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The mean triangle area of a triangle picked inside a regular hexagon with unit area is A^_=289/3888 (Woolhouse 1867, Pfiefer 1989). This is a special case of a general ...

If A, B, and C are three points on one line, D, E, and F are three points on another line, and AE meets BD at X, AF meets CD at Y, and BF meets CE at Z, then the three points ...

A regular polygon is an n-sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular ...

A number of attractive 18-compounds of the regular tetrahedron can be constructed. The compound illustrated above will be implemented in a future version of the Wolfram ...

In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice ...

There are a number of tilings of various shapes by all the 12 order n=6 polyiamonds, summarized in the following table. Several of these (starred in the table below) are also ...

A fusene is a simple planar 2-connected graph embedded in the plane with all vertices of degree 2 or 3, all bounded faces (not necessarily regular) hexagons, and all vertices ...

The Balaban index J is a graph index defined for a graph on n nodes, m edges, and c connected components by J=m/(gamma+1)sum_((i,j) in E(G))(D_iD_j)^(-1/2), where gamma=m-n+c ...

The biggest little polygon with n sides is the convex plane n-gon of unit polygon diameter having largest possible area. Reinhardt (1922) showed that for n odd, the regular ...

The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon ...