Simplex Information
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 sonamed because it represents the simplest possible polytope in any given space.
For example,
 a 0simplex is a point,
 a 1simplex is a line segment,
 a 2simplex is a triangle,
 a 3simplex is a tetrahedron,
 a 4simplex is a 5cell.
Specifically, a ksimplex is a kdimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points are affinely independent, which means are linearly independent. Then, the simplex determined by them is the set of points
A regular simplex^{ [1]} is a simplex that is also a regular polytope. A regular ksimplex may be constructed from a regular (k − 1)simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex ^{ [2]} is the simplex whose vertices are the k standard unit vectors and the origin, or
In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices.
History
The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").^{ [3]}
The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as α_{n}, the other two being the crosspolytope family, labeled as β_{n}, and the hypercubes, labeled as γ_{n}. A fourth family, the tessellation of ndimensional space by infinitely many hypercubes, he labeled as δ_{n}.^{ [4]}
Elements
The convex hull of any nonempty subset of the n + 1 points that define an nsimplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an msimplex, called an mface of the nsimplex. The 0faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1faces are called the edges, the (n − 1)faces are called the facets, and the sole nface is the whole nsimplex itself. In general, the number of mfaces is equal to the binomial coefficient .^{ [5]} Consequently, the number of mfaces of an nsimplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail.
The number of 1faces (edges) of the nsimplex is the nth triangle number, the number of 2faces of the nsimplex is the (n − 1)th tetrahedron number, the number of 3faces of the nsimplex is the (n − 2)th 5cell number, and so on.
Δ^{n}  Name 
Schläfli Coxeter 
0 faces (vertices) 
1 faces (edges) 
2 faces 
3 faces 
4 faces 
5 faces 
6 faces 
7 faces 
8 faces 
9 faces 
10 faces 
Sum = 2^{n+1} − 1 

Δ^{0}  0simplex ( point) 
( ) 
1  1  
Δ^{1}  1simplex ( line segment) 
{ } = ( ) ∨ ( ) = 2 · ( ) 
2  1  3  
Δ^{2}  2simplex ( triangle) 
{3} = 3 · ( ) 
3  3  1  7  
Δ^{3}  3simplex ( tetrahedron) 
{3,3} = 4 · ( ) 
4  6  4  1  15  
Δ^{4}  4simplex ( 5cell) 
{3^{3}} = 5 · ( ) 
5  10  10  5  1  31  
Δ^{5}  5simplex  {3^{4}} = 6 · ( ) 
6  15  20  15  6  1  63  
Δ^{6}  6simplex  {3^{5}} = 7 · ( ) 
7  21  35  35  21  7  1  127  
Δ^{7}  7simplex  {3^{6}} = 8 · ( ) 
8  28  56  70  56  28  8  1  255  
Δ^{8}  8simplex  {3^{7}} = 9 · ( ) 
9  36  84  126  126  84  36  9  1  511  
Δ^{9}  9simplex  {3^{8}} = 10 · ( ) 
10  45  120  210  252  210  120  45  10  1  1023  
Δ^{10}  10simplex  {3^{9}} = 11 · ( ) 
11  55  165  330  462  462  330  165  55  11  1  2047 
In layman's terms, an nsimplex is a simple shape (a polygon) that requires n dimensions. Consider a line segment AB as a "shape" in a 1dimensional space (the 1dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1dimensional space. The triangle is the 2simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2dimensional space (the plane in which the triangle resides). One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2dimensional space. The tetrahedron is the 3simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3dimensional space (the 3space in which the tetrahedron lies). One can place a new point E somewhere outside the 3space. The new shape ABCDE, called a 5cell, requires four dimensions and is called the 4simplex; it cannot fit in the original 3dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1dimensional space to hold it; the line segment is the 1simplex. The line segment itself was formed by starting with a single point in 0dimensional space (this initial point is the 0simplex) and adding a second point, which required the increase to 1dimensional space.
More formally, an (n + 1)simplex can be constructed as a join (∨ operator ) of an nsimplex and a point, ( ). An (m + n + 1)simplex can be constructed as a join of an msimplex and an nsimplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1simplex is the join of two points: ( ) ∨ ( ) = 2 · ( ). A general 2simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An isosceles triangle is the join of a 1simplex and a point: { } ∨ ( ). An equilateral triangle is 3 · ( ) or {3}. A general 3simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4 · ( ) or {3,3} and so on.
In some conventions,^{ [7]} the empty set is defined to be a (−1)simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
Symmetric graphs of regular simplices
These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
The standard simplex
The standard nsimplex (or unit nsimplex) is the subset of R^{n+1} given by
The simplex Δ^{n} lies in the affine hyperplane obtained by removing the restriction t_{i} ≥ 0 in the above definition.
The n + 1 vertices of the standard nsimplex are the points e_{i} ∈ R^{n+1}, where
 e_{0} = (1, 0, 0, ..., 0),
 e_{1} = (0, 1, 0, ..., 0),
 e_{n} = (0, 0, 0, ..., 1).
There is a canonical map from the standard nsimplex to an arbitrary nsimplex with vertices (v_{0}, ..., v_{n}) given by
The coefficients t_{i} are called the barycentric coordinates of a point in the nsimplex. Such a general simplex is often called an affine nsimplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine nsimplex to emphasize that the canonical map may be orientation preserving or reversing.
More generally, there is a canonical map from the standard simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex:
A commonly used function from R^{n} to the interior of the standard simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.
Examples
 Δ^{0} is the point 1 in R^{1}.
 Δ^{1} is the line segment joining (1, 0) and (0, 1) in R^{2}.
 Δ^{2} is the equilateral triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R^{3}.
 Δ^{3} is the regular tetrahedron with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R^{4}.
Increasing coordinates
An alternative coordinate system is given by taking the indefinite sum:
This yields the alternative presentation by order, namely as nondecreasing ntuples between 0 and 1:
Geometrically, this is an ndimensional subset of (maximal dimension, codimension 0) rather than of (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, here correspond to successive coordinates being equal, while the interior corresponds to the inequalities becoming strict (increasing sequences).
A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the ncube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the ncube into mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume Alternatively, the volume can be computed by an iterated integral, whose successive integrands are
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinitedimensional simplex without issues of convergence of sums.
Projection onto the standard simplex
Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given with possibly negative entries, the closest point on the simplex has coordinates
where is chosen such that
can be easily calculated from sorting .^{ [8]} The sorting approach takes complexity, which can be improved to complexity via medianfinding algorithms.^{ [9]} Projecting onto the simplex is computationally similar to projecting onto the ball.
Corner of cube
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
This yields an nsimplex as a corner of the ncube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.
Cartesian coordinates for a regular ndimensional simplex in R^{n}
One way to write down a regular nsimplex in R^{n} is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is ; and the fact that the angle subtended through the center of the simplex by any two vertices is .
It is also possible to directly write down a particular regular nsimplex in R^{n} which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of R^{n} by e_{1} through e_{n}. Begin with the standard (n − 1)simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular nsimplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) for some real number α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular nsimplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for α. Solving this equation shows that there are two choices for the additional vertex:
Either of these, together with the standard basis vectors, yields a regular nsimplex.
The above regular nsimplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
for , and
This simplex is inscribed in a hypersphere of radius .
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
where , and
The side length of this simplex is .
A highly symmetric way to construct a regular nsimplex is to use a representation of the cyclic group Z_{n+1} by orthogonal matrices. This is an n× n orthogonal matrix Q such that Q^{n+1} = I is the identity matrix, but no lower power of Q is. Applying powers of this matrix to an appropriate vector v will produce the vertices of a regular nsimplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix
where each Q_{i} is orthogonal and either 2 × 2 or 1 ÷ 1. In order for Q to have order n + 1, all of these matrices must have order dividing n + 1. Therefore each Q_{i} is either a 1 × 1 matrix whose only entry is 1 or, if n is odd, −1; or it is a 2 × 2 matrix of the form
where each ω_{i} is an integer between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Q_{i} form a basis for the nontrivial irreducible real representations of Z_{n+1}, and the vector being rotated is not stabilized by any of them.
In practical terms, for n even this means that every matrix Q_{i} is 2 × 2, there is an equality of sets
and, for every Q_{i}, the entries of v upon which Q_{i} acts are not both zero. For example, when n = 4, one possible matrix is
Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are
each of which has distance √5 from the others. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a nonzero entry of v; while the remaining diagonal blocks, say Q_{1}, ..., Q_{(n − 1) / 2}, are 2 × 2, there is an equality of sets
and each diagonal block acts upon a pair of entries of v which are not both zero. So, for example, when n = 3, the matrix can be
For the vector (1, 0, 1/√2), the resulting simplex has vertices
each of which has distance 2 from the others.
Geometric properties
Volume
The volume of an nsimplex in ndimensional space with vertices (v_{0}, ..., v_{n}) is
where each column of the n × n determinant is the difference between the vectors representing two vertices.^{ [10]} A more symmetric way to write it is
Another common way of computing the volume of the simplex is via the Cayley–Menger determinant. It can also compute the volume of a simplex embedded in a higherdimensional space, e.g., a triangle in .^{ [11]}
Without the 1/n! it is the formula for the volume of an n parallelotope. This can be understood as follows: Assume that P is an nparallelotope constructed on a basis of . Given a permutation of , call a list of vertices a npath if
(so there are n! npaths and does not depend on the permutation). The following assertions hold:
If P is the unit nhypercube, then the union of the nsimplexes formed by the convex hull of each npath is P, and these simplexes are congruent and pairwise nonoverlapping.^{ [12]} In particular, the volume of such a simplex is
If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the nparallelotope is the image of the unit nhypercube by the linear isomorphism that sends the canonical basis of to . As previously, this implies that the volume of a simplex coming from a npath is:
Conversely, given an nsimplex of , it can be supposed that the vectors form a basis of . Considering the parallelotope constructed from and , one sees that the previous formula is valid for every simplex.
Finally, the formula at the beginning of this section is obtained by observing that
From this formula, it follows immediately that the volume under a standard nsimplex (i.e. between the origin and the simplex in R^{n+1}) is
The volume of a regular nsimplex with unit side length is
as can be seen by multiplying the previous formula by x^{n+1}, to get the volume under the nsimplex as a function of its vertex distance x from the origin, differentiating with respect to x, at (where the nsimplex side length is 1), and normalizing by the length of the increment, , along the normal vector.
Dihedral angles of the regular nsimplex
Any two (n − 1)dimensional faces of a regular ndimensional simplex are themselves regular (n − 1)dimensional simplices, and they have the same dihedral angle of cos^{−1}(1/n).^{ [13]}^{ [14]}
This can be seen by noting that the center of the standard simplex is , and the centers of its faces are coordinate permutations of . Then, by symmetry, the vector pointing from to is perpendicular to the faces. So the vectors normal to the faces are permutations of , from which the dihedral angles are calculated.
Simplices with an "orthogonal corner"
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an ndimensional version of the Pythagorean theorem:
The sum of the squared (n − 1)dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)dimensional volume of the facet opposite of the orthogonal corner.
where are facets being pairwise orthogonal to each other but not orthogonal to , which is the facet opposite the orthogonal corner.
For a 2simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3simplex it is de Gua's theorem for a tetrahedron with an orthogonal corner.
Relation to the (n + 1)hypercube
The Hasse diagram of the face lattice of an nsimplex is isomorphic to the graph of the (n + 1) hypercube's edges, with the hypercube's vertices mapping to each of the nsimplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.
The nsimplex is also the vertex figure of the (n + 1)hypercube. It is also the facet of the (n + 1) orthoplex.
Topology
Topologically, an nsimplex is equivalent to an nball. Every nsimplex is an ndimensional manifold with corners.
Probability
In probability theory, the points of the standard nsimplex in (n + 1)space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.
Compounds
Since all simplices are selfdual, they can form a series of compounds;
 Two triangles form a hexagram {6/2}.
 Two tetrahedra form a compound of two tetrahedra or stella octangula.
 Two 5cells form a compound of two 5cells in four dimensions.
Algebraic topology
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.
A finite set of ksimplexes embedded in an open subset of R^{n} is called an affine kchain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.
Note that each facet of an nsimplex is an affine (n − 1)simplex, and thus the boundary of an nsimplex is an affine (n − 1)chain. Thus, if we denote one positively oriented affine simplex as
with the denoting the vertices, then the boundary of σ is the chain
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
Likewise, the boundary of the boundary of a chain is zero: .
More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map . In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
where the are the integers denoting orientation and multiplicity. For the boundary operator , one has:
where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map).
A continuous map to a topological space X is frequently referred to as a singular nsimplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)^{ [15]}
Algebraic geometry
Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic standard nsimplex is commonly defined as the subset of affine (n + 1)dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
which equals the schemetheoretic description with
the ring of regular functions on the algebraic nsimplex (for any ring ).
By using the same definitions as for the classical nsimplex, the nsimplices for different dimensions n assemble into one simplicial object, while the rings assemble into one cosimplicial object (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).
The algebraic nsimplices are used in higher Ktheory and in the definition of higher Chow groups.
Applications
This section needs expansion. You can help by
adding to it. (December 2009) 
 In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot.
 In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming.^{ [16]}
 In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.
 In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.^{ [17]}
 In chemistry, the hydrides of most elements in the pblock can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
See also
 Aitchison geometry
 Complete graph
 Causal dynamical triangulation
 Distance geometry
 Delaunay triangulation
 Hill tetrahedron
 Other regular n polytopes
 Hypersimplex
 Polytope
 Metcalfe's law
 List of regular polytopes
 Schläfli orthoscheme
 Simplex algorithm – a method for solving optimisation problems with inequalities.
 Simplicial complex
 Simplicial homology
 Simplicial set
 Ternary plot
 3sphere
Notes
 ^ Elte, E.L. (2006) [1912]. "IV. five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Simon & Schuster. ISBN 9781418179687.
 ^ Boyd & Vandenberghe 2004
 ^ Miller, Jeff, "Simplex", Earliest Known Uses of Some of the Words of Mathematics, retrieved 20180108
 ^ Coxeter 1973, pp. 120124, §7.2.
 ^ Coxeter 1973, p. 120.
 ^ Sloane, N. J. A. (ed.). "Sequence A135278 (Pascal's triangle with its lefthand edge removed)". The OnLine Encyclopedia of Integer Sequences. OEIS Foundation.
 ^ Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, SpringerVerlag (Series: Algorithms and Computation in Mathematics)
 ^ Yunmei Chen; Xiaojing Ye (2011). "Projection Onto A Simplex". arXiv: 1101.6081 [ math.OC].
 ^ MacUlan, N.; De Paula, G. G. (1989). "A lineartime medianfinding algorithm for projecting a vector on the simplex of n". Operations Research Letters. 8 (4): 219. doi: 10.1016/01676377(89)900643.
 ^ A derivation of a very similar formula can be found in Stein, P. (1966). "A Note on the Volume of a Simplex". American Mathematical Monthly. 73 (3): 299–301. doi: 10.2307/2315353. JSTOR 2315353.
 ^ Colins, Karen D. "CayleyMenger Determinant". MathWorld.
 ^ Every npath corresponding to a permutation is the image of the npath by the affine isometry that sends to , and whose linear part matches to for all i. hence every two npaths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the npath is the set of points , with and Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are nonoverlapping. The fact that the union of the simplexes is the whole unit nhypercube follows as well, replacing the strict inequalities above by "". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.
 ^ Parks, Harold R.; Wills, Dean C. (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular nSimplex". American Mathematical Monthly. 109 (8): 756–8. doi: 10.2307/3072403. JSTOR 3072403.
 ^ Wills, Harold R.; Parks, Dean C. (June 2009). Connections between combinatorics of permutations and algorithms and geometry (PhD). Oregon State University. hdl: 1957/11929.
 ^ Lee, John M. (2006). Introduction to Topological Manifolds. Springer. pp. 292–3. ISBN 9780387227276.
 ^ Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. ISBN 0471079162.
 ^ Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL9895: 1–32.
References
 Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGrawHill. ISBN 007054235X. (See chapter 10 for a simple review of topological properties.)
 Tanenbaum, Andrew S. (2003). "§2.5.3". Computer Networks (4th ed.). Prentice Hall. ISBN 0130661023.
 Devroye, Luc (1986). NonUniform Random Variate Generation. ISBN 0387963057. Archived from the original on 20090505.

Coxeter, H.S.M. (1973).
Regular Polytopes (3rd ed.). Dover.
ISBN
0486614808.
 pp. 120–121, §7.2. see illustration 72A
 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
 Weisstein, Eric W. "Simplex". MathWorld.
 Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 9781107394001. As PDF
External links
 Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.