This decagonal mandala is split into fifty golden triangles (shown in yellow), and forty golden gnomons (shown in orange).
The golden triangles, in yellow, are acute isosceles triangles with a leg:base ratio which is the golden ratio. Golden gnomons, shown in orange, are related, for they are obtuse isosceles triangles where the golden ratio shows up as the base:leg ratio, which is the reciprocal of the manifestation of the golden ratio which appears in the yellow triangles.
This tessellation can be viewed in at least two ways: it can be seen as being composed of overlapping octagons which are equilateral, but not equiangular — or it can be viewed as a periodically-repeating pattern of golden gnomons, as well as golden triangles of two different sizes. Both golden triangles and golden gnomons are isosceles triangles with sides in the golden ratio, but golden triangles are acute, while golden gnomons are obtuse.
There are two isosceles triangles which are related to the golden ratio, [1 + sqrt(5)]/2, and I used to refer to them as the “golden acute isosceles triangle” and the “golden obtuse isosceles triangle,” before I found out these triangles already have other names –the ones shown above. The golden triangle, especially, shows up in some well-known polyhedra, such as both the great and small stellated dodecahedron. The triangles which form the “points” or “arms” of regular star pentagons (also known as pentagrams) are also golden triangles.
These triangles have sides which are in the golden ratio. For the golden triangle, it is the leg:base ratio which is golden, as shown above. For the golden gnomon, this ratio is reversed: the base:leg ratio is φ, or ~1.61803 — the irrational number known as the golden ratio.
The angle ratios of each of these triangles are also unique. The golden triangle’s angles are in a 1:2:2 ratio, while the angles of the golden gnomon are in a ratio of 1:1:3.
Another interesting fact about these two triangles is that each one can be subdivided into one of each type of triangle. The golden triangle can be split into a golden gnomon, and a smaller golden triangle, while the golden gnomon can be split into a golden triangle, and a smaller golden gnomon, as seen below.
This process can be repeated indefinitely, in each case, creating ever-smaller triangles of each type.
Polyhedra which use these triangles, as either faces or “facelets” (the visible parts of partially-hidden faces) are not uncommon, as previously mentioned. The three most well-known examples are three of the four Kepler-Poinsot solids. In the first two shown below, the small stellated dodecahedron and the great stellated dodecahedron, the actual faces are regular star pentagons which interpenetrate, but the facelets are golden triangles.
The next example is also a Kepler-Poinsot solid: the great dodecahedron. Its actual faces are simply regular pentagons, not star pentagons, but, again, they interpenetrate, hiding much of each face from view. The visible parts, or “facelets,” are golden gnomons.
For another example of a polyhedron made of golden gnomons, I made one myself — meaning that if anyone else has ever seen this polyhedron before, this fact is unknown to me, although I cannot rule it out. I have not given it a name. It has thirty-six faces, all of which are golden gnomons. There are twelve of the larger ones, shown in yellow, and twenty-four of the smaller ones, shown in red. This polyhedron has pyritohedral symmetry (the same type of symmetry seen in the seam-pattern of a typical volleyball), and its convex hull is the icosahedron.
[Picture credits: to create the images in this post, I used both Geometer’s Sketchpad and MS-Paint for the two still, flat pictures found at the top. To make images of the four rotating polyhedra, I used a different program, Stella 4d: Polyhedron Navigator. Stella is available for purchase, with a free trial download available, at http://www.software3d.com/Stella.php.]