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 plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace. Such stars were called 'eutactic' – meaning 'well-situated' or 'well-arranged' – by Schläfli (1901, p. 134) because, for a common scalar multiple, their vectors are projections of an orthonormal basis.A star is here defined as a set of 2s vectors A = ±a1, ..., ±as issuing from a particular origin in a Euclidean space of dimension n ≤ s. A star is eutactic if the ai are the projections onto n dimensions of a set of mutually perpendicular equal vectors b1, ..., bs issuing from a particular origin in Euclidean s-dimensional space. The configuration of 2s vectors in the s-dimensional space B = ±b1, ... , ±bs is known as a cross. Given these definitions, a eutactic star is, concisely, a star produced by the orthogonal projection of a cross.Let T be the symmetric linear transformation defined for vectors x byEutactic 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: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.A star is eutactic if it is transformed to itself by some irreducible group of orthogonal transformations.