Related Polyhedra and Tilings
The planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid. These can be expanded to Platonic solids: five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,9} |
... | (3,∞} |
It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also continuing into the hyperbolic plane.
V3.6.6 |
V4.6.6 |
V5.6.6 |
V6.6.6 |
V7.6.6 |
Read more about this topic: Triangular Tiling
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