Hyperpyramid

2-dimensional Hyperpyramid with a line segment as base

4-dimensional hyperpyramid with a cube as base

Ahyperpyramidis a generalisation of the normalpyramidtondimensions.

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 tondimensions. The base becomes a (n− 1)-polytopein a (n− 1)-dimensionalhyperplane.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 an-dimensional hyperpyramid.

A normaltriangleis a 2-dimensional hyperpyramid, thetriangular pyramidis a 3-dimensional hyperpyramid and thepentachoronor tetrahedral pyramid is a 4-dimensional hyperpyramid with a tetrahedron as base.

Then-dimensional volume of an-dimensional hyperpyramid can be computed as follows:

$V_{n}={\frac {A\cdot h}{n}}$

Here$V_{n}$denotes then-dimensional volume of the hyperpyramid,Athe (n− 1)-dimensional volume of the base andhthe height, that is the distance between the apex and the (n− 1)-dimensional hyperplane containing the baseA.Forn= 2, 3 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,ISBN9789056996819,pp. 41–43 (excerpt,p. 41, atGoogle Books)
• M.G. Kendall:A Course in the Geometry of N Dimensions.Dover Courier, 2004 (reprint),ISBN9780486439273,p. 37 (excerpt,p. 37, atGoogle Books)

External links

• http://www.mathcurve.com/polyedres/hyperpyramide/hyperpyramide.shtml