The golden triangle is an icosceles triangle whose side length (that is, the side congruent to another side of the triangle) is the golden ratio φ* multiplied by the length of its base (the side not congruent). The angles of the golden triangle are equal to 36°, 72°, and 72°. The golden triangle has the interesting property that either one of its base angles can be bisected in order to produce another golden triangle whose side length is φ-1 times the side length of the full triangle, as well as a related triangle called the golden gnomon.

The golden gnomon, so named because of the previously described bisection process producing it as a remainder, is an isosceles triangle whose base length is φ times its side length. The angles of the golden gnomon are equal to 36°, 36°, and 108°. The obtuse angle of a golden gnomon is equal to the angle of a regular pentagon and, in fact, the golden gnomon can also be created by connecting two non-adjacent vertices of a regular pentagon, with a golden triangle also created if both non-adjacent vertices to a specific vertex (which will be adjacent to each other) are joined. In fact, the fully drawn pentagram (an arrangement of the complete graph on five vertices) will produce a large number of both golden triangles and golden gnomons.

The golden gnomon can also be divided in a similar way to the division of the golden triangle; this requires trisecting its obtuse angle and selecting one of the lines created to divide the large gnomon into a golden gnomon scaled by φ-1 as well as a golden triangle. This procedure for both tiles, when done recursively and carefully (there are two ways to perform each division, but only one will preserve the tiling rules), can be used to create the kite-and-dart and rhombus versions of the non-periodic Penrose tiles, and when they are used in this way, they are often called the Robinson tiles.

*defined here as (1+51/2)/2; some writers call φ what we will call φ-1.

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