Tessellation - Number of Sides of A Polygon Versus Number of Sides at A Vertex

Number of Sides of A Polygon Versus Number of Sides At A Vertex

For an infinite tiling, let be the average number of sides of a polygon, and the average number of sides meeting at a vertex. Then . For example, we have the combinations (3, 6), (31/3, 5), (33/4, 42/7), (4, 4), (6, 3), for the tilings in the article Tilings of regular polygons.

A continuation of a side in a straight line beyond a vertex is counted as a separate side. For example, the bricks in the picture are considered hexagons, and we have combination (6, 3). Similarly, for the basketweave tiling often found on bathroom floors, we have (5, 31/3).

For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.

For finite tessellations and polyhedra we have

where is the number of faces and the number of vertices, and is the Euler characteristic (for the plane and for a polyhedron without holes: 2), and, again, in the plane the outside counts as a face.

The formula follows observing that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides. From the two results the formula readily follows.

In most cases the number of sides of a face is the same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner has to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.

A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.

For the Platonic solids we get round numbers, because we take the average over equal numbers: for we get 1, 2, and 3.

From the formula for a finite polyhedron we see that in the case that while expanding to an infinite polyhedron the number of holes (each contributing −2 to the Euler characteristic) grows proportionally with the number of faces and the number of vertices, the limit of is larger than 4. For example, consider one layer of cubes, extending in two directions, with one of every 2 × 2 cubes removed. This has combination (4, 5), with, corresponding to having 10 faces and 8 vertices per hole.

Note that the result does not depend on the edges being line segments and the faces being parts of planes: mathematical rigor to deal with pathological cases aside, they can also be curves and curved surfaces.

As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs. Examples of tessellations of other spaces include:

• Tessellations of n-dimensional Euclidean space. For example, 3-dimensional Euclidean space can be filled with cubes to create the cubic honeycomb.

• Tessellations of n-dimensional elliptic space, either the n-sphere (spherical tiling, spherical polyhedron) or n-dimensional real projective space (elliptic tiling, projective polyhedron).

  For example, projecting the edges of a regular dodecahedron onto its circumsphere creates a tessellation of the 2-dimensional sphere with regular spherical pentagons, while taking the quotient by the antipodal map yields the hemi-dodecahedron, a tiling of the projective plane.

• Tessellations of n-dimensional hyperbolic space. For example, M. C. Escher's Circle Limit III depicts a tessellation of the hyperbolic plane (using the Poincaré disk model) with congruent fish-like shapes. The hyperbolic plane admits a tessellation with regular p-gons meeting in q's whenever ; Circle Limit III may be understood as a tiling of octagons meeting in threes, with all sides replaced with jagged lines and each octagon then cut into four fish.

See (Magnus 1974) for further non-Euclidean examples.

There are also abstract polyhedra which do not correspond to a tessellation of a manifold because they are not locally spherical (locally Euclidean, like a manifold), such as the 11-cell and the 57-cell. These can be seen as tilings of more general spaces.