Regular Polygon Surfaces

Abstract

A regular polygon surface M is a surface graph (Σ, Γ) together with a continuous map ψ from Σ into Euclidean 3-space which maps faces to regular Euclidean polygons. When Σ is homeomorphic to the sphere and the degree of every face of Γ is five, we prove that M can be realized as the boundary of a union of dodecahedra glued together along common facets. Under the same assumptions but when the faces of Γ have degree four or eight, we prove that M can be realized as the boundary of a union of cubes and octagonal prisms glued together along common facets. We exhibit counterexamples showing the failure of both theorems for higher genus surfaces. Contents 1. Introduction 1 2. (5)-RPSs 5 3. (5, 7, 8, 9, 10)-RPSs 9 4. (4, 8)-RPSs 12 5. Examples of higher genus RPSs 20 Acknowledgments 20 References 24

Domes over curves

“…Similarly, recall that the regular octagon Q 8 and decagon Q 10 are spanned by the surfaces of Johnson solids square cupola and pentagonal cupola, respectively, see Figure 3 (right) and [16] for details. 1 In fact, both are cuts of the Archimedean solids, see e.g. [9, p. 88].…”

“…It is best understood in the context of regular polygonal surfaces (see e.g. [1,8]). While we study only the weaker notion (realizations), both the immersed and the embedded surfaces can be considered, as they add further constraints to the domes.…”

“…There are several related rigidity results for surfaces with square and regular pentagonal faces, see e.g. [1,10]. 6.4.…”

“…In Section 3, we follow with a (much shorter) constructive proof of Theorem 1.4, which is almost completely independent of the previous section, except for the earlier analysis of rhombi which can be domed, see §2. 1.…”