![]() Cubes have six sides, and they are pretty useful too.Īnd so on.Six is the smallest composite squarefree integer, and by extension, the first natural number with two distinct prime factors (2 and 3).In a related matter, six is the only number that is both the sum and the product of three consecutive natural numbers (1, 2, and 3).Six is a highly composite number, the second-smallest composite number, and the first perfect number.Let us then take a moment to enumerate some interesting facts about the number six: Such a hexagon would lie on a plane consisting of all points with coordinates that add up to 6, and would bisect a cube of unit length 2 between coordinates 1 and 3.Īnother interesting thing about hexagons-and perhaps the most striking fact about them-is that they do, in fact, have six sides. Hexagons are third-order permutohedra, meaning each vertex of a hexagon can be described with Cartesian coordinates using one of six permutations of the numbers 1, 2, and 3.A consequence of this is that no regular polytope, in any dimension, has hexagonal faces-though many have hexagon-like or hexagonally-symmetric vertices or other elements. The three polygons with fewer sides compose the surfaces of the five platonic solids, but no polygon with six or more sides can be employed for this purpose. Hexagons are the first polygons-when ascending by number of sides-that do not form the faces of a regular convex polyhedron in Euclidean space.In a related fact, hexagons are the unique regular polygon such that the distance between the center and each vertex is equal to the length of each side (sharing this property with the cuboctahedron in 3-space).(As far as I know, hexagons are the only regular polytope of any dimension with this particular property.) Hexagons are the only regular polygon that can be subdivided into another regular polygon.The hexagonal tessellation is combinatorially identical to the close packing of circles on a plane. ![]()
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