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If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there ...
It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.
The vertices and edges on the interior of the hexagon are suppressed. There are five Bravais lattices in two dimensions, related to the parallelogon tessellations by their five symmetry variations. In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not ...
If one considers a pappian plane containing a hexagon as just described but with sides and parallel and also sides and parallel (so that the Pappus line is the line at infinity), one gets the affine version of Pappus's theorem shown in the second diagram.
However, every curve of constant width can be enclosed by at least one regular hexagon with opposite sides on parallel supporting lines. [15] A curve has constant width if and only if, for every pair of parallel supporting lines, it touches those two lines at points whose distance equals the separation between the lines.
In its simplest form, the criterion simply states that any hexagon with a pair of opposite sides that are parallel and congruent will tessellate the plane. [8] In Gardner's article, this is called a type 1 hexagon. [7] This is also true of parallelograms.
[11] [10]: p.11 One way to see this is as a limiting case of Brianchon's theorem, which states that a hexagon all of whose sides are tangent to a single conic section has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of ...
Torus embedding of the order-13 Paley graph, obtained by gluing each pair of parallel sides of a hexagon. The six neighbors of each vertex in the Paley graph of order 13 are connected in a cycle; that is, the graph is locally cyclic.