geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a hyperpyramid is a generalisation of the normal

pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

s. In the case of the pyramid one connects all vertices of the base (a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

in a plane) to a point outside the plane, which is the

peak Peak or The Peak may refer to: Basic meanings Geology * Mountain peak ** Pyramidal peak, a mountaintop that has been sculpted by erosion to form a point Mathematics * Peak hour or rush hour, in traffic congestion * Peak (geometry), an (''n''-3)-d ...

. The pyramid's height is the distance of the peak from the plane. This construction gets generalised to dimensions. The base becomes a -

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

in a -dimensional

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

. 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 -dimensional hyperpyramid. A normal

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

is a 2-dimensional hyperpyramid, the

triangular pyramid In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex ...

is a 3-dimensional hyperpyramid and the

pentachoron In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral ...

or tetrahedral pyramid is a 4-dimensional hyperpyramid with a tetrahedron as base. The -dimensional volume of a -dimensional hyperpyramid can be computed as follows:

V_n = \frac

Here denotes the -dimensional volume of the hyperpyramid, the -dimensional volume of the base and the height, that is the distance between the apex and the -dimensional hyperplane containing the base . For the formula above yields the standard formulas for the area of a triangle and the volume of a pyramid.

Elements

A hyperpyramid with a

polyhedral In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...

base with

f-vector Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral com ...

(''v'',''e'',''f'') (vertices, edges, and faces), will have a new f-vector (1+''v'', ''v''+''e'', ''e''+''f'', ''f''+1). For example, a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

has f-vector (8,12,6), 8 vertices, 12 edges, and 6 square faces. Its

cubic pyramid In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cell (mathematics), cells which meet at the Apex (geometry), apex. Since a cube has a circumradius divided by edge length less than one, the squ ...

has 1+8 vertices, 20 edges (8 lateral, and 12 base), 18 faces (12 lateral triangles and 6 base squares), and 7 cells (6 lateral

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

s and 1 cubic base). In a general for ''n''-

s, extended f-vector elements expand like

polynomial expansion In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that ...

. An extended f-vector will be (1,f₀,f₁,f₂,f_''n''-1,1) transform to polynomial coefficients, (1+f₀⋅''x''+f₁⋅''x''²+f₂⋅''x''³+...+''x''^''n'')(1+''x''). Abstractly, this is a join operator, making new elements by joining all permutations of cross elements, including 1

nullitope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

(f₀), and 1 body (f_''n''). The apex point has no proper elements, just the 1 nullitope and 1 body. A ''n''-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

is a recursive pyramid of points (1-simplex, base point)

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

to (2-simplex, base segment)

to (3-simplex, base triangle)

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

to (4-simplex, base tetrahedron)

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

, etc. Their extended f-vector follows

Pascal triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

, like coefficients (1+''x'')^''n''+1 (joining ''n''+1 points). The first and last coefficients are the 1 nullitope and 1 body.

References

*A. M. Mathai: ''An Introduction to Geometrical Probability''. CRC Press, 1999, , pp. 41–43 () *M.G. Kendall: ''A Course in the Geometry of N Dimensions''. Dover Courier, 2004 (reprint), , p. 37 ()

External links

*http://www.mathcurve.com/polyedres/hyperpyramide/hyperpyramide.shtml {{dimension topics Polytopes Multi-dimensional geometry Pyramids (geometry)