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A parallelogon is constructed by two or three pairs of parallel line segments. ... Every convex parallelogon is a zonogon, but hexagonal parallelogons enable the ...
In two dimensions the analogous figure to a parallelohedron is a parallelogon, a polygon that can tile the plane edge-to-edge by translation. There are two kinds of parallelogons: the parallelograms and the hexagons in which each pair of opposite sides is parallel and of equal length.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form. [ 3 ] Every 2 n {\displaystyle 2n} -sided zonogon can be tiled by ( n 2 ) {\displaystyle {\tbinom {n}{2}}} parallelograms . [ 4 ] (
An example of a convex polygon: a regular pentagon. In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). [1]
§9.1 Tilings by polygons, triangular tiling, quadrilteral tiling, pentagonal tiling, hexagonal tiling, parallelogon, §9.2 non-convex polygon tilings, ...
Conway was likely inspired by Martin Gardner's July 1975 column in Scientific American that discussed which convex polygons can tile the plane. [6] In August 1975, Gardner revealed that Conway had discovered his criterion while trying to find an efficient way to determine which of the 108 heptominoes tile the plane.
In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases ...