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Plane Fundamental Domains with Minimal Perimeters

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Abstract

In his paper Á. G. Horváth posed two isoperimetric type questions for extremal polyhedra with respect to a given lattice L. He solved the problems in the case of the plane.

In this paper we continue the investigations and generalize the questions. The first one is: Which fundamental domains of space groups have minimal surface area for a given space group with fixed affine parameters? And the second one is: Which values of affine parameters serve the fundamental domain having the minimal surface area for a given space group class? In this sense the results of Á. G. Horváth correspond to the solutions for the plane group p1.

We shall give the solutions of these two problems for every plane groups using the concept of fundamental planigon and we calculate the IQ (isoperimetric quotient) for every "optimal" fundamental domain.

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Authors and Affiliations

  1. Dept. of Geometry, Budapest University of Technology and Economics (BME), H-1521, Budapest, Hungary

    Attila Bölcskei

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  1. Attila Bölcskei
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Bölcskei, A. Plane Fundamental Domains with Minimal Perimeters. Periodica Mathematica Hungarica 40, 147–165 (2000). https://doi.org/10.1023/A:1010387526327

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  • DOI: https://doi.org/10.1023/A:1010387526327

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