Schläfli orthoscheme
Encyclopedia
In geometry
, Schläfli orthoscheme is a type of simplex
. They are defined by a sequence of edges
that are mutually orthogonal. These were introduced by Ludwig Schläfli
, who called them orthoschemes and studied their volume
in the Euclidean
, Lobachevsky
and the spherical geometry
. H.S.M. Coxeter later named them after Schläfli. J.-P. Sydler
and Børge Jessen
studied them extensively in connection with Hilbert's third problem
.
Orthoschemes, also called path-simplices in the applied mathematics
literature, are a special case of a more general class of simplices studied by , and later rediscovered by . These simplices are the convex hull
s of trees
in which all edges are mutually perpendicular. In the orthoscheme, the underlying tree is a path
. In three dimensions, an orthoscheme is also called a birectangular tetrahedron.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, Schläfli orthoscheme is a type of simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
. They are defined by a sequence of edges
that are mutually orthogonal. These were introduced by Ludwig SchläfliLudwig Schläfli
Ludwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...
, who called them orthoschemes and studied their volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
in the Euclidean
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, Lobachevsky
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
and the spherical geometry
Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
. H.S.M. Coxeter later named them after Schläfli. J.-P. Sydler
Jean-Pierre Sydler
Jean-Pierre Sydler is a Swiss mathematician and a librarian, well known for his work in geometry, most notably on Hilbert's third problem.-Biography :...
and Børge Jessen
Børge Jessen
Børge Christian Jessen was a Danish mathematician best known for his work in analysis, specifically on zeta function, and in geometry, specifically on Hilbert's third problem....
studied them extensively in connection with Hilbert's third problem
Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
.
Orthoschemes, also called path-simplices in the applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
literature, are a special case of a more general class of simplices studied by , and later rediscovered by . These simplices are the convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
s of trees
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
in which all edges are mutually perpendicular. In the orthoscheme, the underlying tree is a path
Path graph
In the mathematical field of graph theory, a path graph or linear graph is a particularly simple example of a tree, namely a tree with two or more vertices that is not branched at all, that is, contains only vertices of degree 2 and 1...
. In three dimensions, an orthoscheme is also called a birectangular tetrahedron.
Properties
- All 2-facesFace (geometry)In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
are right triangles. - All facets of a d-dimensional orthoscheme are (d − 1)-dimensional orthoschemes.
- The midpointMidpointThe midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...
of the longest edgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
is the center of the circumscribed sphereCircumscribed sphereIn geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing...
. - The case when
is a generalized Hill tetrahedronHill tetrahedronIn geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M.J.M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube.- Construction :...
. - In 3- and 4-dimensional Euclidean space, every convex polytopeConvex polytopeA convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...
is scissor congruentHilbert's third problemThe third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
to an orthoscheme. - (H.S.M. Coxeter) Every orthoscheme in a three-dimensional space of constant curvature can be dissected into three orthoschemes.
- (H.E. Debrunner) Every orthoscheme in a d-dimensional space of constant curvature can be dissected into d + 1 orthoschemes.
- Every hypercubeHypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . 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.An...
in d-dimensional space can be dissected into d! congruent orthoschemes. - Every d-dimensional boxHyperrectangleIn geometry, an orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals....
can be dissected into d! (not necessarily congruent) orthoschemes. - In 3-dimensional hyperbolic and spherical spaces, the volume of orthoschemes can be expressed in terms of the Lobachevsky function, or in terms of dilogarithms, see e.g. .