2017/03/07 by Emil Molnár, Molnár, Emil, István Prok +3
Chemistry · Engineering · #52C17 #57M07 #57M60 #FOS: Mathematics #Fullerene Chemistry and Applications #Metric Geometry (math.MG) #Structural Analysis and Optimization #Systems Engineering Methodologies and Applications
paper · pdf · doi:10.48550/arxiv.1703.02264
openalex publication_date 2017/03/07 · openalex created_date 2023/03/01 · openalex updated_date 2026/07/28
This presentation starts with the regular polygons, of course, then with the\nPlatonic and Archimedean solids. The latter ones are whose symmetry groups are\ntransitive on the vertices, and in addition, whose faces are regular polygons\n(see only I. Prok's home page [11] for them). Then there come these symmetry\ngroups themselves (starting with the cube and octahedron, of course, then\nicosahedron and dodecahedron). Then come the space filling properties: Namely\nthe cube is a space filler for the Euclidean space E3. Then we jump for the\nother regular solids that cannot fil E3, but can hyperbolic space H3, a new\nspace. These can be understood better if we start regular polygons, of course,\nthat cannot fil E2 in general, but can fil the new plane H2, as hyperbolic or\nBolyai-Lobachevsky plane. Now it raises the problem, whether a football\npolyhedron - with its congruent copies - fil a space. It turns out that E3 is\nexcluded (it remains an open problem - for you, of course, in other aspects),\nbut H3 can be filled as a schematic construction show this (Fig. 5), far from\nelementary. Then we mention some stories on Buckminster Fuller, an architect,\nwho imagined first time fullerens as such crystal structures. Many problems\nremain open, of course, we are just in the middle of living science.\n