(Circle Limit IV)
All M.C. Escher works (c) 2003 Cordon Art - Holland. All rights reserved. Used by permission.
| Heaven
and Hell (Circle Limit IV) |
All M.C. Escher works (c) 2003 Cordon Art - Holland. All rights reserved. Used by permission. |
| I. R. Aitchison and J. H. Rubinstein, Heaven & Hell, Proceedings of the Sixth International Colloquium on Differential Geometry (Santiago de Compostela, 1988), 5--24, Cursos Congr. Univ. Santiago de Compostela, 61, Univ. Santiago de Compostela, Santiago de Compostela, 1989. |
This paper is, at least partly, an attempt to relate some mathematics to the world of art and even to some mystical vision of reality. The Heaven and Hell of the title is the famous work of M. C. Escher, known also as Circle Limit IV. The authors associate with it a tessellation of the hyperbolic plane by hexagons with colored edges, and consider the group G3 of symmetries of this tessellation. They define a generic (abstract) polyhedron to be a 3-ball with a 3-valent graph embedded in its boundary. Regions of the boundary complementary to this graph are called faces. Then the authors prove that to every decomposition of a 3-manifold into generic polyhedra is associated a unique conjugacy class of torsion-free, orientation-preserving subgroups of G3. Several related topics are also discussed, including pseudo-Anosov maps, decompositions of 3-manifolds into Platonic solids, singular geometry of 3-manifolds, and others. For the "mystic" reference, see the "Final remark" section (numbered 13!).
(Review by N. V. Ivanov, AMS Mathematical Reviews 91e:57024) Download the paper as a PDF file
| [1]
H. M. S. Coxeter, The trigonometry of
Escher's woodcut "Circle Limit III", Mathematical
Intelligencer 18 (1996), 42-46. [2] H. M. S. Coxeter, The non-Eucliedan symmetry of Escher's picture "Circle Limit III", Leonardo 12 (1979), 19-25. |
Review of [2]: Of all Escher's pictures with a mathematical background, the most sophisticated is his 1959 woodcut, Circle Limit III, which uses four colours in addition to black and white. Queues of fishes of each colour are swimming along white arcs that cut the peripheral circle at a certain angle. After discussing the kind of symmetry that is involved and the underlying regular tessellations (so cleverly disguised), the author explains why the above-mentioned angle is not 90 but 80 degrees.
(AMS Mathematical Reviews 81j:51017) For a complete copy of the Coxeter paper [1] and further information click here
For further interesting information on hyperbolic geometry and Escher's hyperbolic patterns see this page.