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Title:  Repcubes: Unfolding and Dissection of Cubes 
Authors:  Xu, Dawei Horiyama, Takashi Uehara, Ryuhei 
Keywords:  repcube paper folding pythagorean triple 
Issue Date:  20170726 
Publisher:  CCCG 
Magazine name:  Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG 2017) 
Start page:  62 
End page:  67 
Abstract:  Last year, a new notion of repcube was proposed. A repcube is a polyomino that is a net of a cube, and it can be divided into some polyominoes such that each of them can be folded to a cube. This notion was inspired by the notions of polyomino and reptile, which were introduced by Solomon W. Golomb. It was proved that there are infinitely many distinct repcubes. In this paper, we investigate this new notion and obtain three new results. First, we prove that there does not exist a regular repcube of order 3, which solves an open question proposed in the paper. Next, we enumerate all regular repcubes of order 2 and 4. For example, there are 33 repcubes of order 2; that is, there are 33 dodecominoes that can fold to a cube of size √<2> x √<2> x √<2> and each of them can be divided into two nets of unit cube. Similarly, there are 7185 repcubes of order 4. Lastly, we focus on pythagorean triples that consist of three positive integers (a, b,c) with a^2+b^2=c^2. For each of these triples、 we can consider a repcube problem that asks whether a net of a cube of size c x c x c can be divided into two nets of two cubes of size a x a x a and b x b x b. We give a partial answer to this natural open question by dividing into more than two pieces. For any given pythagorean triple (a, b, c), we construct five polyominoes that form a net of a cube of size c x c x c and two nets of two cubes of size a x a x a and b x b x b. 
Rights:  Copyright (C) 2017 Authors. Dawei Xu, Takashi Horiyama, and Ryuhei Uehara, Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG 2017), 2017, 6267. 
URI:  http://hdl.handle.net/10119/15126 
Material Type:  publisher 
Appears in Collections:  b111. 会議発表論文・発表資料 (Conference Papers)

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