Rep-Tiles: Replicating Figures ------------------------------ Golomb defines Rep-tiles: A geometric figure such that a finite number of identical copies of it can be assembled to form an enlarged scale model of it. Scherer defines Irrep-tiles: A geometric figure such that a finite number of similar copies of it can be assembled to form an enlarged scale model of it. Examples: reptiles of order 4 1 1 1 2 3 2 2 4 3 3 4 4 1 1 1 2 1 2 3 3 3 4 2 2 3 4 4 4 a irreptile of order 6 1 1 2 2 1 1 1 1 2 x x 3 x x 3 x x 3 3 --> x x 3 3 4 4 x x x x 5 x x x x 5 4 x x x x 5 5 x x x x 5 5 This gives a sequences of irreptiles. a irreptile of order 10 1 2 2 2 2 1 1 2 2 2 1 1 1 2 2 1 1 1 1 2 x x x x 3 x x x x 3 3 4 4 4 4 x x x x 3 3 3 4 4 4 x x x x 3 3 3 3 4 4 x x x x x x x x 5 4 x x x x x x x x 5 5 x x x x x x x x 5 5 5 6 6 6 6 x x x x x x x x 5 5 5 5 6 6 6 x x x x x x x x x x x x 7 6 6 x x x x x x x x x x x x 7 7 6 x x x x x x x x x x x x 7 7 7 x x x x x x x x x x x x 7 7 7 7 8 8 8 8 x x x x x x x x x x x x x x x x 9 8 8 8 x x x x x x x x x x x x x x x x 9 9 8 8 x x x x x x x x x x x x x x x x 9 9 9 8 x x x x x x x x x x x x x x x x 9 9 9 9 All these examples can be sheared. In [Scherer] you can find hundrests of reptiles and irreptiles. Some reptiles and irreptiles which pack rectangles. I tried to find some of L-shape. type 0: x x x x x x x make a rectangle: x x y y y y y x x x x x y y part1: Karl Scherer 1.1 x x x x x x x x x -> 4*9 make a type 0 L-piece x x x x x y x x y y y y x x y y y y make a rectangle: x x x y y y y y y x x x y y y y y y x x x x x x y y y x x x x x x y y y part2: Karl Scherer 1. Karl Scherer 4.4 F and 5.5 A irreptile x x x x x x make a type 0 L-piece x x y x x y x x y x x y y y z z z z x x x x x x x x z x x x x x x x x z these two types of L-pieces gives a 11 piece square: 1 2 2 2 x x 1 1 1 2 x x 3 3 3 2 x x 3 4 4 2 x x 3 4 x x x x 3 4 x x x x part3: Torsten Sillke (1997 Sep) irreptile x x x x x x x x x x x x x x x x x x gives not a L-piece of type 0: x x x x y y x x x x y y x x x x y y x x x x y y x x x x y y y y y z z z z z x x x x y y y y y z z z z z x x x x x x x x x x x x z z x x x x x x x x x x x x z z x x x x x x x x x x x x z z x x x x x x x x x x x x z z gives a rectangle: x x x x y y w w w w w w w w w w w w x x x x y y w w w w w w w w w w w w x x x x y y w w w w w w w w w w w w x x x x y y w w w w w w w w w w w w x x x x y y y y y z z z z z w w w w x x x x y y y y y z z z z z w w w w x x x x x x x x x x x x z z w w w w x x x x x x x x x x x x z z w w w w x x x x x x x x x x x x z z w w w w x x x x x x x x x x x x z z w w w w special L-partitions: Klarner 1969 x x x x x x x -> 14*14 Rex Marshall 1990 x x x x x x x x x x x -> order 50 Michael Reid 1995 x x x x x x x x x x x -> 30*154 References: - Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions Simon & Schuster (1968) Chapter 19: Rep-Tiles: Replicating Figures on the Plane - Solomon W. Golomb, Replicating Figures in the Plane, Mathematical Gazette 48 no. 366 (Dec. 1964) 403-412 - Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals privatly distributed Address: 11 Utting Str., Birkdale, Auckland, New Zealand Telephon: 0064-9-483-4211 (private) mailto:karl@kiwi.gen.nz http://www.kiwi.gen.nz/~karl/ -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/