gasilclass.blogg.se - Triangular tessellation

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. If you want to try a more complicated version, cut two different squiggles out of two different sides, and move them both.The following is a brief survey of some elemental properties of hexagons, and why they might be useful.Color in your basic shape to look like something - an animal? a flower? a colorful blob? Add color and design throughout the tessellation to transform it into your own Escher-like drawing.

The shape will still tessellate, so go ahead and fill up your paper.Then move it the same way you moved the squiggle (translate or rotate) so that the squiggle fits in exactly where you cut it out. On a large piece of paper, trace around your tile. Tape the squiggle into its new location.It’s important that the cut-out lines up along the new edge in the same place that it appeared on its original edge.You can either translate it straight across or rotate it. Cut out the squiggle, and move it to another side of your shape.Draw a “squiggle” on one side of your basic tile.The first time you do this, it’s easiest to start with a simple shape that you know will tessellate, like an equilateral triangle, a square, or a regular hexagon. Here’s how you can create your own Escher-like drawings. Work on the following exercises on your own or with a partner.