Voderberg tiling

Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg [de] (1911–1945).^[1] Karl August Reinhardt posed the question, "Is there 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]

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

Voderberg tiling is monohedral, consisting of a single 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. For example, 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, Voderberg tiling is technically non-periodic, even though it exhibits an obvious repeating pattern and its prototile can also be used to create a periodic tiling.

Vorderberg 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]