Common Unfoldings of Polyominoes and Polycubes Greg Aloupis123 Prosenjit K

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cOMMON uNFOLDINGS oF pOLYOMINOES aND pOLYCUBES Greg Aloupis123 Prosenjit K. Bose2 Sebastien Collette3 Erik D. Demaine45 Martin Demaine4 Karim Douïeb2 Vida Dujmovic´2 John Iacono56 Stefan Langerman3 Pat Morin2

Abstract Common planar unfoldings of various classes For boxes, i.e., convex polycubes, Biedl et al. [1] of polyominoes and polycubes are studied. A common found two polyomino unfoldings that folds into two dis- unfolding of all tree-like tetracubes was exhaustively tinct boxes. Mitani and Uehara [3] have shown that there discovered by Knuth and Miller; we show here that no are an infinite number of such polyomino unfoldings. common unfolding exists for all tree-like pentacubes. The problem of finding a polyomino unfolding that folds 1 Introduction into more than two distinct boxes is still open. Related to this topic is the problem of finding an Polyominoes A polyomino or k-omino is an orthogo- edge unfolding of some classes of orthogonal polyhedra; nal polygon made from k squares of unit size arranged see [4] for a survey. with coincident sides. The coincident sides of two ad- jacent squares either overlap or are joined together. A 2 Unfolding polyominoes polyomino is tree-like if its dual graph is a tree (each The perimeter of a k-omino forms a closed chain of node a square, arcs corresponding to joined sides) and 2k + 2 unit length segments. An unfolding of a k-omino is incision-free if it does not have overlapping sides; is defined as follows: open the chain, cut the k-omino overlapping corners are allowed. into smaller polyominoid pieces partially attached to the chain and straighten the chain. This definition is motivated by the problem of unfolding tree-like planar polycubes (see §3.2). A valid unfolding does not overlap and every piece lies on the same side of the chain. A triv- ial unfolding exists for all polyominoes: open the chain, keep the polyomino identical and leave it connected to an arbitrary segment of the chain. We ask: Is there a common unfolding of all tree-like k-ominoes? Figure 1. Three octakaidecominoes: non-tree-like (left), tree-like (middle), and incision-free tree-like (right). Polycubes The 3D analogues of polyominoes are poly- Figure 2. A possible unfolding of a tree-like 6-omino. cubes which are orthogonal polyhedra made from unit 2.1 Path-like polyominoes A polyomino is path-like cubes arranged with coincident faces which either over- if its dual graph is a path. Let c , c , . . . , c be { 1 2 2k+2} lap or are joined together. A polycube is tree-like if its the ordered sequence of segments composing the un- dual graph is a tree and is incision-free if it does not folded chain C of a k-omino where c1 is the left most have overlapping faces; corner overlap is allowed. A segment. A caterpillar unfolding of a k-omino is a chain planar polycube has its dual graph lying on one plane, where one unit size square is attached to each segment i.e., it is only one cube thick. A chiral polycube has a c2i+ 2i/k+1 for 1 < i k (see Fig. 3). Note that b c ≤ non-superimposable-under-rigid-motion mirror image. the caterpillar unfolding cannot fold into all path-like Chiral twins, chiral polycubes that are the mirror image polyominoes but we show that it can fold into any non- of each other, are considered to be distinct. spiraling path-like polyomino. The difference between 1.1 Related work Cubigami7 [2] is a commercialized the number of left and right turns in a non-spiraling path- puzzle developed by Donald Knuth and George Miller like polyomino does not exceed 3 if k is even and does which solves this problem for k 4. Miller posed the not exceed 4 otherwise. ≤ question, and Knuth exhaustively tried all unfoldings of We have proven the following (proofs omitted): all tree-like tetracubes. Nothing is known for k > 5. Theorem 1 For any positive value of k, the caterpillar unfolding is a common unfolding of all one-sided non-spiraling path-like k-ominoes. Corollary 2 For any odd positive integer k, there Cubigami7: exists an unfolding that folds to at least C(k 1)/2 one- − sided k-ominoes (or at least C(k 1)/2/2 free k-ominoes), − 1Academica Sinica 2Carleton University 3Université Libre de Bruxelles 4MIT 5MADALGO, Aaurhus University 6Poly:NYU

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There is an unique free polyomino (see below) un- where Cn is the nth Catalan number. Planar 7 ofThere 492 is an unique two-sided polyomino unfolding 1 •• The notion of a caterpillar unfolding and the results foldingof all tree-like of all tree-like incision-free incision-free non-planar non-planar pentacubes pentacubes. This unfolding can fold to a total of 22 1 of2 Theorem3 14 and Corollary5 6 2 can be extended to path- (Fig 4 #12-27). This unfolding, shown below, can 2 like planar polycubes. pentacubes.fold to a total of 22 pentacubes. 1 2 3 4 5 6 3 ...... 7 8 9 10 11 1 2 3 Figure4 5 3. Caterpillar6 unfolding of k-ominoes for k even 4 7 8 (top)9 or10 odd (bottom).11 1 2 3 4 We5 have6 proven... the following (proofs omitted):... There are 492 common one-sided polyomino un- 5 12 13 14 15 16 17 • 7 8 9 10Theorem11 1 For any positive value of k, the caterpillar foldingsThere are of492 all tree-like common incision-free one-sided polyomino planar pen- un- 1 2 3 4 5 6 •tacubes. None of them can fold to a non-planar 12 13 unfolding14 15 is a16 common17 unfolding of all one-sided non- foldings of all tree-like incision-free planar pen- 6 1 Figure2 3.3 Caterpillar4 5 unfolding6 of k-ominoes for k even 7 8 9 10 spiraling11 path-like k-ominoes. pentacube.tacubes (Fig One 4 of #1-11). them is:None of them can fold to a 1 2 3 4 5 186 (top)19 or20 odd (bottom).21For any22 odd23 positive integer k, there exists 12 13 14 Corollary15 16 1 17 non-planar pentacube. One of them is: 1 7 2 3 4 5 6 7 8 9 10 11 an unfolding that folds to at least C one-sided k- 18 19 2.220 Tree-like21 22 polyminoes23 The following(k 1)/2 lemma has 7 8 9 10 11 − 12 13 14 15 ominoes16 17 (or at least C /2 free k-ominoes), where 8 been verified experimentally:(k 1)/2 1 2 3 254 5 6 − 7 8 9 10 11 24 C is the26 nth27 Catalan number. 18 19 20 21 nProposition22 23 3 For 0 < k 6, there exists a com- 7 8 9 10 11 ≤ 12 13 14 15 16 17 The notion of a caterpillar unfolding and the results of 1 2 243 9 254 mon265 unfolding276 of all tree-like incision-free k-ominoes. 12 13 14 15 16 17 18 19 20 121 ThereTheorem222 is323 no 1 common and4 Corollary5 unfolding 16 can ofbe all extended tree-like to path-likeincision- 7 8 9 10 11 12 13 14 15 16 17 There is no common free polyomino unfolding of 10 planar polycubes. 1 2 243 254 265 free276 heptominoes. • 12 13 14 15 16 17 18 19 20 121 222 2.2323 Tree-like4 5 polyminoes6 The following lemma has all path-like incision-free pentacubes. 7 8 9 10 11 The question remains open for k 8. There is no common two-sided polyomino unfold- 18 19 20 21 22 23 ≥ •The smallest subset of pentacubes that have no com- 24 25 26 11 727 been8 verified9 10 experimentally:11 ing of all path-like incision-free pentacubes. 12 13 14 15 16 17 • 18 19 20 121 222 323 3Lemma4 Unfolding5 1 For6 polycubes0 8. One open problem is is there some such that for all 12 13 overlapping14 15 faces.16 We17 study the problem of finding com- 18 19 20 121 222 323 4 5 6 k l,have there no are common always two polyomino incision-free unfolding.k-polyominoes 7 8 9 14 10 mon11 polyomino unfoldings of all tree-like k-cubes. A 18 193 Unfolding20 21 polycubes22 23 ≥ 1 2 3 244 255 266 27 without a common unfolding? 12 13 14 common15 16 unfolding17 of several polyominoes is one-sided 18 19 20 121 222 A323 polyomino4 5 unfolding6 of a polycube is a cutting of a It remains open if there is there some l such that 7 8 9 10 15 11 3.2 Tree-like planar polycubes 18 19 if20 all polyominoes21 22 can23 be folded with the same side of k l k 24 25 26 727 collection8 9 of10 edges11 on the surface of a polycube so that for all , there are always two incision-free - 12 13 14 15 16 17 ≥ 24 the25 unfolding26 on27 the exterior. A mutual unfolding which 0 ≤6. 12 13 14 15 16 17 common one-sided polyomino unfoldings of all tree-like 24 25 26 27 1 2 3 4 18 5 investigate6 the problem of finding common polyomino Nothing is known about 12 131 common142 153 unfolding164 17 of5 all incision-free6 quadracubes. We 3.3 Non-planar polycubes 18 19 20 21 22 23 incision-free planar k-cubes. 7 8 9 10 unfoldingsinvestigate11 of the all problem incision-free of finding pentacubes common (Fig. polyomino 4). The finding a common polyomino unfolding of all non-planar 18 19 20 21 22 23 24 25 26 27 The question is open for k > 6. 1 2 3 4 5 19 6 12 131 142 resultsunfoldings153 described164 of175 all in incision-free this6 section pentacubes are experimental. (Fig. 4). Using The k-cubes for k>5. It is open if all tree-like incision-free 18 19 20 21 22 23 3.3 Non-planar polycubes Nothing is known about 7 8 9 10 11 aresults ULB supercomputer, described in this 1,099,511,627,776 section are experimental. possible We un- polycubes have at least one polyomino unfolding. 18 197 820 921 1022 1123 24 25 26 27 20 finding a common polyomino unfolding of all non-planar 12 131 142 153 foldingsexhaustively164 175 of the6 generated 5-tube (Fig. 1,099,511,627,776 4, #1) were generated, possible and un- 24 25 26 27 References 7 8 9 10 11 k-cubes for k > 5. It is open if all tree-like incision-free 18 197 820 forfoldings921 each1022 generated of the1123 5-tube, unfolding and for it was each determined generated unfolding which of 24 25 26 27 21 [1]polycubesT. Biedl, have T. Chan, at least E. one Demaine, polyomino M. Demaine, unfolding. A. Lu- 12 131 142 153 164 theit175 27was incision-free6 determined pentacubes which of the could 27 be incision-free folded to. This pen- 24 1225 1326 1427 15 16 17 biw, J. I. Munro, and J. Shallit. Notes from the Univ. 18 19 20 21 tacubes22 23 could be folded to. Here are the results: 7 8 9 yielded10 11 the following discoveries: References 22 of Waterloo algorithmic problem session. Sept. 8 12 13 14 15 16 17 24 1225 1326 1427 There15 is16 no common17 two-sidedfree polyomino polyomino unfolding unfold- of [1] 1999.T. Biedl, T. Chan, E. Demaine, M. Demaine, A. Lu- 18 197 820 921 1022 1123 • 23 18 19 ingall20 tree-like of all21 tree-like incision-free22 incision-free23 pentacubes pentacubes (even (evenwhen [2] G.biw, Miller.J. I. Munro, and Cubigami. J. Shallit. Noteshttp://www. from the Univ. 24 1225 1326 1427 15 16 17 puzzlepalace.com/resources/t9of Waterloo algorithmic problem session.. Sept. 8 18 19 20 21 22 23 whenchiral chiral twins aretwins considered are considered identical). identical). 18 24 19 20 There21 are22 several23 two-sidedfree sided polyomino unfoldings [3] J.1999. Mitani and R. Uehara. Polygons Folding to Plural 24 1225 1326 1427 15 16 • 17 24 25 that26 can can27 fold fold to to 23 23 pentacubes pentacubes (one (one is pictured is pictured be- [2] IncongruentG. Miller. Orthogonal Cubigami. Boxes. In http://www.CCCG, pages 18 19 25 20 21 low).22 None23 can fold to more than 23. 31–34,puzzlepalace.com/resources/t9 2008. . 24 25 26 27 below). None fold to 24. 24 25 26 27 [4][3] J.J. O’Rourke.Mitani and R.Unfolding Uehara. orthogonal Polygons Folding polyhedra. to PluralIn 18 19 20 26 21 22 23 J.Incongruent E. Goodman, Orthogonal J. Pach, Boxes. and R. Pollack, In CCCG editors,, pages 24 25 26 27 Surveys31–34, 2008. on Discrete and Computational Geometry 27 [4] —J. TwentyO’Rourke. Years Unfolding later, Contemporary orthogonal Mathematics polyhedra. In, 24 25 26 27 Figure 4 453:307–318,J. E. Goodman, 2008. J. Pach, and R. Pollack, editors, All 27 There are 27 one-sided polyomino unfoldings that Surveys on Discrete and Computational Geometry • incision- can fold to 22 pentacubes. None can fold to more — Twenty Years later, Contemporary Mathematics, free than 22. 453:307–318, 2008. pentacubes 2