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 f 1 2 2k+2g 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 cater- pillar 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 1 488058892672 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 pen- tacubes. 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 <k 6, there exists a common un- mon polyomino unfolding has a size of at most 4. 7 8 9 10 11 181346714828 Non-planar 1 of 2 The smallest subset of pentacubes that have no Copyright 2010 John Iacono. All rights reserved. Do not distribute. 18 19 20 21 22 23 ≤ 24 25 26 72712 8 Afolding9 polyomino10 of all11 unfolding tree-like incision-free of a polycubek-ominoes. is a cutting There of is a •For example pentacubes 1, 9, 27 and any of 6, 7, 12, 12 13 14 15 16 17 common polyomino unfolding has a size of at 1 2 3 244 no255 common266 unfolding27 of all tree-like incision-free hep- 13, 24 or 26 have no common polyomino unfolding. 12 collection13 14 of edges15 on16 the17 surface of a polycube so that 18 19 20 21 22 23 most 4. For example pentacubes (numbered in 24 25 26 727 8 13 9 the10tominoes. surface11 may be flattened into a polyomino without 12 13 14 15 16 17 Fig. 4) 1, 9, 27 and any of 6, 7, 12, 13, 24 or 26 1 2 3 4 5 6 l 24 25 26 The27 question remains open for k>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.