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Golden Bee


GoldenBee

The golden bee is a concave hexagon with five right angles and one reflex angle that can be dissected into two smaller similar copies of unequal size, as illustrated above. The name golden bee is used by Akiyama (2012) and Durand et al. (2020). The name Ammann chair is used in the Tilings Encyclopedia (Frettlöh et al. n.d.). The shape was found by Robert Ammann and independently rediscovered by Karl Scherer (Akiyama 2012). The word "bee" refers to the shape's resemblance to the lowercase letter "b" (Akiyama 2012).

The two copies have linear scale factors phi^(-1/2) and phi^(-1) relative to the original, where phi is the golden ratio. Their area fractions are therefore phi^(-1) and phi^(-2), whose sum is 1 (Durand et al. 2020). This makes the golden bee an irreptile, using a definition that allows unequal-size pieces. The displayed two-piece dissection does not satisfy the equal-size requirement in the definition of a rep-tile.

The golden bee occurs in two sizes in Ammann A2 substitution tilings (Durand et al. 2020).


See also

Golden Ratio, Hexagon, Irreptile, Rep-Tile, Substitution Tiling

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References

Akiyama, S. "A Note on Aperiodic Ammann Tiles." Disc. Comput. Geom. 48, 702-710, 2012. https://doi.org/10.1007/s00454-012-9418-4.Durand, B.; Shen, A.; and Vereshchagin, N. "On the Structure of Ammann A2 Tilings." Disc. Comput. Geom. 63, 577-606, 2020. https://doi.org/10.1007/s00454-019-00074-1.Frettlöh, D.; Gähler, F.; and Harriss, E. "Ammann Chair." Tilings Encyclopedia. n.d. https://tilings.math.uni-bielefeld.de/substitution/ammann-chair/.Grünbaum, B. and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1987.

Cite this as:

Weisstein, Eric W. "Golden Bee." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoldenBee.html

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