for an octagon of side a. Standard coordinates The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are: (±1, ±(1+)) (±(1+), ±1). Dissectibility Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings. Skew A skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes. A regular skew octagon is vertex-transitive with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D4d, [2+,8] symmetry, order 16. Petrie polygons The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes. Symmetry The regular octagon has Dih8 symmetry, order 16. There are three dihedral subgroups: Dih4, Dih2, and Dih1, and four cyclic subgroups: Z8, Z4, Z2, and Z1, the last implying no symmetry. On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as r16. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r16 and no symmetry is labeled a1. The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8, an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order