Voderberg tiling

A partial Voderberg tiling. Note that all of the colored tiles are congruent.

The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg [de] (1911-1945).[1] Karl August Reinhardt asked the question of whether there is a tile such that two copies can completely enclose a third copy. Voderberg, his student, answered in the affirmative with Form eines Neunecks eine Lösung zu einem Problem von Reinhardt ["On a nonagon as a solution to a problem of Reinhardt"].[2][3]

It is a monohedral tiling: it consists only of one shape that tessellates the plane with congruent copies of itself. In this case, the prototile is an elongated irregular nonagon, or nine-sided figure. The most interesting feature of this polygon is the fact that two copies of it can fully enclose a third one. E.g., the lowest purple nonagon is enclosed by two yellow ones, all three of identical shape.[4] Before Voderberg's discovery, mathematicians had questioned whether this could be possible.

Because it has no translational symmetries, the Voderberg tiling is technically non-periodic, even though it exhibits an obvious repeating pattern. This tiling was the first spiral tiling to be devised,[5] preceding later work by Branko Grünbaum and Geoffrey C. Shephard in the 1970s.[1] A spiral tiling is depicted on the cover of Grünbaum and Shephard's 1987 book Tilings and patterns.[6]

Wikimedia Commons has media related to Voderberg spiral tiling.
  1. ^ a b Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 372. ISBN 9781402757969. Retrieved 24 March 2015.
  2. ^ Adams, Jadie; Lopez, Gabriel; Mann, Casey; Tran, Nhi (2020-03-14). "Your Friendly Neighborhood Voderberg Tile". Mathematics Magazine. 93 (2): 83–90. doi:10.1080/0025570X.2020.1708685. ISSN 0025-570X.
  3. ^ "Karl Reinhardt - Biography". Maths History. Retrieved 2023-07-09.
  4. ^ Voderberg, Heinz (1936). "Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente". Jahresbericht der Deutschen Mathematiker-Vereinigung. 46: 229–231.
  5. ^ Dutch, Steven (29 July 1999). "Some Special Radial and Spiral Tilings". University of Wisconsin, Green Bay. Archived from the original on 5 March 2016. Retrieved 24 March 2015.
  6. ^ Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, Section 9.5, "Spiral Tilings," p. 512, ISBN 0-7167-1193-1.