Regular polytopes

equiangular and equilateral polygon

In geometry, a hypercube is an n-dimensional analogue of a square (two-dimensional|) and a cube (Three-dimensional|); the special case for Four-dimensional space| is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to \sqrt{n}.

polytope whose symmetry group acts transitively on its flags

{| class="wikitable floatright" |+ Cross-polytopes of dimension 2 to 5 |- |style="text-align: center;"|120px|A 2-dimensional cross-polytope |style="text-align: center;"|120px|A 3-dimensional cross-polytope |- |style="text-align: center;"|2 dimensionssquare |style="text-align: center;"|3 dimensionsoctahedron |- |style="text-align: center;"|120px|A 4-dimensional cross-polytope |style="text-align: center;"|120px|A 5-dimensional cross-polytope |- |style="text-align: center;"|4 dimensions16-cell |style="text-align: center;"|5 dimensions5-orthoplex |}

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