Journal of Information Processing Vol.22 No.4 1–6 (July 2014) [DOI: 10.2197/ipsjjip.22.1] Regular Paper Unfolding and Dissection of Multiple Cubes, Tetrahedra, and Doubly Covered Squares Zachary Abel1,a) Brad Ballinger2,b) Erik D. Demaine3,c) Martin L. Demaine3,d) Jeff Erickson4,e) Adam Hesterberg3,f) Hiro Ito5,g) Irina Kostitsyna6,h) Jayson Lynch3,i) Ryuhei Uehara7,j) Received: March 4, 2014, Accepted: August 1, 2014 Abstract: In this paper, we propose a new notion of rep-cube which is a net of a cube, and it can be divided into some polygons such that each of them can be folded to a cube. This notion is inspired by polyomino and rep-tile; both are introduced by Solomon W. Golomb, and well investigated in the recreational mathematics society. We prove that there are infintely many distinct rep-cubes. We also extend this notion to doubly covered squares and regular tetrahedra. Keywords: Dissection, folding and unfolding, polyomino, rep-cube, rep-tile. polyominoes such that each of them can be folded to a (smaller) 1. Introduction cube for some k? That is, a polyomino is a rep-cube of order k A polyomino is a “simply connected” set of unit squares in- if it is a net of a cube, and it can be divided into k polyominoes troduced by Solomon W. Golomb in 1954. Since then, a set of such that each of them can be folded to a cube. If each of these k polyominoes has been playing an important role in puzzle soci- polyominoes has the same size, we call the original polyomino a ety (see, e.g., [7], [9]). In Figure 82 in [7], it is shown that a set of regular rep-cube of order k. We note that we do not define crease 12 pentominoes exactly covers a cube that is the square root of 10 lines for these nets. That is, although each polyomino consists of units on the side. In this context, there are series of results about unit squares, we may fold along the line that is not orthogonal to the set of polyominoes that covers a cube in recreational math the unit squares when we fold to a cube. Simple examples can society; see [4], [5], [6], [10], [12], [13], [14], [15], [16], [17]. be found in Fig. 1. The first figurep indicatesp p two T-shapes that There is a comprehensive survey on a web page maintained by can fold into one cube of size 2 × 2 × 2. In this net, each Haubrich [11]. T-shape consists of six unit squares and it can fold to a unit cube. In 1962, Golomb also proposed an interesting notion called On the other hand, gluing these two T-shapesp togetherp asp shown “rep-tile”: a polygon is a rep-tile of order k if it can be divided in the figure, we can fold to a cube of size 2 × 2 × 2. The into k replicas congruent to one another and similar to the original dotted lines arep not ap part ofp the polyomino which is the net of the × × (see [8], Chap 19). cube of size 2 2 2, butp they arep just illustrated to help These notions lead us to the following natural question: is there to understand the squares of size 2 × 2. any polyomino that can be folded to a cube and divided into k From this viewpoint, we can find some affirmative examples in the previous results. In [14], [15], [16], we can observe that seven 1 Mathematics Department, MIT, USA. out of eleven developments of a cube have the following prop- 2 Department of Mathematics, Humboldt State University, USA. 3 Computer Science and Artificial Intelligence Laboratory, MIT, USA. erty: five copies of each can cover the surface of a cube without 4 Department of Computer Science, University of Illinois at Urbana- overlapping and holes. In our words, there are seven regular rep- Champaign, USA. cubes of order 5. In [13], Torbijn also investigated the same no- 5 Graduate School of Informatics and Engineering, University of Electro- Communications, Japan. tion, which was called cubic hexomino cubes, and showed some 6 Computer Science Department, Universite´ libre de Bruxelles, Belgium. examples for each k = 4; 5; 7; 9; 10*1. In this paper, we inves- 7 School of Information Science, Japan Advanced Institute of Science and tigate this notion and show more general results. We first give Technology, Japan. a) [email protected] some regular rep-cubes of order k for some specific k. Based on b) [email protected] this idea, we give a constructive proof for a series of regular rep- c) [email protected] cubes of order 36gk02 for any positive integer k0 and an integer g d) [email protected] e) jeff[email protected] in f2; 4; 5; 8; 9; 36; 50; 64g. That is, there are infinitely many k that f) [email protected] allow regular rep-cube of order k. We also give some non-regular g) [email protected] h) [email protected] i) [email protected] *1 In a preliminary version of this paper presented at JCDCGGG, we have j) [email protected] not yet found this article which dealt with the same notion. c 2014 Information Processing Society of Japan 1 Journal of Information Processing Vol.22 No.4 1–6 (July 2014) a b b a b a b a c c Fig. 1 Rep-cubes of order k = 2; 4; 5. b a b a c c Fig. 2 Rep-cubes of order k = 8; 9. rep-cubes and its variants. shown in Fig. 5. It is worth mentioning that this pattern contains Moreover, we also extend this notion to other dimensions and all eleven edge unfoldings of a cube, while the rep-cube of order solids, where each polygon is no longer a polyomino. First, we k = 9 in Fig. 2 consists of only one kind of them. = show the universal result for a doubly covered square.P That is, for Based on the solution for k 36 in Theorem 1, we obtain the ; ; ;:::; = any positive real numbers A a1 a2 ak such that i ai A, following theorem: there is a net of a doubly-covered square with area A that can Theorem 2 There exists a regular rep-cube of order 02 0 be cut into k polygons with areas a1; a2;:::; ak such that each of 36gk for any positive integer k and an integer g in them can be folded into a doubly-covered square. Next, we also f2; 4; 5; 8; 9; 36; 50; 64g. That is, there exists an infinite number of show this result can be extended to a regular tetrahedron. regular rep-cubes. Proof. We first choose one pattern in the proof of Theorem 1 ac- 2. Results on Rep-Cubes cording to the value of g. Next we split each unit square in the We first show some specific solutions. pattern into k02 small squares. Then we replace each small square Theorem 1 There exists a regular rep-cube of order k for by the pattern for k = 36 in Fig. 3. It is not difficult to see that the k = 2; 4; 5; 8; 9; 36; 50; 64. notches match with each other since they are arranged properly Proof. For each of k = 2; 4; 5; 8; 9, we give a regular rep-cube in in the pattern. Therefore, we obtain the theorem. Fig.s 1 and 2. It is not difficult to see that they satisfy the condi- One may think that non-regular rep-cubes are more difficult tion of rep-cubes. than regular ones. So far, we have found some: For k = 36, we use six copies of the pattern given in Fig. 3. Theorem 3 There exists a non-regular rep-cube of order k for Using this pattern, we can combine them into any one of eleven k = 2; 10. = nets of a cube. Proof. For k 2, the rep-cubep p is givenp in Fig. 6(left): this itself For k = 64, we use one copy of the left pattern in Fig. 4 for the folds to a cube of size 5 × 5 × 5, and it can be cut into bottom of a big cube, four copies of the center pattern in Fig. 4, two pieces such that one folds into a cube of size 2 × 2 × 2, and and one copy of the right pattern in Fig. 4 for the top of the big the otherp folds into a unit cube. We note that these areas satisfy cube. The consistency can be easily observed. 6 × ( 5)2 = 6 × 12 + 6 × 22 = 30. For k = 50, we make a program for finding packings of nets of For k = 10, the rep-cube is given in Fig. 6(right): this pat- unit cubes on twisted grids on bigger cubes by exhaustive search. tern contains 150 unit squares. It is easy to see that nice nets ; We foundp a packingp on ap (7 1) twist, i.e., a dissection of the sur- of unit cube use 54 unit squares in total. The remaining 96 face of a 50 × 50 × 50 cube into 50 nets of unit cubes as squares form a net of cube of size 4 × 4 × 4. Moreover, this c 2014 Information Processing Society of Japan 2 Journal of Information Processing Vol.22 No.4 1–6 (July 2014) Fig. 4 Patterns for rep-cubes of order k = 64. Fig. 3 Pattern for rep-cubes of order k = 36.