About the partitions of a truncated icosahedron for parquet-hedra

E. S. Karpova ; A. V. Timofeenko

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 447-476.

Résumé

The study of parquet-facets began immediately after the classification of convex polyhedra with the regular faces half a century ago was completed. Parquet-Hedron will be called a convex polyhedron with regular or parquet faces.Recall, a parquet is a convex polygon made up of a finite and larger unit of the number of equiangular polygons. Parquet polygons are classified: there are 23 of their type. Four of them can be represented by regular polygons, and five more have equilateral representatives, composed as from regular polygons, that each vertex of such a regular polygon serves as the vertex of the parquet. About ten years ago, all parquet-facets became known to within the similarity, which apart from the right ones can also have the five parquet faces. A hypothesis has been put forward, which leads to the finding of all equilateral parquethedra. Without consideration of joints on the same faces, it is impossible to obtain all types of parquet facets, i.e. to close the main problem: "What are all types of parquethedra?" In this paper, we consider some of the connections required for the solution of this problem of a regular-angled pentagonal pyramid $M_3$ with single edges, truncated along the middle lines of the lateral triangular faces of the pyramid $M_{3a}$, bodies $M_{19a}$ and $M_{19b}$, obtained from the truncated icosahedron $ M_ {19} $ by cutting off $M_{3a}$ by two and three planes, respectively. The edges of the last three bodies and the edges of the junction have lengths one and two. At present, this result may be of independent interest for quasicrystallography. In particular, the Archimedean body $M_{19}$ with regular pentagons and two hexagons at each vertex is a representative of fullerenes. In addition, the amount of the calculations already done shows the need to attract programming and computer graphics for substantially larger scales, for which the work done will serve as a good test.

Détail

Keywords: parquet polygon, parquet-hedron, symmetry group, truncated icosahedron.

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