Simplex (math)

Hypertenrahedron is a redirect to this article. For the so-called 4-dimensional special case, see Pentachoron .

A 3 simplex or tetrahedron

When simplex or n -simplex , occasionally, n -dimensional hyper tetrahedron , is called in the geometry of a particular n -dimensional polytope .

A simplex is the simplest form of a polytope . Every -dimensional simplex has corners. A -Simplex is created from a -Simplex by adding an affine independent point (see below) and connecting all corners of the lower-dimensional simplex with this point in the form of a cone formation by stretching . Thus, with increasing dimensions, there is the series point , line , triangle , tetrahedron . A simplex is the continuation of this series on dimensions. ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle (n\in \mathbb {N} )}$ ${\displaystyle n}$

Definitions

Affine independence

Let and are finitely many points of a -vector space . These points are called affine independent if it holds for the scalars that it follows from with that . ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle v_{0},\ldots ,v_{k}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle t_{0},\ldots ,t_{k}\in \mathbb {R} }$ ${\displaystyle v_{0}t_{0}+\cdots +v_{k}t_{k}=0}$ ${\displaystyle t_{0}+\cdots +t_{k}=0}$ ${\displaystyle t_{0}=\cdots =t_{k}=0}$

In other words, there is no -dimensional affine subspace in which the points lie. An equivalent formulation is: The amount is linearly independent. In this case each of the points is affine independent of the other points and just as of the affine subspace spanned by them . ${\displaystyle (k-1)}$ ${\displaystyle V_{0}\subset V}$ ${\displaystyle k+1}$ ${\displaystyle \{v_{1}-v_{0},\ldots ,v_{k}-v_{0}\}}$ ${\displaystyle v_{j}(j=0,1,\ldots ,k)}$ ${\displaystyle v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{k}}$ ${\displaystyle v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{k}}$

A set of points of a -dimensional vector space over ( ) is called in general position if every subset consisting of at most points is affine independent . ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n+1}$

simplex

Be and are affine independent points of the (or a n -dimensional vector space over given), then the by spanned (or generated) simplex equal to the following quantity: ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle v_{0},\ldots ,v_{k}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle v_{0},\ldots ,v_{k}}$ ${\displaystyle \Delta }$

${\displaystyle \Delta =\left\{x\in \mathbb {R} ^{n}:x=\sum _{i=0}^{k}t_{i}v_{i}\ {\text{mit}}\ 0\leq t_{i}\leq 1\ {\text{und}}\ \sum _{i=0}^{k}t_{i}=1\right\}}$.

The points are called vertices of and barycentric coordinates . The number is the dimension of the simplex . A simplex of dimension is also briefly simplex called. A simplex is therefore nothing more than the convex hull of finitely many affine independent points im , which are then the corner points of this simplex. ${\displaystyle v_{i}}$ ${\displaystyle \Delta }$ ${\displaystyle (t_{0},...,t_{k})\in [0,1]^{k+1}}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \mathbb {R} ^{n}}$

Side faces and edge

It is a simplex. Each in given Simplex, which by a non-empty subset of the vertices of spanned called side (rare facet or sub Simplex ) of . The zero-dimensional sides (facets) are precisely the corner points or corners , the 1-sides (or 1-facets) are the edges and the -sides or facets are called side surfaces . The union of the side surfaces is called the edge of the simplex : ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle (k-1)}$ ${\displaystyle (k-1)}$ ${\displaystyle \partial \Delta }$ ${\displaystyle \Delta }$

${\displaystyle \partial \Delta =\left\{x\in \mathbb {R} ^{n}:x=\sum _{i=0}^{k}t_{i}v_{i}\ {\text{mit}}\ 0\leq t_{i}\leq 1\ {\text{und mindestens einem}}\ t_{i}=0,{\text{sowie}}\ \sum _{i=0}^{k}t_{i}=1\right\}}$

The number of sides (or facets) of the simplex is equal to the binomial coefficient . ${\displaystyle d}$ ${\displaystyle d}$ ${\displaystyle k}$ ${\displaystyle {\tbinom {k+1}{d+1}}}$

The -Simplex is the simplest -dimensional polytope , measured by the number of corners. The simplex method from linear optimization and the downhill simplex method from nonlinear optimization are named after the simplex . ${\displaystyle n}$ ${\displaystyle n}$

example

• A 0 simplex is a point .
• A 1-simplex is a line .
• A 2-simplex is a triangle .
• A 3-simplex is a tetrahedron (four corners, four sides of triangles, six edges); it is generated from a triangle (2-simplex), to which a point that is not in the plane of the triangle is added and connected to all corners of the triangle.
• A 4-simplex is also called a pentachoron .
• An example of a -Simplex im (namely one with a right-angled corner at the origin) is through ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \left\{x\in \mathbb {R} ^{n}\mid x_{i}\geq 0,\,\sum \limits _{i=1}^{n}x_{i}\leq 1\right\}}$

given. This simplex is called the unit simplex . It is spanned by the zero vector and the unit vectors of the standard basis of and has the volume with the length of the unit vectors . ${\displaystyle e_{1},\dotsc ,e_{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle c=1}$ ${\displaystyle c^{n}/n!\,}$

volume

The volume of the unit simplex is . If points are des , then reads the affine mapping which transforms the unit simplex to the simplex spanned by ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle 1/n!}$ ${\displaystyle v_{0},\ldots ,v_{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle v_{0},\ldots ,v_{n}}$

${\displaystyle x\mapsto v_{0}+Ax\quad {\text{mit }}A=\left({v_{1}-v_{0},\dots ,v_{n}-v_{0}}\right)\in \mathbb {R} ^{n\times n}}$

and the volume of the simplex is given by . ${\displaystyle \left|\det A\right|/n!}$

Standard simplex

In the algebraic topology , especially the definition of the singular homology , the so-called standard simplexes play an important role.

The -dimensional standard simplex is that of the unit vectors , i.e. of the corners ${\displaystyle n}$ ${\displaystyle \Delta ^{n}}$ ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle e_{1},\dots ,e_{n+1}}$

${\displaystyle v_{0}=(1,0,0,\ldots ,0),v_{1}=(0,1,0,\ldots ,0),v_{2}=(0,0,1,\ldots ,0),\ldots ,v_{n}=(0,0,0,\ldots ,1)}$

spanned simplex. The standard simplex thus corresponds to the largest side area of ​​a unit simplex. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle (n+1)}$

A singular simplex ${\displaystyle n}$ is by definition a continuous mapping of the standard simplex in a topological space , see singular homology . ${\displaystyle \Delta ^{n}}$ ${\displaystyle X}$

Simplexes with a right corner

A right-angled corner means that every 2 edges converging in this corner form a right angle . In other words, the -Simplex has a corner at which its -dimensional hypersurfaces adjoining it are orthogonal to one another. Such a simplex represents a generalization of right triangles and in it a -dimensional version of the Pythagorean theorem applies . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

The sum of the squared -dimensional volumes of the hypersurfaces adjacent to the right-angled corner is equal to the squared -dimensional volume of the hypersurface opposite the right-angled corner. The following applies: ${\displaystyle (n-1)}$ ${\displaystyle (n-1)}$

${\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}.}$

Here, the hypersurfaces are in pairs orthogonal to one another but not orthogonal to the hypersurface that is opposite the right-angled corner. ${\displaystyle A_{1}\ldots A_{n}}$ ${\displaystyle A_{0}}$

In the case of a 2-simplex, this corresponds to a right-angled triangle and the Pythagorean theorem, and in the case of a 3 -simplex, it corresponds to a tetrahedron with a cube corner and de Gua's theorem .

Basic homeomorphism properties

1. Two simplexes and the same dimension are always homeomorphic . Such a homeomorphy is present if the vertex sets of both simplexes have an identical number. ${\displaystyle \Delta \subset \mathbb {R} ^{n}}$ ${\displaystyle {\Delta }^{*}\subset \mathbb {R} ^{m}}$
2. A -Simplex im is always homeomorphic to the closed k-dimensional unit sphere . Hence every simplex of Euclidean space is a compact set . ${\displaystyle k}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\overline {B_{\mathbb {R} ^{k}}}}=\{v\in \mathbb {R} ^{k}:\|v\|_{2}\leq 1\}\subset \mathbb {R} ^{k}}$

Euclidean simplicial complex

A Euclidean simplicial complex (engl. Euclidean simplicial complex ), in German literature mostly simplicial complex called, is a family of simplices in with the following characteristics: ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \mathbb {R} ^{n}}$

1. With every simplex , every side of belongs to . ${\displaystyle \Delta \in {\mathcal {K}}}$ ${\displaystyle \Delta }$ ${\displaystyle {\mathcal {K}}}$
2. The intersection of two simplexes of is empty or common side of both simplexes. ${\displaystyle {\mathcal {K}}}$
3. Every point of a simplex has (with regard to the standard topology of ) an environment which cuts out at most a finite number of simplexes ( local finiteness ). ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {K}}}$

The union formed of all the simplices of and provided with the by deriving subspace topology , it means to corresponding polyhedra . The associated family is then also called a triangulation or simplicial decomposition of . If one exists, it is called triangulable . ${\displaystyle \textstyle \Sigma =\bigcup {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \Sigma }$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \Sigma }$  

A polyhedron that is triangulated by a finite simplicial complex is always a compact subset of the . ${\displaystyle \mathbb {R} ^{n}}$

Abstract simplicial complex

→ Main article : Simplicial complex

An abstract simplicial complex (. English abstract simplicial complex ) is a family of non-empty , finite sets, which (abstract) simplexes met are called, and the following property: ${\displaystyle {\mathcal {K}}}$

• With always, any non-empty subset of in included. ${\displaystyle \Delta \in {\mathcal {K}}}$ ${\displaystyle \Delta }$ ${\displaystyle {\mathcal {K}}}$

Every element of a simplex is called a corner and every non-empty subset is called a side (or facet). The dimension of an (abstract) simplex with corners is defined as . The dimension of a simplicial complex is defined as the maximum of the dimensions of all simplexes occurring in it, provided that this maximum exists. In this case, the simplicial complex is called finite-dimensional and said maximum is its dimension . If the dimensions of the simplexes of the simplicial complex grow beyond all limits, then the simplicial complex is called infinite-dimensional . ${\displaystyle k+1}$ ${\displaystyle k}$

application

One application is found in the downhill simplex method . This is an optimization method in which one wants to find parameter values ​​by varying them until the deviation between measured values ​​and a theory function that depends on these parameters is minimal. For this purpose, a simplex of parameter sets is set up in the -dimensional parameter space, the error function is calculated for each point of the simplex and then in the course of the algorithm the "worst" of these points is replaced by a (hopefully) "better" one (with a smaller error value) for as long until a convergence or other termination criterion is met. A simplex with a right-angled corner (as explained above) is usually used as the initial configuration . ${\displaystyle n}$ ${\displaystyle n}$

Simplex , simplicial complexes and polyhedra are also widely used in topology . One of the outstanding examples of application here is Brouwer's Fixed Point Theorem , by which Bronisław Knaster , Kazimierz Kuratowski and Stefan Mazurkiewicz showed in 1929 that this theorem and related theorems of topology within the framework of simplex theory with elementary combinatorial methods, especially when using them of Sperner's lemma , can be derived.

literature

items

• Bronisław Knaster , Casimir Kuratowski , Stefan Mazurkiewicz : A proof of the fixed point theorem for n-dimensional simplexes. In: Fundamenta Mathematicae . Volume 14, No. 1, 1929, pp. 132-137 ( online ).

Monographs

• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• John M. Lee: Introduction to Topological Manifolds (=  Graduate Texts in Mathematics, 202 ). 2nd Edition. Springer-Verlag, New York et al. 2011, ISBN 978-1-4419-7939-1 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 .

Web links

Wiktionary: Simplex  - explanations of meanings, word origins, synonyms, translations

Commons : Simplex  collection of images

Individual evidence

1. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 20th ff .
2. ^ ^{A b} E. Harzheim: Introduction to combinatorial topology . 1978, p. 4 .
3. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 5 .
4. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 26 .
5. ^ H. Schubert: Topology . 1975, p. 165 .
6. ↑ IM James: Handbook of Algebraic Topology . Elsevier Science, 1995, ISBN 978-0-08-053298-1 , pp. 3 .
7. ^ AK Austin, RJ Webster: 3147. A Note on Pythagoras' Theorem. In: The Mathematical Gazette , Volume 50, No. 372, 1966, p. 171, doi: 10.2307 / 3611958 ( JSTOR ).
8. ^ H. Schubert: Topology . 1975, p. 165 .
9. ^ H. Schubert: Topology . 1975, p. 166 .
10. ^ JM Lee: Introduction to Topological Manifolds . 2011, p. 149 .
11. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 34 .
12. ^ H. Schubert: Topology . 1975, p. 167 .
13. ^ Often, as for example in Harzheim, p. 34, or in Schubert, p. 167, it is even required that only a finite number of simplexes occur in the simplicial complex.
14. ^ H. Schubert: Topology . 1975, p. 167 .
15. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 26 .
16. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 37 .
17. ^ JM Lee: Introduction to Topological Manifolds . 2011, p. 153 .
18. ^ JM Lee: Introduction to Topological Manifolds . 2011, p. 153 ff .
19. ↑ Schubert, p. 169, speaks of a “simplcial scheme”. Schubert calls an abstract simplex an excellent set . He also demands that each element of the basic set be contained in a distinguished set, i.e. an abstract simplex.
20. ^ E. Harzheim: Introduction to combinatorial topology . 1978, p. 56-65, 317 .
21. ↑ B. Knaster, C. Kuratowski, S. Mazurkiewicz: A proof of the fixed point theorem for -dimensional simplexes ${\displaystyle n}$ . 1929, p. 132 ff .