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.
Hexagons are one of only three regular polygons to tessellate the Euclidean plane (along with squares and triangles).At first glance, several facts about them stand out: The properties of hexagons are numerous and interesting. It has six rotational symmetries and six reflection symmetries, making up the dihedral group D 6. A regular hexagon is a convex figure with sides of the same length, and internal angles of 120 degrees. When describing things as "hexagonal" I am often referring in a very broad sense to all hexagonal and hexagon-like symmetries, and not necessarily to regular hexagons per se.Ī hexagon is a closed plane figure with six edges and six vertices. In addition, I tend to speak rather loosely about "hexagonal" this and that. Bear in mind that only a very small fraction of the interesting properties of hexagons are explored in this article, and it is hoped that a more complete view of their qualities will emerge through the sum of diverse material available on this site.Ī note about terminology: As is my general custom, and unless otherwise noted, "hexagon" refers to regular hexagons only. I intend to replace or at least supplement it with a more comprehensive and eloquent survey of hexagonal concepts at some point. This article is very much a work in progress, and is not really "done" in any meaningful sense. I have avoided discussing hexagons as they pertain to human culture, religion, history, and other "local" concerns, though there are many fascinating instances of hexagonality and sixness in these areas, and they will no doubt be treated more fully elsewhere at another time. My specific concern here is with the mathematical properties of hexagons, and, to an extent, their role in the natural world. It is not intended to be a comprehensive treatment of the subject.
Using (a) and (b), find all possible pairs $(m,n)$įor a regular tessellation of the plane.The following is a brief survey of some elemental properties of hexagons, and why they might be useful. Show that for any such tesselation, we must have $m \geq 3$ and, using part (a), that $n \leq 6$. In this problem you will discover some very strong restrictions on possible tesselations of the plane, stemming from the fact that that each interior angle of an $n$ sided regular polygon measures $\frac\right) = 360. Of a regular tessellation which can be continued indefinitely in all directions: The checkerboard pattern below is an example If any two polygons in the tessellation either do not meet, share a vertex only, If all polygons in the tessellation are congruent regular polygons and For example, part of a tessellation with rectangles is A tessellation of the plane is an arrangement of polygons which cover the plane without gaps or overlapping.
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