Traditionally, the P3 Penrose tiling is made of thin and thick rhombi. However, by raising the vertices in three dimensions, it is possible to force all of the rhombi to be congruent. The resulting surface is known as a Wieringa roof. Due to similarities with three-dimensional quasicrystals, you can see rhombic triacontahedra and hexecontahedra hidden in the tiling.
The vertices of the rhombi in the Wieringa roof lie in four horizontal planes. If the side length of each rhombus in the two-dimensional Penrose tiling is 2 units, these planes must have the form , , , and . This increases the edge length of each rhombus to units. The diagonals of each rhombus are in the ratio .
Reference
[1] B. Grunbaum and G. C. Shephard, Tilings and Patterns, New York: W. H. Freeman & Co., 1987 p. 582.
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