Tetrahedral chains and a curious semigroup
doi:10.17398/2605-5686.34.1.99
Keywords:
tetrahedral chain, free product, semigroup, density, equidistribution, spherical harmonic, Cayley graphAbstract
In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. Świerczkowski gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group R isomorphic to the free product Z2 ∗ Z2 ∗ Z2 ∗ Z2 . We relate the reflections to elements of a semigroup of 3 × 3 matrices over the finite field Z3 , whose structure provides a simple and transparent new proof that R is a free product. We deduce the non-existence of a closed tetrahedral chain, prove that R is dense in the orthogonal group O(3), and show that every R-orbit on the 2-sphere is equidistributed.
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References
T.M. Apostol, Kronecker’s approximation theorem: the one-dimensional case, in “ Modular Functions and Dirichlet Series in Number Theory ”, Second edition, Graduate Texts in Mathematics, 41, Springer-Verlag, New York, 1990, 148 – 155.
V.I. Arnold, A.L. Krylov, Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain, Dokl. Akad. Nauk SSSR 148 (1963), 9 – 12 (Russian). English translation: Soviet Math. Dokl. 4 (1963), 1 – 5.
H. Babiker, S. Janeczko, Combinatorial representation of tetrahedrachains, Commun. Inf. Syst. 15 (2015), 331 – 359.
A.H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Research Rep. 7 (1952), 303 – 313.
A. Cayley, Desiderata and suggestions: No. 2. The theory of groups: graphical representation, Amer. J. Math. 1 (1878), 174- 176. Collected Mathematical Papers 10 (reprint), Scholarly Publishing Office, University of Michigan Library, Michigan, 2005, 403 – 405.
J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396 – 414.
H.S.M. Coxeter, “ Regular Complex Polytopes ”, Cambridge University Press, London-New York, 1974.
T.J. Dekker, On reflections in Euclidean spaces generating free products, Nieuw Arch. Wisk. (3) 7 (1959), 57 – 60.
M. Elgersma, S. Wagon, Closing a Platonic gap, Math. Intelligencer 37 (1) (2015), 54 – 61.
M. Elgersma, S. Wagon, The quadrahelix: a nearly perfect loop of tetrahedra, arXiv:1610.00280 [math.MG].
M. Elgersma, S. Wagon, An asymptotically closed loop of tetrahedra, Math. Intelligencer 39 (3) (2017), 40 – 45.
R.B. Fuller, “ Synergetics ”, Macmillan, New York, 1975.
A. Gorodnik, A. Nevo, On Arnold’s and Kazhdan’s equidistribution problems, Ergodic Theory Dynam. Systems 32 (2012), 1972 – 1990.
M. Golubitsky, I.N. Stewart, D.G. Schaeffer, “ Singularities and Groups in Bifurcation Theory ”, Vol. II, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988.
A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen (German), Ann. of Math. 34 (1933), 147 – 169.
O. Hanner, On the uniform convexity of Lp and `p , Ark. Mat. 3 (1956), 239 – 244.
L. Kronecker, Näherungsweise ganzzahlige Auflösung linearer Gleichungen, Berl. Ber. (1884), 1179 – 1193, 1271 – 1299.
W. Magnus, A. Karrass, D. Solitar, “ Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations ”, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966; (2nd ed., Dover, New York, 2005).
J.H. Mason, Can regular tetrahedra be glued together face to face to form a ring? Math. Gaz. 56 (1972), 194 – 197.
W. Rudin, “ Functional Analysis ”, McGraw-Hill, New York, 1973.
H. Steinhaus, Problème 175, Colloq. Math. 4 (1957), 243.
M.H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375 – 481.
M.H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167 – 184, 237 – 254.
S. Świerczkowski, On a free group of rotations of the Euclidean space, Indag. Math. 20 (1958), 376 – 378.
S. Świerczkowski, On chains of regular tetrahedra, Colloq. Math. 7 (1959), 9 – 10.
S. Świerczkowski, “ Looking Astern ”, unpublished memoir.
G. Tomkowicz, S. Wagon, “ The Banach–Tarski Paradox ” (2nd. ed.), Encyclopedia of Mathematics and its Applications 163, Cambridge University Press, New York, 2016.
S. Ulam, The Scottish Book (typed English translation sent by Stan Ulam to Professor Copson in Edinburgh on January 28, 1958), http://kielich.amu.edu.pl/Stefan Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf.
K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsber. Königlich Preussischen Akad. Wiss. Berlin 2 (1885), 633 –639, 789 – 805.
The Lviv Scottish Book, http://www.math.lviv.ua/szkocka/view.php.
