![]() ![]() There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. f (x) f ( x) is concave down on an interval I I if all of the tangents to the curve on I I are above the graph of f (x) f ( x). Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Definition 1 Given the function f (x) f ( x) then f (x) f ( x) is concave up on an interval I I if all of the tangents to the curve on I I are below the graph of f (x) f ( x). The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.Hypercell or Teron, a 4-dimensional elementįor example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body. There are no nonconvex Euclidean regular tessellations in any number of dimensions. This table shows a summary of regular polytope counts by dimension. ( April 2018) ( Learn how and when to remove this template message) Unsourced material may be challenged and removed. ( talk) Please help improve this article by adding citations to reliable sources in this section. This section needs additional citations for verification. ![]() Monkey saddle (saddle-like surface for 3 legs.).Hyperbolic paraboloid (a ruled surface).Curves with genus greater than one Ĭurve families with variable genus Ĭurves generated by other curves ![]()
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