TRIANGULAR TILING

In geometry, the 'triangular tiling' is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
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.
There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 11222, 112122, 121212, 121213, 121314)
This tiling is topologically related as a part of sequence of regular polyhedra with vertex figure (3ⁿ), and continues into the hyperbolic plane.

(33)

(34)

(35)

(36)

(37)

It is also topologically related as a part of sequence of Catalan solids with face configuration V(n.6.6).

(V3.6.6)

(V4.6.6)

(V5.6.6)

(V6.6.6) tiling

(V7.6.6) tiling

Contents

See also

References

External links

See also

★ Tilings of regular polygons

★ List of uniform tilings

★ Platonic solid

References

★ Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs

★ Tilings and Patterns, Grünbaum, Branko ; and Shephard, G. C., , , W. H. Freeman, 1987, ISBN 0-716-71193-1 (Chapter 2.1: ''Regular and uniform tilings'', p.58-65)

★ Williams, Robert ''The Geometrical Foundation of Natural Structure: A Source Book of Design'' New York: Dover, 1979. p35

External links

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