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A non-convex regular polygon is a regular star polygon. The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. [4] Each vertex has edges evenly spaced around it.
This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
1–18: 5 convex regular and 13 convex semiregular; 20–22, 41: 4 non-convex regular; 19–66: Special 48 stellations/compounds (Nonregulars not given on this list) 67–109: 43 non-convex non-snub uniform; 110–119: 10 non-convex snub uniform; Chi: the Euler characteristic, χ. Uniform tilings on the plane correspond to a torus topology ...
The regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after the Platonic solids: Tetrahedral; Octahedral (or cubic) Icosahedral (or dodecahedral) Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.