In
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 Schläfli orthoscheme is a type of
simplex. The orthoscheme is the generalization of the
right triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees).
The side opposite to the right angle i ...
to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of
edges that are mutually
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
. They were introduced by
Ludwig Schläfli, who called them ''orthoschemes'' and studied their
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
in
Euclidean,
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
, and
spherical geometries.
H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for
trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
, orthoschemes form the basis of a trigonometry of ''n'' dimensions, as developed by
Schoute who called it
polygonometry.
J.-P. Sydler and
Børge Jessen studied orthoschemes extensively in connection with
Hilbert's third problem
The third of Hilbert's problems, Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedron, polyhedra of equal volume, is it always possible t ...
.
Orthoschemes, also called path-simplices in the
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
literature, are a special case of a more general class of simplices studied by
Fiedler, and later rediscovered by
Coxeter. These simplices are the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
s of
trees in which all edges are mutually perpendicular. In an orthoscheme, the underlying tree is a
path.
In three dimensions, an orthoscheme is also called a
birectangular tetrahedron (because its path makes two right angles at vertices each having two right angles) or a
quadrirectangular tetrahedron
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 conv ...
(because it contains four right angles).
Properties

* All
2-faces are
right triangles.
* All
facets of a ''d''-dimensional orthoscheme are (''d'' − 1)-dimensional orthoschemes.
* The
dihedral angles that are disjoint from edges of the path have
acute angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
s; the remaining
dihedral angles are all
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s.
* The
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dim ...
of the longest
edge is the center of the
circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's Vertex (geometry), vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the te ...
.
* The case when
is a generalized
Hill tetrahedron.
* Every
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
in ''d''-dimensional space can be dissected into ''d''! congruent orthoschemes. A similar dissection into the same number of orthoschemes applies more generally to every
hyperrectangle but in this case the orthoschemes may not be congruent.
* Every
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
can be dissected radially into ''g'' congruent orthoschemes that meet at its center, where ''g'' is the ''order'' of the regular polytope's symmetry group.
* Every orthoscheme can be trisected into three smaller 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.
Dissection into orthoschemes
Hugo Hadwiger conjectured in 1956 that every simplex can be
dissected into finitely many orthoschemes. The conjecture has been proven in spaces of five or fewer dimensions, but remains unsolved in higher dimensions.
Hadwiger's conjecture implies that every convex polytope can be dissected into orthoschemes.
Characteristic simplex of the general regular polytope
Coxeter identifies various orthoschemes as the characteristic simplexes of the polytopes they generate by reflections. The characteristic simplex is the polytope's fundamental building block. It can be replicated by reflections or rotations to construct the polytope, just as the polytope can be dissected into some integral number of it. The characteristic simplex is
chiral
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is dist ...
(it comes in two mirror-image forms which are different), and the polytope is dissected into an equal number of left- and right-hand instances of it. It has dissimilar edge lengths and faces, instead of the equilateral triangle faces of the regular simplex. When the polytope is regular, its characteristic simplex is an orthoscheme, a simplex with only right triangle faces.
Every regular polytope has its characteristic orthoscheme which is its
fundamental region, the irregular simplex which has exactly the same
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
characteristics as the regular polytope but captures them all without repetition. For a regular ''k''-polytope, the
Coxeter-Dynkin diagram of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the
generating point ring. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
. This is a
barycentric subdivision.
See also
*
3-orthoscheme (
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 ...
with right-triangle faces)
*
4-orthoscheme (
5-cell with right-triangle faces)
*
Goursat tetrahedron
In geometry, a Goursat tetrahedron is a tetrahedron, tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-spa ...
*
Order polytope
References
{{DEFAULTSORT:Schlafli orthoscheme
Polytopes