Combinations of Regular Polygons That Can Meet At A Vertex
The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular -gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex. Only eleven of these can occur in a uniform tiling of regular polygons. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.
With 3 polygons at a vertex:
- 3.7.42 (cannot appear in any tiling of regular polygons)
- 3.8.24 (cannot appear in any tiling of regular polygons)
- 3.9.18 (cannot appear in any tiling of regular polygons)
- 3.10.15 (cannot appear in any tiling of regular polygons)
- 3.122 - semi-regular, truncated hexagonal tiling
- 4.5.20 (cannot appear in any tiling of regular polygons)
- 4.6.12 - semi-regular, truncated trihexagonal tiling
- 4.82 - semi-regular, truncated square tiling
- 52.10 (cannot appear in any tiling of regular polygons)
- 63 - regular, hexagonal tiling
Below are diagrams of such vertices:
-
3.7.42
-
3.8.24
-
3.9.18
-
3.10.15
-
3.12.12
-
4.5.20
-
4.6.12
-
4.8.8
-
5.5.10
-
6.6.6
With 4 polygons at a vertex:
- 32.4.12 - not uniform, has two different types of vertices 32.4.12 and 36
- 3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
- 32.62 - not uniform, occurs in two patterns with vertices 32.62/36 and 32.62/3.6.3.6.
- 3.6.3.6 - semi-regular, trihexagonal tiling
- 44 - regular, square tiling
- 3.42.6 - not uniform, has vertices 3.42.6 and 3.6.3.6.
- 3.4.6.4 - semi-regular, rhombitrihexagonal tiling
Below are diagrams of such vertices:
-
3.3.4.12
-
3.4.3.12
-
3.3.6.6
-
3.6.3.6
-
4.4.4.4
-
3.4.4.6
-
3.4.6.4
With 5 polygons at a vertex:
- 34.6 - semi-regular, Snub hexagonal tiling, comes in two entianomorphic forms. The vertex figures of the two entianomorphs are the same, but the resulting tilings are different.
- 33.42 - semi-regular, Elongated triangular tiling
- 32.4.3.4 - semi-regular, Snub square tiling
Below are diagrams of such vertices:
-
3.3.3.3.6
-
3.3.3.4.4
-
3.3.4.3.4
With 6 polygons at a vertex:
- 36 - regular, Triangular tiling
Below is a diagram of such a vertex:
-
3.3.3.3.3.3
Read more about this topic: Tiling By Regular Polygons
Famous quotes containing the words combinations of, combinations, regular and/or meet:
“You should try to understand every thing you see and hear; to act and judge for yourselves; to remember you each have a soul of your own to account for; M a mind of your own to improve. When you once get these ideas fixed, and learn to act upon them, no man or set of men, no laws, customs, or combinations of them can seriously oppress you.”
—Jane Grey Swisshelm (1815–1884)
“Europe has a set of primary interests, which to us have none, or a very remote relation. Hence she must be engaged in frequent controversies, the causes of which are essentially foreign to our concerns. Hence, therefore, it must be unwise in us to implicate ourselves, by artificial ties, in the ordinary vicissitudes of her politics or the ordinary combinations and collisions of her friendships or enmities.”
—George Washington (1732–1799)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)
“As we try to change, we will discover within us a fierce struggle between our loyalty to that battle-scarred victim of his own childhood, our father, and the father we want to be. We must meet our childhood father at close range: get to know him, learn to forgive him, and somehow, go beyond him.”
—Augustus Y. Napier (20th century)