Eutactic star

File:Eutactic star.png

A star is here defined as a set of 2s vectors A = ±a₁, ..., ±a_s issuing from a particular origin in a Euclidean space of dimension n ≤ s. A star is eutactic if the a_i are the projections onto n dimensions of a set of mutually perpendicular equal vectors b₁, ..., b_s issuing from a particular origin in Euclidean s-dimensional space.{{cite book|title=Regular polytopes|url=https://archive.org/details/regularpolytopes0000coxe|url-access=registration|last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter |publisher=Courier Dover Publications|year=1973|isbn=0-486-61480-8|page=[https://archive.org/details/regularpolytopes0000coxe/page/251 251]}} The configuration of 2s vectors in the s-dimensional space B = ±b₁, ... , ±b_s is known as a cross. Given these definitions, a eutactic star is, concisely, a star produced by the orthogonal projection of a cross.

An equivalent definition, first mentioned by Schläfli,{{Cite book | last1=Schläfli | first1=Ludwig | title=Collected mathematical works | publisher=Birkhäuser Verlag | language=German | year=1949 | volume=I | chapter=Theorie der vielfachen Kontinuität|zbl=0035.21902 }}

stipulates that a star is eutactic if a constant ζ exists such that

:$\backslash sum\_\{k=1\}^s(\backslash mathbf\{a\}\_k\; \backslash cdot\; \backslash mathbf\{v\})^2\; =\; \backslash zeta\; |\backslash mathbf\{v\}|^2$

for every vector v. The existence of such a constant requires that the sum of the squares of the orthogonal projections of A on a line be equal in all directions.{{Cite journal | doi=10.4153/CJM-1951-045-8 | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Extreme forms | mr=0044580 | year=1951 | journal=Canadian Journal of Mathematics| issn=0008-414X | volume=3 | pages=391–441 | s2cid=247197697 | doi-access=free }} In general,

:$\backslash zeta\; =\; \backslash sum\_\{k=1\}^s\; |\backslash mathbf\{a\}\_k|^2/n.$

A normalised eutactic star is a projected cross composed of unit vectors.{{cite web|url=http://mathworld.wolfram.com/EutacticStar.html|title=Eutactic Star – MathWorld|author=E. W. Weisstein|accessdate=2009-08-28}} Eutactic stars are often considered in n = 3 dimensions because of their connection with the study of regular polyhedra.

Let T be the symmetric linear transformation defined for vectors x by

:$T\backslash mathbf\{x\}\; =\; \backslash sum\_\{k=1\}^s\; \backslash mathbf\{a\}\_k\; (\backslash mathbf\{a\}\_k\backslash cdot\backslash mathbf\{x\}),$

where the a_j form any collection of s vectors in the n-dimensional Euclidean space. Hadwiger's principal theorem states that the vectors ±a₁, ..., ±a_s form a eutactic star if and only if there is a constant ζ such that Tx = ζx for every x.{{cite web|url=http://mathworld.wolfram.com/HadwigersPrincipalTheorem.html|title=Hadwiger's Principal Theorem – MathWorld|author=E. W. Weisstein|accessdate=2009-08-28}} The vectors form a normalized eutactic star precisely when T is the identity operator – when ζ = 1.

Equivalently, the star is normalized eutactic if and only if the matrix A = [a₁ ... a_s], whose columns are the vectors a_k, has orthonormal rows. A proof may be given in one direction by completing the rows of this matrix to an orthonormal basis of $\backslash mathbb\{R\}^s$, and in the other by orthogonally projecting onto the n-dimensional subspace spanned by the first n Cartesian coordinate vectors.

Hadwiger's theorem implies the equivalence of Schläfli's stipulation and the geometrical definition of a eutactic star, by the polarization identity. Furthermore, both Schläfli's identity and Hadwiger's theorem give the same value of the constant ζ.

Eutactic stars are useful largely because of their relationship with the geometry of polytopes and groups of orthogonal transformations. Schläfli showed early on that the vectors from the center of any regular polytope to its vertices form a eutactic star. Brauer and Coxeter proved the following generalization:{{Cite journal | last1=Brauer | first1=R. | author1-link=Richard Brauer | last2=Coxeter | first2=Harold Scott MacDonald | author2-link=Harold Scott MacDonald Coxeter | title=A generalization of theorems of Schönhardt and Mehmke on polytopes | mr=0002869 | year=1940 | journal=Trans. Roy. Soc. Canada. Sect. III. (3) | volume=34 | pages=29–34 }}.

:

A star is eutactic if it is transformed to itself by some irreducible group of orthogonal transformations that acts transitively on pairs of opposite vectors.

An irreducible group here means a group that does not leave any nontrivial proper subspace invariant (see irreducible representation). Since the set theoretic union of two eutactic stars is itself eutactic (a consequence of Hadwiger's principal theorem), it can be concluded that, in general:

:

A star is eutactic if it is transformed to itself by some irreducible group of orthogonal transformations.

Eutactic stars may be used to validate the eutaxy of any form in general. According to H. S. M. Coxeter: "A form is eutactic if and only if its minimal vectors are parallel to the vectors of a eutactic star."

• Eutactic lattice
• Tight frame{{Cite web |url=http://www.biblioteca.juriquilla.unam.mx/WONDERLAND/ParaJaviero/cfata/import/GomezA2007.pdf |title=Archived copy |access-date=2014-01-11 |archive-url=https://web.archive.org/web/20140112032659/http://www.biblioteca.juriquilla.unam.mx/WONDERLAND/ParaJaviero/cfata/import/GomezA2007.pdf |archive-date=2014-01-12 |url-status=dead }}

{{reflist}}

• {{Citation | last1=Schläfli | first1=Ludwig | editor1-last=Graf | editor1-first=J. H. | title=Theorie der vielfachen Kontinuität | orig-year=1852 | url=https://books.google.com/books?id=foIUAQAAMAAJ | publisher=Zürich, Basel: Georg & Co. | language=German | series=Republished by Cornell University Library historical math monographs 2010 | isbn=978-1-4297-0481-6 | year=1901}}

Category:Polytopes

Category:Vectors (mathematics and physics)