Balancing polyhedra

Domokos, Gábor [Domokos, Gábor (Nemlineáris mecha...), szerző] Szilárdságtani és Tartószerkezeti Tanszék (BME / ÉPK); Kovács, Flórián [Kovács, Flórián (Kinematika), szerző] Tartószerkezetek Mechanikája Tanszék (BME / ÉMK); MTA-BME Szilárd testek morfodinamikája Kutatócs... (BME / ÉPK / SZTT); Lángi, Zsolt ✉ [Lángi, Zsolt (Geometria), szerző] Geometria Tanszék (BME / TTK / MI); Regős, Krisztina [Regős, Krisztina (Geometria), szerző]; Varga, Tamás Péter

Angol nyelvű Szakcikk (Folyóiratcikk) Tudományos
Megjelent: ARS MATHEMATICA CONTEMPORANEA 1855-3966 1855-3974 19 (1) pp. 95-124 2020
  • SJR Scopus - Algebra and Number Theory: Q2
Azonosítók
We define the mechanical complexity C(P) of a 3-dimensional convex polyhedron P, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria; and the mechanical complexity C(S, U) of primary equilibrium classes (S, U)E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U)E with S, U > 1 is the minimum of 2(f + v − S − U) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class (S, U)E is zero if and only if there exists a convex polyhedron with S faces and U vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes (1, U)E and (S, 1)E, and offer a complexity-dependent prize for the complexity of the Gömböc-class (1, 1)E. Dedicated to the memory of John Horton Conway.
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2025-02-12 17:57