![]() ![]() ![]() A skew zig-zag octagon has vertices alternating between two parallel planes.Ī regular skew octagon is vertex-transitive with equal edge lengths. The interior of such an octagon is not generally defined. These squares and rhombs are used in the Ammann–Beenker tilings.Ī regular skew octagon seen as edges of a square antiprism, symmetry D 4d,, (2*4), order 16.Ī skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The list (sequence A006245 in the OEIS) defines the number of solutions as eight, by the eight orientations of this one dissection. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:ĭissectibility 8-cube projectionĬoxeter states that every zonogon (a 2 m-gon whose opposite sides are parallel and of equal length) can be dissected into m( m-1)/2 parallelograms. A regular octagon is represented by the Schläfli symbol It has eight lines of reflective symmetry and rotational symmetry of order 8. 10 Regularity Ī regular octagon is a closed figure with sides of the same length and internal angles of the same size. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other). As with all polygons, the external angles total 360°. The sum of all the internal angles of any octagon is 1080°. ![]() The diagonals of the green quadrilateral are equal in length and at right angles to each other ![]()
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