Pentagonal tiling

Figure 1-11. Attempts of pentagonal tiling by (a) Diirer (after Crowe) [32] (b) Shubnikov (after Mackay) and (c) Kepler (after Danzer et al.) [33, 34],...

Figure 1-10. Aucmpts of pentagonal tiling by, from top to bottom, Dtircr (after Crowe 11-24 ), Kepler (after Dan/.er t l at. (I-2. >D and Shubnikov (after Mackay [1-26]).

Figure 9-63. Tiling with regular pentagons after Mackay [132],...

While the pentagon does not tile the plane, it does indeed tile the sphere. Twelve pentagons make up ti e pentagonal dodecahedron whose vertices lie on the sphere. Similarly, the icosahedron is a triangular tiling of the sphere with fivefold rotational symmetry. 

Figure 3.16 A periodic tiling using equilateral (not regular) pentagons (a) and the unit of structure (b) which will tile the area by translation alone. This tiling has been known and used since ancient times...

Fig. 2. Demonstration that it is not possible to tile a plane with pentagonal units (left) and that pentagons are ideal units for the construction of spherical patterns like the dodecahedron (right)...

Figure 9-64. Tiling with regular pentagons, after Mackay [9-73].

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