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121:, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 tetracubes are achiral and one is chiral, giving a count of 7 or 8 tetracubes respectively. Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, because one cannot turn a polycube over to reflect it as one can a polyomino given three dimensions. In particular, the
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98:
361:
333:) were first enumerated by W. F. Lunnon in 1972. Most polycubes are asymmetric, but many have more complex symmetry groups, all the way up to the full symmetry group of the cube with 48 elements. Numerous other symmetries are possible; for example, there are seven possible forms of 8-fold symmetry.
472:
If a polycube has the additional property that its complement (the set of integer cubes that do not belong to the polycube) is connected by paths of cubes meeting square-to-square, then the boundary squares of the polycube are necessarily also connected by paths of squares meeting edge-to-edge. That
351:
A polycube may have up to 24 orientations in the cubic lattice, or 48, if reflection is allowed. Of the pentacubes, 2 flats (5-1-1 and the cross) have mirror symmetry in all three axes; these have only three orientations. 10 have one mirror symmetry; these have 12 orientations. Each of the remaining
464:
Although the cubes of a polycube are required to be connected square-to-square, the squares of its boundary are not required to be connected edge-to-edge. For instance, the 26-cube formed by making a 3×3×3 grid of cubes and then removing the center cube is a valid polycube, in which the boundary of
751:
Robert
Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube
534:
The structure of a polycube can be visualized by means of a "dual graph" that has a vertex for each cube and an edge for each two cubes that share a square. This is different from the similarly-named notions of a
392:: it consists of four cubes stacked one on top of each other, with another four cubes attached to the exposed square faces of the second-from-top cube of the stack, to form a three-dimensional
526:
whether every polycube with a connected boundary can be unfolded to a polyomino, or whether this can always be done with the additional condition that the polyomino tiles the plane.
832:
803:
652:"Dirichlet convolution and enumeration of pyramid polycubes", C. Carré, N. Debroux, M. Deneufchâtel, J. Dubernard, C. Hillairet, J. Luque, O. Mallet; November 19, 2013
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As with polyominoes, polycubes may be classified according to how many symmetries they have. Polycube symmetries (conjugacy classes of subgroups of the achiral
310:
182:
161:
494:
966:. See in particular Lemma 3.9, p. 924, which states a generalization of this boundary connectivity property to higher-dimensional polycubes.
469:. For instance, one of the pentacubes has two cubes that meet edge-to-edge, so that the edge between them is the side of four boundary squares.
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Dual graphs have also been used to define and study special subclasses of the polycubes, such as the ones whose dual graph is a tree.
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All 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
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388:, the tesseract can be unfolded into an octacube. One unfolding, in particular, mimics the well-known unfolding of a cube into a
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the interior void is not connected to the exterior boundary. It is also not required that the boundary of a polycube form a
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Barequet, Ronnie; Barequet, Gill; Rote, Günter (2010), "Formulae and growth rates of high-dimensional polycubes",
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640:"Enumeration of Specific Classes of Polycubes", Jean-Marc Champarnaud et al, Université de Rouen, France
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Lunnon, W. F. (1972), "Symmetry of
Cubical and General Polyominoes", in Read, Ronald C. (ed.),
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to a polyomino? If so, can every such polycube be unfolded to a polyomino that tiles the plane?
1283:
879:
19th Japan
Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016)
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The bounding boxes of the pentacubes have sizes 5×1×1, 4×2×1, 3×3×1, 3×2×1, 3×2×2, and 2×2×2.
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in 1966), out of all 3811 different free octacubes, 261 are unfoldings of the tesseract.
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717:
Fowler, David (2010), "Mathematics in
Science Fiction: Mathematics as Science Fiction",
694:
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1066:, Lecture Notes in Comput. Sci., vol. 7033, Springer, Heidelberg, pp. 44–54,
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345:. 5 of the remaining 17 have mirror symmetry, and the other 12 form 6 chiral pairs.
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Fixed polycubes (both reflections and rotations counted as distinct (sequence
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923:; Scheideler, Christian (2006), "The effect of faults on network expansion",
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face to face. Polycubes are the three-dimensional analogues of the planar
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321:=16. More recently, specific families of polycubes have been investigated.
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Polycubes are classified according to how many cubical cells they have:
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Program (with Lua source code) to fill boxes with polycubes using
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36:
30:"Tetracube" redirects here. For the four-dimensional object, see
113:, polycubes can be enumerated in two ways, depending on whether
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A puzzle involving arranging nine L tricubes into a 3×3×3 cube
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177:
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1057:(2011), "Common unfoldings of polyominoes and polycubes",
919:
Bagchi, Amitabha; Bhargava, Ankur; Chaudhary, Amitabh;
414:". In honor of Dalí, this octacube has been called the
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Unlike in three dimensions in which distances between
313:)) and one-sided polycubes have been enumerated up to
57:
is a solid figure formed by joining one or more equal
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782:
679:
Kemp, Martin (1 January 1998), "Dali's dimensions",
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872:"Polycube unfoldings satisfying Conway's criterion"
448:states that the analogue in four dimensions yields
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892:Turney, Peter (1984), "Unfolding the tesseract",
440:of a polycube with unit edges excludes √7 due to
487:Can every polycube with a connected boundary be
1060:Computational geometry, graphs and applications
317:=22. Free polycubes have been enumerated up to
598:, New York: Academic Press, pp. 101–108,
522:to a polyomino that tiles the plane. It is an
425:More generally (answering a question posed by
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580:Weisstein, Eric W. "Polycube." From MathWorld
341:12 pentacubes are flat and correspond to the
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117:pairs of polycubes (those equivalent by
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495:(more unsolved problems in mathematics)
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473:is, in this case the boundary forms a
629:Kevin Gong's enumeration of polycubes
400:used this shape in his 1954 painting
27:Shape made from cubes joined together
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894:Journal of Recreational Mathematics
352:17 pentacubes has 24 orientations.
25:
356:Octacube and hypercube unfoldings
152:(reflections counted as distinct)
1349:
1340:
827:{\displaystyle \mathbb {R} ^{2}}
798:{\displaystyle \mathbb {R} ^{3}}
511:as well as the Dalí cross (with
1105:Wooden hexacube puzzle by Kadon
776:Hypercube unfoldings that tile
482:Unsolved problem in mathematics
442:Legendre's three-square theorem
403:Crucifixion (Corpus Hypercubus)
446:Lagrange's four-square theorem
380:, and just as the cube can be
173:(reflections counted together)
1:
543:of a surface-embedded graph.
618:Polycubes, at The Poly Pages
412:And He Built a Crooked House
1072:10.1007/978-3-642-24983-9_5
925:Theory of Computing Systems
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79:Slothouber–Graatsma puzzle
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1000:10.1007/s00493-010-2448-8
947:10.1007/s00224-006-1349-0
376:) has eight cubes as its
337:Properties of pentacubes
1033:; Collette, Sébastien;
406:and it is described in
325:Symmetries of polycubes
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719:World Literature Today
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460:Boundary connectivity
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410:'s 1940 short story "
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93:Enumerating polycubes
48:
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146:Number of one-sided
89:based on polycubes.
1110:Polycube Symmetries
851:2015arXiv151202086D
695:1998Natur.391...27K
1039:Demaine, Martin L.
1031:Bose, Prosenjit K.
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408:Robert A. Heinlein
372:(four-dimensional
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1380:Discrete geometry
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1221:Higher dimensions
1051:Langerman, Stefan
1041:; Douïeb, Karim;
868:Langerman, Stefan
669:. From MathWorld.
605:978-1-48325-512-5
556:Herzberger Quader
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119:mirror reflection
16:(Redirected from
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1024:
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752:(tesseract).
750:
725:(3): 48–52,
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524:open problem
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450:square roots
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394:double cross
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54:
52:
1346:WikiProject
1254:Polydrafter
1228:Polyominoid
1164:Polyominoes
1119:Algorithm X
900:(1): 1–16,
667:"Pentacube"
504:-cube with
475:polyominoid
390:Latin cross
343:pentominoes
111:polyominoes
71:Bedlam cube
63:polyominoes
1369:Categories
1306:Snake cube
1264:Polyiamond
1055:Morin, Pat
938:cs/0404029
842:1512.02086
689:(27): 27,
567:References
541:dual graph
530:Dual graph
420:tile space
416:Dalí cross
277:heptacube
249:pentacube
235:tetracube
175:(sequence
171:-polycubes
154:(sequence
150:-polycubes
143:-polycube
81:, and the
18:Dali cross
1375:Polyforms
1301:Soma cube
1274:Polystick
1249:Polyabolo
1197:Heptomino
1187:Pentomino
1182:Tetromino
1156:Polyforms
986:CiteSeerX
747:115769478
518:) can be
452:of every
418:. It can
374:hypercube
370:tesseract
291:octacube
263:hexacube
193:monocube
123:Soma cube
105:pentacube
67:Soma cube
32:tesseract
1316:Hexastix
1233:Polycube
1212:Decomino
1207:Nonomino
1202:Octomino
1192:Hexomino
1016:18571788
774:(2015),
739:27871086
550:See also
520:unfolded
489:unfolded
467:manifold
438:vertices
386:hexomino
382:unfolded
221:tricube
139:Name of
55:polycube
1322:Tantrix
1311:Tangram
1288:puzzles
1259:Polyhex
1177:Tromino
1088:2927309
1008:2728490
963:9332443
955:2279081
906:0765344
847:Bibcode
691:Bibcode
396:shape.
384:into a
309:in the
306:A001931
207:dicube
181:in the
178:A038119
160:in the
157:A000162
65:. The
1356:Portal
1329:Tetris
1296:Blokus
1242:Others
1172:Domino
1086:
1014:
1006:
988:
961:
953:
904:
745:
737:
682:Nature
602:
509:< 7
500:Every
378:facets
115:chiral
103:chiral
77:, the
73:, the
69:, the
1284:Games
1064:(PDF)
1012:S2CID
959:S2CID
933:arXiv
875:(PDF)
837:arXiv
743:S2CID
735:JSTOR
297:3811
294:6922
280:1023
109:Like
59:cubes
1286:and
805:and
600:ISBN
368:The
311:OEIS
283:607
269:112
266:166
183:OEIS
162:OEIS
1076:hdl
1068:doi
996:doi
943:doi
727:doi
699:doi
687:391
654:PDF
642:PDF
516:= 8
255:23
252:29
1371::
1084:MR
1082:,
1074:,
1053:;
1049:;
1045:;
1037:;
1010:,
1004:MR
1002:,
994:,
982:30
980:,
957:,
951:MR
949:,
941:,
929:39
927:,
902:MR
898:17
896:,
877:,
859:^
845:,
835:,
760:^
749:,
741:,
733:,
723:84
721:,
697:,
685:,
586:^
477:.
444:,
422:.
288:8
274:7
260:6
246:5
241:7
238:8
232:4
227:2
224:2
218:3
213:1
210:1
204:2
199:1
196:1
190:1
185:)
164:)
101:A
53:A
1148:e
1141:t
1134:v
1121:.
1091:.
1078::
1070::
1019:.
998::
945::
935::
909:.
882:.
854:.
849::
839::
820:2
815:R
791:3
786:R
755:.
729::
701::
693::
514:k
507:k
502:k
484::
319:n
315:n
169:n
148:n
141:n
135:n
34:.
20:)
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