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256:. In addition to seven ordinary setisets (i.e., loops of length 1) they found a bewildering variety of loops of every length up to a maximum of 14. The total number of loops identified was nearly one and a half million. More research in this area remains to be done, but it seems safe to suppose that other shapes may also entail loops.
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Alternatively, there exists a method whereby multiple copies of a rep-tile can be dissected in certain ways so as to yield shapes that create setisets. Figures 7 and 8 show setisets produced by this means, in which each piece is the union of 2 and 3 rep-tiles, respectively. In Figure 8 can be seen
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can be dissected or combined so as to yield smaller or larger duplicates of themselves. Clearly, the twin actions of forming still larger and larger copies (known as inflation), or still smaller and smaller dissections (deflation), can be repeated indefinitely. In this way, setisets can produce
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how the 9 pieces above together tile the 3 rep-tile shapes below, while each of the 9 pieces is itself formed by the union of 3 such rep-tile shapes. Hence each shape can be tiled with smaller duplicates of the entire set of 9.
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Besides self-tiling tile sets, which can be interpreted as loops of length 1, there exist longer loops, or closed chains of sets, in which every set tiles its successor. Figure 6 shows a pair of mutually tiling sets of
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different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for
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The fewest pieces in a setiset is two. Figure 4 encapsulates an infinite family of order 2 setisets each composed of two triangles,
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regions. Disjoint pieces composed of two or more separated islands are also permitted. Such pieces are described as
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To date, two methods have been used for producing setisets. In the case of sets composed of shapes such as
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shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of
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non-periodic tilings. However, none of the non-periodic tilings thus far discovered qualify as
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A setiset of order 4 using octominoes. Two stages of inflation are shown.
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72:= 4 was asked decades previously by C. Dudley Langford, and examples for
84:(discovered by Maurice J. Povah) were previously published by Gardner.
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identical pieces is the same thing as a 'self-replicating tile' or
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From the above definition it follows that a setiset composed of
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Sallows, Lee (April 2014). "More On Self-Tiling Tile Sets".
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Sallows, Lee (December 2012). "On Self-Tiling Tile Sets".
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The properties of setisets mean that their pieces form
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128:because two of the component shapes are the same.
68:in 2012, but the problem of finding such sets for
24:A 'perfect' self-tiling tile set of order 4
116:distinct shapes, such as Figure 1, are called
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131:The shapes employed in a setiset need not be
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371:, by Martin Gardner, Knopf, 1977, pp 146-159
354:Alejandro Erickson on Self-tiling tile sets
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177:A setiset showing weakly-connected pieces.
781:Dividing a square into similar rectangles
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189:An infinite family of order 2 setisets.
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307:A rep-tile-based setiset of order 9.
295:A rep-tile-based setiset of order 4.
243:A loop of length 2 using decominoes.
80:, Wade E. Philpott and others) and
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416:by Jean-Paul Delahaye in Scilogs
120:. Figure 2 shows an example for
100:A setiset with duplicated piece.
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806:Regular Division of the Plane
443:Self-Tiling Tile Sets website
428:Self-Tiling Tile Sets website
60:= 4 using distinctly shaped
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612:Aperiodic set of prototiles
52:shapes can be assembled in
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365:Polyhexes and Polyaboloes
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88:Examples and definitions
369:Mathematical Magic Show
260:Methods of construction
194:Inflation and deflation
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414:Geometric Hidden Gems
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48:shapes. That is, the
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30:self-tiling tile set
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437:External links
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394:(2): 100–112.
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124:= 4 which is
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1550:Tessellation
817:Substitution
812:Regular grid
804:
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651:Quaquaversal
549:Kisrhombille
479:Tessellation
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137:disconnected
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41:
40:is a set of
37:
33:
29:
27:
847:vertex type
705:Anisohedral
660:Self-tiling
503:Pythagorean
266:polyominoes
202:Figure 5:
185:Figure 4:
173:Figure 3:
82:polyominoes
74:polyaboloes
66:Lee Sallows
36:, of order
1544:Categories
751:Pentagonal
312:References
303:Figure 8:
291:Figure 7:
254:octominoes
250:decominoes
239:Figure 6:
220:prototiles
96:Figure 2:
62:decominoes
20:Figure 1:
859:Spherical
827:Voderberg
788:Prototile
755:Problems
731:Honeycomb
709:Isohedral
596:Aperiodic
534:honeycomb
518:Rectangle
508:Rhombille
225:aperiodic
133:connected
126:imperfect
941:V3.4.3.4
776:Squaring
771:Heesch's
736:Isotoxal
656:Rep-tile
646:Pinwheel
539:Coloring
492:Periodic
110:rep-tile
1401:6.4.8.4
1356:5.4.6.4
1316:4.12.16
1306:4.10.12
1276:V4.8.10
1251:V4.6.16
1241:V4.6.14
1141:3.6.4.6
1136:3.4.∞.4
1131:3.4.8.4
1126:3.4.7.4
1121:3.4.6.4
1071:3.∞.3.∞
1066:3.4.3.4
1061:3.8.3.8
1056:3.7.3.7
1051:3.6.3.8
1046:3.6.3.6
1041:3.5.3.6
1036:3.5.3.5
1031:3.4.3.∞
1026:3.4.3.8
1021:3.4.3.7
1016:3.4.3.6
1011:3.4.3.5
966:3.4.6.4
936:3.4.3.4
929:regular
896:Regular
822:Voronoi
746:Packing
677:Truchet
672:Socolar
641:Penrose
636:Gilbert
561:Wythoff
118:perfect
34:setiset
1291:4.8.16
1286:4.8.14
1281:4.8.12
1271:4.8.10
1246:4.6.16
1236:4.6.14
1231:4.6.12
1001:Hyper-
986:4.6.12
759:Domino
665:Sphinx
544:Convex
523:Domino
305:
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22:
1406:(6.8)
1361:(5.6)
1296:4.8.∞
1266:(4.8)
1261:(4.7)
1256:4.6.∞
1226:(4.6)
1221:(4.5)
1191:4.∞.4
1186:4.8.4
1181:4.7.4
1176:4.6.4
1171:4.5.4
1151:(3.8)
1146:(3.7)
1116:(3.4)
1111:(3.4)
1003:bolic
971:(3.6)
927:Semi-
798:Girih
695:Other
231:Loops
214:, or
139:, or
32:, or
1491:8.16
1486:8.12
1456:7.14
1426:6.16
1421:6.12
1416:6.10
1376:5.12
1371:5.10
1326:4.16
1321:4.14
1311:4.12
1301:4.10
1161:3.16
1156:3.14
976:3.12
961:V3.6
887:V4.n
877:V3.n
764:Wang
741:List
707:and
658:and
617:List
532:and
150:and
1521:∞.8
1516:∞.6
1481:8.6
1451:7.8
1446:7.6
1411:6.8
1366:5.8
1331:4.∞
1166:3.∞
1091:3.4
1086:3.∞
1081:3.8
1076:3.7
991:4.8
981:4.∞
956:3.6
951:3.∞
946:3.4
882:4.n
872:3.n
845:By
396:doi
367:in
337:doi
158:or
1546::
392:87
390:.
376:^
333:85
331:.
166:.
28:A
1511:∞
1506:∞
1501:∞
1496:∞
1476:8
1471:8
1466:8
1461:8
1441:7
1436:7
1431:7
1396:6
1391:6
1386:6
1381:6
1351:5
1346:5
1341:5
1336:5
1216:4
1211:4
1206:4
1201:4
1196:4
1106:3
1101:3
1096:3
918:6
913:4
908:3
903:2
867:2
471:e
464:t
457:v
402:.
398::
343:.
339::
274:n
270:n
160:Q
156:P
152:Q
148:P
122:n
114:n
106:n
70:n
58:n
54:n
50:n
46:n
42:n
38:n
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