SafekipediaIn geometry, a skew polygon is a special shape made of straight lines connected end-to-end to form a closed loop. Unlike regular polygons, which lie flat on one plane, a skew polygon has points that are not all on the same flat surface. This means its lines and points make a curved path through space, making it a three-dimensional shape even though it uses straight lines.
A skew polygon needs at least four points, called vertices, with lines called edges connecting them. Because the vertices are not all on one plane, the shape does not have a single, clear inside space like a flat polygon. This special feature makes skew polygons interesting for mathematicians who study more complex three-dimensional shapes.
There are special kinds of skew polygons, such as the zig-zag skew polygon or antiprismatic polygon, where the vertices switch back and forth between two parallel flat surfaces. These shapes always have an even number of sides. In three dimensions, some regular skew polygons follow this zig-zag pattern, making them both symmetrical and three-dimensional.
Skew polygons in three dimensions
A regular skew polygon is a special shape that exists in three dimensions. Unlike regular flat polygons, its points move between two parallel flat surfaces. Imagine a polygon that has been lifted up and down between two layers.
These shapes can be described using something called a Schläfli symbol, which combines a regular polygon with a straight line. They appear in shapes like uniform square and pentagon antiprisms, and even in the five Platonic solids, which have skew polygons with 4, 6, and 10 sides.
| Skew square | Skew hexagon | Skew octagon | Skew decagon | Skew dodecagon | ||
| {4}#{ } | {6}#{ } | {8}#{ } | {10}#{ } | {5}#{ } | {5/2}#{ } | {12}#{ } |
| s{2,4} | s{2,6} | s{2,8} | s{2,10} | sr{2,5/2} | s{2,10/3} | s{2,12} |
Regular skew polygon as vertex figure of regular skew polyhedron
A regular skew polyhedron has regular polygon faces, and a regular skew polygon is used to show its vertex figure.
There are three endless regular skew polyhedra that fill all of 3-dimensional space, and other regular skew polyhedra exist in 4-dimensional space, sometimes as part of uniform 4-polytopes.
| {4,6|4} | {6,4|4} | {6,6|3} |
|---|---|---|
Regular skew hexagon {3}#{ } | Regular skew square {2}#{ } | Regular skew hexagon {3}#{ } |
Regular skew polygons in four dimensions
In 4 dimensions, a regular skew polygon can have its vertices on a Clifford torus and be moved using a Clifford displacement. These special polygons can have an odd number of sides, which is different from some other types.
The Petrie polygons of regular 4-polytopes are examples of regular zig-zag skew polygons. The number of sides in these polygons is given by the Coxeter number for each coxeter group symmetry. For example, they have 5 sides for a 5-cell, 8 sides for a tesseract and 16-cell, 12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell. When shown in a certain way, these polygons look like regular shapes in a flat space.
The n-n duoprisms and dual duopyramids also have special Petrie polygons with many sides. (The tesseract is a 4-4 duoprism, and the 16-cell is a 4-4 duopyramid.)
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