The Truncated Icosahedron as an Inflatable Ball

  • Tibor Tarnai Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary
  • András Lengyel Department of Structural Mechanics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary

Abstract

In the late 1930s, an inflatable truncated icosahedral beach-ball was made such that its hexagonal faces were coloured with five different colours. This ball was an unnoticed invention. It appeared more than twenty years earlier than the first truncated icosahedral soccer ball. In connection with the colouring of this beach-ball, the present paper investigates the following problem: How many colourings of the dodecahedron with five colours exist such that all vertices of each face are coloured differently? The paper shows that four ways of colouring exist and refers to other colouring problems, pointing out a defect in the colouring of the original beach-ball.

Keywords: polyhedron, truncated icosahedron, compound of five tetrahedra, coloring of polyhedra, permutation, inflatable ball
Published online
2018-10-29
How to Cite
Tarnai, T. and Lengyel, A. (2018) “The Truncated Icosahedron as an Inflatable Ball”, Periodica Polytechnica Architecture, 49(2), pp. 99-108. doi: https://doi.org/10.3311/PPar.12375.
Section
Articles