Върху 18-ия проблем на Хилберт
On Hilbert’s 18th problem

Summary/Abstract: The paper is dedicated to Hilbert’s 18th problem and its 3 parts in the contain. The first part considers the finite number of the essential groups of symmetries of a repeating model in 3-dimensional space. Crystals are included in the second part in connection with the existence of a solid, that completes the 3-dimensial space anisohedrally. The appearance of quasicrystals is justified apart from this. The part is related to plane tessellations. A review is proposed on convex polygon tessellations, namely on 15 types of pentagons. A concrete realization of a special pentagon is included together with its application to mosaic art. The third part considers the famous Kepler’s hypothesis and the problem for the densest packing of spheres.

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