Faces Of Polyhedra Research Articles

A Thirteenth-Century Dodecahedron in Central Anatolia: Geometric Patterns and Polyhedral Geometry

The lace-like geometric patterns of medieval Seljuk art cover mostly planar surfaces, extending to the borders in a uniform manner. Some are applied on engaged columns, interiors of concave surfaces, and exteriors of half-spheres. There are also rare examples on faces of polyhedra. The construction of either the tessellation or a polyhedron is already demanding, and combining the two poses an additional mathematical challenge. One such case is that of the tomb of Gomec Hatun in Konya, where a dodecahedral column capital was, as evidence suggests, covered with a mosaic tiling. A visual geometric investigation of the capital and its historical precedents sheds light on how the artisan’s knowledge of polyhedral and polygonal geometries may have been essential in the design and its materialization.

Assembled molecular face-rotating polyhedra to transfer chirality from two to three dimensions.

In nature, protein subunits on the capsids of many icosahedral viruses form rotational patterns, and mathematicians also incorporate asymmetric patterns into faces of polyhedra. Chemists have constructed molecular polyhedra with vacant or highly symmetric faces, but very little is known about constructing polyhedra with asymmetric faces. Here we report a strategy to embellish a C3h truxene unit with rotational patterns into the faces of an octahedron, forming chiral octahedra that exhibit the largest molar ellipticity ever reported, to the best of our knowledge. The directionalities of the facial rotations can be controlled by vertices to achieve identical rotational directionality on each face, resembling the homo-directionality of virus capsids. Investigations of the kinetics and mechanism reveal that non-covalent interaction among the faces is essential to the facial homo-directionality.

Stanovení odchylky stěn těles pravidelných v jeden vrchol se sbíhajících od roviny, která prochází koncovými body hran v témže vrcholu se protínajících

Stanovení odchylky stěn těles pravidelných v jeden vrchol se sbíhajících od roviny, která prochází koncovými body hran v témže vrcholu se protínajících