Hyperpyramid
In geometry, a hyperpyramid is a generalisation of the normal pyramid to Template:Mvar dimensions.
In the case of the pyramid one connects all vertices of the base (a polygon in a plane) to a point outside the plane, which is the peak. The pyramid's height is the distance of the peak from the plane. This construction gets generalised to Template:Mvar dimensions. The base becomes a Template:Math-polytope in a Template:Math-dimensional hyperplane. A point called apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height. This construct is called a Template:Mvar-dimensional hyperpyramid.
A normal triangle is a 2-dimensional hyperpyramid, the triangular pyramid is a 3-dimensional hyperpyramid and the pentachoron or tetrahedral pyramid is a 4-dimensional hyperpyramid with a tetrahedron as base.
The Template:Mvar-dimensional volume of a Template:Mvar-dimensional hyperpyramid can be computed as follows: Here Template:Mvar denotes the Template:Mvar-dimensional volume of the hyperpyramid, Template:Mvar the Template:Math-dimensional volume of the base and Template:Mvar the height, that is the distance between the apex and the Template:Math-dimensional hyperplane containing the base Template:Mvar. For Template:Math the formula above yields the standard formulas for the area of a triangle and the volume of a pyramid.
References
- A. M. Mathai: An Introduction to Geometrical Probability. CRC Press, 1999, Template:ISBN, pp. 41–43 (Template:Google books)
- M.G. Kendall: A Course in the Geometry of N Dimensions. Dover Courier, 2004 (reprint), Template:ISBN, p. 37 (Template:Google books)