does not distort the base of this pyramid,  nor the red hexagonal cross section which is regular.  Its six edges are parallel to those of the outline not shrinked by the projection, by disregarding the reduction of the whole projection relative to the first one.

On the left at the image bottom, two curved arrows represent two direct similarities.  Their composition enlarges a smallest element of tilings.  Their two successive ratios are written near the arrows.  Why a ${\displaystyle {\text{ratio of φ}}\,{\text{, }}}$ the golden ratio?  Because ${\displaystyle {\text{φ }}}$ is the largest length ratio of two edges of any element of tilings,  which is a golden triangle.  Why  square root of 5: ${\displaystyle \;{\sqrt {5}}\,{\text{? }}}$ Because this second similarity multiplies the area by 5, like on the right at the image bottom,  where all triangular tilings have their true shape,  including the largest one.  See also such enlargements on another image.