Eutactic star
In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors (or arms) issuing from a central origin. A star is eutactic if it is the orthogonal projection of a crosspolytope from a higherdimensional space onto a subspace. Such stars were called "eutactic" – meaning "wellsituated" or "wellarranged" – by Schläfli (1901, p. 134) because, for a common scalar multiple, their vectors are projections of an orthonormal basis.^{[1]}
Definition
A star is here defined as a set of 2s vectors A = ±a_{1}, ..., ±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 b_{1}, ..., b_{s} issuing from a particular origin in Euclidean sdimensional space.^{[2]} The configuration of 2s vectors in the sdimensional space B = ±b_{1}, ... , ±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,^{[3]} stipulates that a star is eutactic if a constant ζ exists such that
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.^{[4]} In general,
A normalised eutactic star is a projected cross composed of unit vectors.^{[2]}^{[5]} Eutactic stars are often considered in n = 3 dimensions because of their connection with the study of regular polyhedra.
Hadwiger's principal theorem
Let T be the symmetric linear transformation defined for vectors x by
where the a_{j} form any collection of s vectors in the ndimensional Euclidean space. Hadwiger's principal theorem states that the vectors ±a_{1}, ..., ±a_{s} form a eutactic star if and only if there is a constant ζ such that Tx = ζx for every x.^{[2]}^{[6]} 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 = [a_{1} ... 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 , and in the other by orthogonally projecting onto the ndimensional 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.
Applications
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:^{[7]}

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:^{[4]}

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."^{[4]}
See also
 Eutactic lattice
 Tight frame^{[8]}
References
 ↑ D. M. Cvetković; P. Rowlinson; S. Simić (1997). Eigenspaces of graphs. Cambridge University Press. p. 151. ISBN 0521573521.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{2.0} ^{2.1} ^{2.2} Coxeter, Harold Scott MacDonald (1973). Regular polytopes. Courier Dover Publications. p. 251. ISBN 0486614808.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Schläfli, Ludwig (1949). "Theorie der vielfachen Kontinuität". Collected mathematical works (in German). I. Birkhäuser Verlag. Zbl 0035.21902.CS1 maint: unrecognized language (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{4.0} ^{4.1} ^{4.2} Coxeter, Harold Scott MacDonald (1951). "Extreme forms". Canadian Journal of Mathematics. 3: 391–441. doi:10.4153/CJM19510458. ISSN 0008414X. MR 0044580.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ E. W. Weisstein. "Eutactic Star – MathWorld". Retrieved 20090828.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ E. W. Weisstein. "Hadwiger's Principal Theorem – MathWorld". Retrieved 20090828.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Brauer, R.; Coxeter, Harold Scott MacDonald (1940). "A generalization of theorems of Schönhardt and Mehmke on polytopes". Trans. Roy. Soc. Canada. Sect. III. (3). 34: 29–34. MR 0002869.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ http://www.biblioteca.juriquilla.unam.mx/WONDERLAND/ParaJaviero/cfata/import/GomezA2007.pdf
 Schläfli, Ludwig (1901) [1852], Graf, J. H. (ed.), Theorie der vielfachen Kontinuität, Republished by Cornell University Library historical math monographs 2010 (in German), Zürich, Basel: Georg & Co., ISBN 9781429704816CS1 maint: unrecognized language (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>