|
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The ''interior'' surface (or area) of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all collinear. A zig-zag skew polygon or antiprismatic polygon〔Regular complex polytopes , p. 6〕 has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygon in 3-dimensions (and regular skew apeirogons in 2-dimensions) are always zig-zag. == Antiprismatic skew polygon in 3-dimensions== A regular skew polygon is isogonal with equal edge lengths. In 3-dimensions a regular skew polygon is a zig-zag skew (or antiprismatic polygon), with vertices alternating between two parallel planes. The sides of an n-antiprism can define a regular skew 2''n''-gons. A regular skew n-gonal can be given a symbol # as a ''blend'' of a regular polygon, and an orthogonal line segment, .〔Abstract Regular Polytopes, p.217〕 The symmetry operation between sequential vertices is glide reflection. Examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top and bottom polygons. |# |# |# |# |# |- align=center |160px |100px |120px |120px |100px |120px |- align=center |sr |sr |sr |sr |sr |sr |} A regular compound skew 2''n''-gon can be similarly constructed by adding a second skew polygon by a rotation. These shares the same vertices as the prismatic compound of antiprisms. |Three # |Two # |Two # |- align=center |100px |100px |100px |120px |} Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes. For example, the 5 Platonic solids have 4, 6, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around the projective envelope. The tetrahedron and octahedron include all the vertices in the zig-zag skew polygon and can be seen as a digonal and a triangular antiprisms respectively. The regular skew polyhedron have regular faces, and regular skew polygon vertex figures. Three are infinite space-filling in 3-space and others exist in 4-space, some within the uniform 4-polytope. ! |- align=center valign=top | Regular skew hexagon # | Regular skew square # | Regular skew hexagon # |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Skew polygon」の詳細全文を読む スポンサード リンク
|
