Some applications of the formula for volume of octahedron
by
Idjad Sabitov
Moscow State University
Coauthors: Galiulin R.V., Mikhalyov S.N.

In [1] a way for finding of the canonical polynomial for the volume (CPV) of any octahedron is suggested. The degree of the CPV is equal 16, but in the general case the number of its monoms is so big (more than 10¹⁰) that in fact it cannot be computed even if one use very powerfull computers. On the other hand, in practice there is a necessity to find this polinomial in the cases when some of edges are equal between them. For instance, the formulae for the volume of octahedra with some symmetry could be usefull in cristallography .

In this work we give the complete classification of the symmetries of octahedra and compute CPV for different cases of symmetry.

We will distinguish several kinds of symmetry.

1) Combinatorial symmetry. The group G of automorphisms of the set of vertices keeping the incidence of edges and vertices can be obviously considered as a subgroup of the group of permutations S₆. It turns out that the order of G equals 48. Thus in the set of vertices of a octahedron there exist 48 automorphisms. We call any automorphism from G as "combinatorial symmetry". The group G, in its turn, has a considerable number of subgroups.

2) Metric symmetry. Now let's suppose that for a octahedron we khow also the lengths of its edges. Any automorphism g+AFw-in G induces a mapping of edges too. We say that the octahedron is metrically symmetric with respect to a subgroup G₁+AFw-in G, if any automorphism g+AFw-in G₁ maps any edge of the octahedron to an equal edge.

3) Spatial symmetry. Let's now consider octahedra in R³. In the case of an octahedron having a centre of inversion, (inverse) axis of symmetry, or (inverse) plane of symmetry, we say that the octahedron has the corresponding spatial symmetry. Evidently, any spatial transformation of symmetry induces an automorphism g+AFw-in G. It's also obvious that any spatially symmetric octahedron is metrically symmetric. In general, the inverse statement is not true: it's possible that a realization in R³ of a metrically symmetric octahedron has no elements of symmetry (such as a centre, axis or plane of symmetry).

In this work we, however, consider only different classes of metrically symmetric octahedra without paying attention to the question whether they can be realized in R³ as spatially symmetric octahedra or not. For different subgroups of G we point out which edges of octahedron must have the same lenghts, and for the corresponding octahedra we compute CPV (using method, suggested in [1]). Of course, it is impossible to present here these polynomials for all the cases: for different groups of symmetry there are polynomials of different complexity having from several monoms to several thousand ones.

The work of S.N.Mikhalyov and Idjad Sabitov is supported by INTAS, grant No. IR-97-1778, the work of R.V. Galiulin is supported by RFBR, grant No. 99-01-00867.

Reference: [1] Astrelin A.V., Sabitov I.Kh. A minimal-degree polynomial for determining the volume of an octahedron from its metric, Uspekhi Mat. Nauk, 50(4) (1995), 245-246.

Date received: February 21, 2000