Voderberg Tiles with Unique Properties
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1936: Heinz Voderberg from University of Greifswald, Germany, published a remarkable article.
He described the construction of tiles with weak (2, 1)- and (2, 2)-enclosure property.
Definition:
A tile has the weak (2, r)-enclosure property, if there exists a tiling patch, where two copies TA and TB of the tile surround
r other copies T1, T2, T3, ..., and Tr in a way, that ...
(1) no tiles overlap and
(2) the patch has no holes and
(3) each tile Ti has at most two points in common with the boundary of the patch for i = 1, 2, 3, ..., r.
See English translation Voderberg-English of Voderberg's article,
or a German copy Voderberg-Abschrift.
1955: Michael Grünbaum derived tiles with weak (2, r)-enclosure property for all positive integers r from Voderberg's nonagon.
2002: Casey Mann derived tiles with strong (2, r)-enclosure property from Voderberg's and Grünbaum's tiles.
Definition:
A tile has the strong (2, r)-enclosure property, if it has the weak (2, r)-enclosure property
and no inner tile Ti has a point in common with the boundary of the patch for i = 1, 2, 3, ..., r.
Opposite to Voderberg's and Grünbaum's tiles, the tiles of Casey Mann do not admit a tiling of the plane.
The author of this website could find tiles with strong (2, 2)-enclosure property in 2020 and with strong (2, 1)-enclosure property in 2024,
which additionally can tile the plane. These tiles are called Voderberg-Dragons or V-Dragons (this name was suggested by several people).
Walter Trump, Nürnberg, Germany, ms(at)trump.de, © 2025-02-06 (last modified: 2025-03-24)