Hyper pyramid

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2-dimensional hyperpyramid with a segment as a base
3-dimensional hyperpyramid with a square as a base
4-dimensional hyperpyramid with a cube as a base

A hyperpyramid is a generalization of the three-dimensional concept of a pyramid to n dimensions .

construction

In a three-dimensional pyramid, all corner points of a (two-dimensional) polygon in the plane, the base, are connected to a point in space, the pyramid tip. This construction is now extended to n dimensions for the hyper pyramid. The base, i.e. the polygon in the plane, becomes an (n-1) -polytope in an (n-1) -dimensional hyperplane , the corner points of which are now connected to a point in the n-dimensional space outside the hyperplane. The resulting body is called an n-dimensional hyperpyramid. The distance from the pyramid tip to the base or to the hyperplane in which the base is embedded is referred to as the height, as in the three-dimensional case.

A one-dimensional hyperpyramid is a line, a two-dimensional hyperpyramid is a triangle and a three-dimensional hyperpyramid is the ordinary pyramid. The pentachoron is a four-dimensional hyperpyramid with a tetrahedron as a base.

The n-dimensional volume of an n-dimensional hyperpyramid is:

Here, the n-dimensional volume of the hyperpyramid, A the (n-1) -dimensional volume of its base and h its height. For the cases n = 2 and n = 3, the above formula provides the known formulas for the triangular area and the pyramid volume.

4-dimensional examples

Remarks:
Representation as a Schlegel diagram (this places the - actually extradimensional - pyramid tip in the center).
Analogous to the duality of cube ↔ octahedron and dodecahedron ↔ icosahedron, the corresponding pyramids are also dual (the tetrahedron pyramid is self-dual).

literature

Web links