1000:
851:
1472:
883:
873:
861:
31:
1012:
731:
625:
651:
113:
599:
130:
962:
794:
562:
490:
714:
942:
952:
704:
635:
661:
609:
206:
411:
1479:
554:
480:
465:
437:
691:
including the one formed from three equilateral triangles, for three axis-parallel hexagons (the L-tromino, L-tetromino, and P-pentomino), and the sphinx hexiamond. In addition, many rep-tiles, particularly those with higher
510:
is a triangle containing one right angle of 90°. Two particular forms of right triangle have attracted the attention of rep-tile researchers, the 45°-90°-45° triangle and the 30°-60°-90° triangle.
538:. An infinite number of them are rep-tiles. Indeed, the simplest of all rep-tiles is a single isosceles right triangle. It is rep-2 when divided by a single line bisecting the right angle to the
542:. Rep-2 rep-tiles are also rep-2 and the rep-4,8,16+ triangles yield further rep-tiles. These are found by discarding half of the sub-copies and permutating the remainder until they are
696:, can be self-tiled in different ways. For example, the rep-9 L-tetramino has at least fourteen different rep-tilings. The rep-9 sphinx hexiamond can also be tiled in different ways.
256:
is irrep-7: six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake.
982:
can be dissected into six equilateral triangles, each of which can be dissected into a regular hexagon and three more equilateral triangles. This is the basis for an infinite
1011:
999:
823:
at smaller and smaller scales. A rep-tile fractal is formed by subdividing the rep-tile, removing one or more copies of the subdivided shape, and then continuing
738:
The most familiar rep-tiles are polygons with a finite number of sides, but some shapes with an infinite number of sides can also be rep-tiles. For example, the
903:
are often self-similar on smaller and smaller scales, many may be decomposed into copies of themselves like a rep-tile. However, if the fractal has an empty
359:
related by mirror-reflection. Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image.
546:
within a right triangle. In other words, two copies will tile a right triangle. One of these new rep-tiles is reminiscent of the fish formed from three
407:, which are created from eight squares. Two copies of some octominoes will tile a square; therefore these octominoes are also rep-16 rep-tiles.
774:
have found more examples, including a double-pyramid and an elongated version of the sphinx. These pentagonal rep-tiles are illustrated on the
197:
even smaller tiles. The order of a shape, whether using rep-tiles or irrep-tiles is the smallest possible number of tiles which will suffice.
1388:
978:
Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves. However, a regular
157:. A rep-tile dissection using different sizes of the original shape is called an irregular rep-tile or irreptile. If the dissection uses
2412:
1677:
775:
1610:
915:
is rep-8, tiled with eight copies of itself, but repetition of these decompositions does not form a tiling. On the other hand, the
782:. However, the sphinx and its extended versions are the only known pentagons that can be rep-tiled with equal copies. See Clarke's
2417:
1632:
1366:
835:
is formed from a rep-tiling of an equilateral triangle into four smaller triangles. When one sub-copy is discarded, a rep-4 L-
2227:
2062:
493:
A rocket-like rep-tile created from a dodeciamond, or twelve equilateral triangles laid edge-to-edge (and corner-to-corner)
165:. If all these sub-tiles are of different sizes then the tiling is additionally described as perfect. A shape that is rep-
2377:
2352:
2342:
2312:
2267:
2217:
2197:
2012:
1897:
754:
Triangular and quadrilateral (four-sided) rep-tiles are common, but pentagonal rep-tiles are rare. For a long time, the
2387:
2382:
2322:
2317:
2272:
2222:
2207:
2407:
2192:
1440:
399:, then it will also be a rep-tile, because the rectangle will have an integer side length ratio and will thus tile a
2247:
2182:
2167:
2002:
1622:
2347:
2307:
2262:
2202:
2187:
2177:
2152:
1513:
2212:
2132:
1987:
2142:
2127:
2087:
2017:
1967:
1882:
1702:
302:
times the short side of the paper. Rectangles in this shape are rep-2. A rectangle (or parallelogram) is rep-
2112:
2077:
2067:
1927:
2252:
2082:
2072:
2052:
2032:
2007:
1952:
1932:
1917:
1907:
1842:
1508:
86:
2402:
2397:
2392:
2297:
2057:
2022:
1982:
1962:
1937:
1922:
1912:
1872:
1359:
263:
with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic
75:
2337:
2332:
2242:
2237:
2232:
2027:
1997:
1992:
1972:
1957:
1947:
1942:
1862:
1503:
17:
410:
2372:
2367:
2362:
2292:
2287:
2282:
2277:
1977:
1857:
1852:
931:
1471:
850:
2037:
1887:
1837:
1525:
1047:
2157:
2147:
2117:
1799:
1414:
840:
268:
2446:
2257:
2162:
2122:
2107:
2102:
2097:
2092:
1847:
1637:
1352:
1137:
882:
872:
63:
2302:
2042:
1755:
1743:
1627:
1556:
1532:
1457:
1248:
1220:
1042:
860:
547:
380:
336:
105:
96:
1290:
783:
30:
8:
2047:
1867:
1713:
1672:
1667:
1547:
1124:
920:
908:
904:
832:
91:
730:
1832:
1601:
1399:
1334:
1200:
519:
320:
299:
574:
Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1 :
249:
2451:
2327:
1877:
1804:
1647:
1430:
1298:
1284:
912:
831:
is formed in this way from a rep-tiling of a square into 27 smaller squares, and the
828:
553:
479:
344:
78:
1101:
907:, this decomposition may not lead to a tiling of the entire plane. For example, the
2357:
2172:
2137:
1814:
1778:
1723:
1689:
1642:
1616:
1605:
1520:
1492:
1435:
1409:
1404:
1315:
1236:
1192:
1037:
575:
526:
154:
1718:
1542:
1452:
1244:
1216:
987:
820:
779:
264:
210:
927:
is rep-7, formed from the space-filling Gosper curve, and again forms a tiling.
464:
1655:
1568:
1537:
1426:
1171:
1159:
1018:
923:
with a non-empty interior; it is rep-4, and does form a tiling. Similarly, the
771:
561:
522:
507:
260:
253:
82:
1301:
1240:
489:
2440:
1809:
1773:
1573:
1561:
1419:
1066:
941:
755:
348:
238:
226:
117:
39:
945:
A Sierpinski triangle based on three smaller copies of a
Sierpinski triangle
650:
624:
1708:
1445:
1375:
983:
951:
924:
916:
839:
can be used to create four fractals, two of which are identical except for
543:
307:
134:
112:
51:
1164:
The
Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems
598:
1694:
793:
763:
584:
100:
1339:
955:
A Sierpinski carpet based on eight smaller copies of a
Sierpinski carpet
129:
1763:
1204:
961:
588:
539:
458:
372:
272:
241:
35:
1067:
A Gardner's Dozen—Martin's Scientific
American Cover Stories
713:
1783:
1768:
1684:
1660:
1306:
824:
703:
688:
535:
396:
376:
356:
352:
222:
150:
1196:
1215:, Providence, RI: American Mathematical Society, pp. 205–217,
1174:(1991), "Chapter 19: Rep-Tiles, Replicating Figures on the Plane",
836:
742:
triangle, or horned triangle, is rep-4. It is also an example of a
445:
441:
430:
426:
422:
415:
404:
368:
340:
234:
71:
47:
979:
900:
816:
759:
739:
286:
230:
628:
A tetradrafter, or shape created from four 30°-60°-90° triangles
602:
A tridrafter, or shape created by three triangles of 30°-60°-90°
298:, in which the long side of a rectangular sheet of paper is the
244:(illustrated above) is rep-4 and rep-9, and is one of few known
66:
into smaller copies of the same shape. The term was coined as a
1344:
1032:
660:
634:
400:
384:
332:
218:
608:
436:
205:
189: > 1, by replacing the smallest tile in the rep-
1478:
654:
A hexadrafter, or shape created by six 30°-60°-90° triangles
565:
A fish-like rep-tile based on four isosceles right triangles
461:
tiles an equilateral triangle, it will also be a rep-tile.
1265:
The
Penguin Dictionary of Curious and Interesting Geometry
986:
of the hexagon with hexagons. The hexagon is therefore an
965:
A dragon curve based on 4 smaller copies of a dragon curve
758:
was widely believed to be the only example known, but the
1178:, Chicago, IL: Chicago University Press, pp. 222–233
767:
483:
A fish-like rep-tile based on three equilateral triangles
275:, or sloping side of the rep-5 triangle, has a length of
67:
1176:
The
Unexpected Hanging and Other Mathematical Diversions
1291:
http://www.recmath.com/PolyPages/PolyPages/Reptiles.htm
395:
If a polyomino is rectifiable, that is, able to tile a
1322:
1257:
A Puzzling
Journey to the Reptiles and Related Animals
1183:
Langford, C. D. (1940), "Uses of a
Geometric Puzzle",
525:, with sides in the ratio 1 : 1 :
911:
is rep-3, tiled with three copies of itself, and the
42:
can be put together as shown to make a larger sphinx.
1005:
Regular hexagon tiled with infinite copies of itself
213:) by repeatedly dissecting and inflating a rep-tile.
672:
1285:http://mathematicscentre.com/taskcentre/sphinx.htm
1083:
1071:
934:of n contracting maps of the same ratio is rep-n.
1296:
1227:Sallows, Lee (2012), "On self-tiling tile sets",
1106:
717:Variant rep-tilings of the rep-9 sphinx hexiamond
133:The chair substitution (left) and a portion of a
2438:
1316:http://www.uwgb.edu/dutchs/symmetry/reptile1.htm
797:A pentagonal rep-tile discovered by Karl Scherer
725:
1125:Math Magic, Problem of the Month (October 2002)
854:Geometrical dissection of an L-triomino (rep-4)
95:. In 2012 a generalization of rep-tiles called
1211:Niţică, Viorel (2003), "Rep-tiles revisited",
1021:(Siamese) tiled with infinite copies of itself
1360:
778:pages overseen by the American mathematician
707:Variant rep-tilings of the rep-9 L-tetromino
468:Rep-tiles created from equilateral triangles
153:for a tiling of the plane, in many cases an
1095:
930:By construction, any fractal defined by an
149:copies. Such a shape necessarily forms the
1367:
1353:
1135:
743:
362:
1678:Dividing a square into similar rectangles
1335:Math Magic - Problem of the Month 10/2002
1321:IFStile - program for finding rep-tiles:
1118:
805:
326:
245:
1182:
1166:, New York: W. W. Norton, pp. 46–58
960:
950:
940:
894:
881:
871:
864:A fractal based on an L-triomino (rep-4)
859:
849:
810:
792:
729:
712:
702:
683:for all positive integer values of
659:
649:
633:
623:
607:
597:
569:
560:
552:
513:
488:
478:
463:
452:
435:
409:
289:defines sizes of paper sheets using the
204:
128:
116:A selection of rep-tiles, including the
111:
29:
1254:
1226:
1170:
1158:
1089:
1077:
749:
687:. In particular this is true for three
14:
2439:
1210:
1112:
886:Another fractal based on an L-triomino
876:Another fractal based on an L-triomino
161:copies, the shape is said to be irrep-
27:Shape subdivided into copies of itself
1740:
1590:
1490:
1386:
1348:
1297:
1262:
734:Horned triangle or teragonic triangle
677:Many of the common rep-tiles are rep-
429:will tile a square, therefore these
1267:, London: Penguin, pp. 213–214
440:Rep-tiles created from rectifiable
24:
1741:
973:
638:The same tetradrafter as a reptile
557:Rep-tiles based on right triangles
501:
209:Defining an aperiodic tiling (the
89:" column in the May 1963 issue of
25:
2463:
1283:Mathematics Centre Sphinx Album:
1272:
1138:"Tartapelago. Arte tassellazione"
664:The same hexadrafter as a reptile
1477:
1470:
1374:
1010:
998:
815:Rep-tiles can be used to create
673:Multiple and variant rep-tilings
612:The same tridrafter as a reptile
770:and the American mathematician
414:Rep-tiles based on rectifiable
1340:Tanya Khovanova - L-Irreptiles
1328:
1129:
1060:
379:, or shapes created by laying
323:right triangle is also rep-2.
124:
13:
1:
1703:Regular Division of the Plane
1491:
1152:
990:or irrep-infinity irreptile.
726:Rep-tiles with infinite sides
583: : 2, are known as
120:, two fish and the 5-triangle
38:rep-tile. Four copies of the
1387:
1277:
1136:Pietrocola, Giorgio (2005).
367:Some rep-tiles are based on
7:
1611:Architectonic and catoptric
1509:Aperiodic set of prototiles
1026:
433:are also rep-36 rep-tiles.
285:The international standard
200:
141:A rep-tile is labelled rep-
10:
2468:
1289:Clarke, A. L. "Reptiles."
938:
869:
847:
790:
700:
647:
621:
595:
476:
403:. This can be seen in the
390:
343:and remain identical when
246:self-replicating pentagons
76:recreational mathematician
1896:
1823:
1792:
1754:
1750:
1736:
1597:
1591:
1586:
1499:
1486:
1468:
1395:
1382:
1241:10.4169/math.mag.85.5.323
331:Some rep-tiles, like the
173:is trivially also irrep-(
1185:The Mathematical Gazette
1053:
932:iterated function system
587:. Some are identical to
1102:Polydrafter Irreptiling
1048:Reptiles (M. C. Escher)
363:Rep-tiles and polyforms
145:if the dissection uses
62:is a shape that can be
1255:Scherer, Karl (1987),
966:
956:
946:
887:
877:
865:
855:
806:Rep-tiles and fractals
798:
735:
718:
708:
665:
655:
639:
629:
613:
603:
566:
558:
494:
484:
469:
449:
418:
327:Rep-tiles and symmetry
214:
138:
121:
43:
1162:(2001), "Rep-Tiles",
964:
954:
944:
895:Fractals as rep-tiles
885:
875:
863:
853:
819:, or shapes that are
811:Rep-tiles as fractals
796:
733:
716:
706:
663:
653:
637:
627:
611:
601:
570:30°-60°-90° triangles
564:
556:
548:equilateral triangles
514:45°-90°-45° triangles
492:
482:
467:
453:Equilateral triangles
439:
413:
381:equilateral triangles
345:reflected in a mirror
208:
132:
115:
97:self-tiling tile sets
33:
1229:Mathematics Magazine
1043:Self-tiling tile set
827:. For instance, the
750:Pentagonal rep-tiles
421:Four copies of some
337:equilateral triangle
106:Mathematics Magazine
1323:https://ifstile.com
921:space-filling curve
909:Sierpinski triangle
833:Sierpinski triangle
518:Polyforms based on
347:. Others, like the
269:Pythagoras' theorem
177: −
92:Scientific American
81:and popularized by
1299:Weisstein, Eric W.
1263:Wells, D. (1991),
1017:Fractal elongated
967:
957:
947:
888:
878:
866:
856:
799:
736:
719:
709:
666:
656:
640:
630:
614:
604:
567:
559:
544:mirror-symmetrical
495:
485:
470:
450:
419:
357:two distinct forms
300:square root of two
215:
139:
122:
99:was introduced by
87:Mathematical Games
44:
2434:
2433:
2430:
2429:
2426:
2425:
1732:
1731:
1623:Computer graphics
1582:
1581:
1466:
1465:
971:
970:
913:Sierpinski carpet
892:
891:
829:Sierpinski carpet
803:
802:
723:
722:
670:
669:
644:
643:
618:
617:
499:
498:
79:Solomon W. Golomb
16:(Redirected from
2459:
1752:
1751:
1738:
1737:
1690:Conway criterion
1617:Circle Limit III
1588:
1587:
1521:Einstein problem
1488:
1487:
1481:
1474:
1410:Schwarz triangle
1384:
1383:
1369:
1362:
1355:
1346:
1345:
1312:
1311:
1268:
1259:
1251:
1223:
1207:
1191:(260): 209–211,
1179:
1167:
1146:
1145:
1133:
1127:
1122:
1116:
1110:
1104:
1099:
1093:
1087:
1081:
1075:
1069:
1064:
1038:Self-replication
1014:
1002:
937:
936:
846:
845:
789:
788:
772:George Sicherman
744:fractal rep-tile
699:
698:
686:
682:
646:
645:
620:
619:
594:
593:
581:
580:
532:
531:
475:
474:
457:Similarly, if a
318:
317:
297:
296:
295:
281:
280:
155:aperiodic tiling
21:
2467:
2466:
2462:
2461:
2460:
2458:
2457:
2456:
2437:
2436:
2435:
2422:
1899:
1892:
1825:
1819:
1788:
1746:
1728:
1593:
1578:
1495:
1482:
1476:
1475:
1462:
1453:Wallpaper group
1391:
1378:
1373:
1331:
1280:
1275:
1197:10.2307/3605717
1155:
1150:
1149:
1134:
1130:
1123:
1119:
1111:
1107:
1100:
1096:
1088:
1084:
1076:
1072:
1065:
1061:
1056:
1029:
1022:
1015:
1006:
1003:
993:
976:
974:Infinite tiling
897:
813:
808:
752:
728:
684:
678:
675:
578:
576:
572:
534:, are known as
529:
527:
523:right triangles
516:
504:
502:Right triangles
472:
455:
393:
365:
329:
313:
311:
293:
291:
290:
278:
276:
265:pinwheel tiling
211:pinwheel tiling
203:
127:
28:
23:
22:
15:
12:
11:
5:
2465:
2455:
2454:
2449:
2432:
2431:
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2415:
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2300:
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2260:
2255:
2250:
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2240:
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2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
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2175:
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2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2050:
2045:
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2035:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1995:
1990:
1985:
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1965:
1960:
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1571:
1566:
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1564:
1550:
1545:
1540:
1535:
1530:
1529:
1528:
1526:Socolar–Taylor
1518:
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254:Koch snowflake
252:is rep-7. The
237:is rep-4. The
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34:The "sphinx"
32:
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2447:Tessellation
1714:Substitution
1709:Regular grid
1701:
1615:
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1548:Quaquaversal
1446:Kisrhombille
1376:Tessellation
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1213:MASS selecta
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821:self-similar
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768:Karl Scherer
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135:chair tiling
104:
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1744:vertex type
1602:Anisohedral
1557:Self-tiling
1400:Pythagorean
1329:Irrep-tiles
1172:Gardner, M.
1160:Gardner, M.
841:orientation
825:recursively
764:New-Zealand
589:polyiamonds
448:(nonakings)
446:9-polykings
377:polyominoes
373:polyiamonds
341:symmetrical
125:Terminology
101:Lee Sallows
2441:Categories
1648:Pentagonal
1302:"Rep-Tile"
1153:References
776:Math Magic
689:trapezoids
540:hypotenuse
536:polyabolos
459:polyiamond
442:nonominoes
423:nonominoes
416:octominoes
405:octominoes
273:hypotenuse
185:) for any
70:on animal
36:polyiamond
1756:Spherical
1724:Voderberg
1685:Prototile
1652:Problems
1628:Honeycomb
1606:Isohedral
1493:Aperiodic
1431:honeycomb
1415:Rectangle
1405:Rhombille
1307:MathWorld
1278:Rep-tiles
740:teragonic
520:isosceles
431:polyforms
427:nonakings
397:rectangle
369:polyforms
321:isosceles
242:hexiamond
223:rectangle
169:or irrep-
151:prototile
64:dissected
18:Rep-tiles
2452:Fractals
1838:V3.4.3.4
1673:Squaring
1668:Heesch's
1633:Isotoxal
1553:Rep-tile
1543:Pinwheel
1436:Coloring
1389:Periodic
1027:See also
905:interior
901:fractals
899:Because
837:triomino
817:fractals
235:triangle
201:Examples
85:in his "
72:reptiles
56:rep-tile
48:geometry
2298:6.4.8.4
2253:5.4.6.4
2213:4.12.16
2203:4.10.12
2173:V4.8.10
2148:V4.6.16
2138:V4.6.14
2038:3.6.4.6
2033:3.4.∞.4
2028:3.4.8.4
2023:3.4.7.4
2018:3.4.6.4
1968:3.∞.3.∞
1963:3.4.3.4
1958:3.8.3.8
1953:3.7.3.7
1948:3.6.3.8
1943:3.6.3.6
1938:3.5.3.6
1933:3.5.3.5
1928:3.4.3.∞
1923:3.4.3.8
1918:3.4.3.7
1913:3.4.3.6
1908:3.4.3.5
1863:3.4.6.4
1833:3.4.3.4
1826:regular
1793:Regular
1719:Voronoi
1643:Packing
1574:Truchet
1569:Socolar
1538:Penrose
1533:Gilbert
1458:Wythoff
1249:3007213
1221:2027179
1205:3605717
988:irrep-∞
980:hexagon
577:√
528:√
391:Squares
385:squares
319::1. An
312:√
306:if its
292:√
287:ISO 216
277:√
231:rhombus
137:(right)
60:reptile
46:In the
2188:4.8.16
2183:4.8.14
2178:4.8.12
2168:4.8.10
2143:4.6.16
2133:4.6.14
2128:4.6.12
1898:Hyper-
1883:4.6.12
1656:Domino
1562:Sphinx
1441:Convex
1420:Domino
1318:(1999)
1247:
1219:
1203:
1142:Maecla
1033:Mosaic
984:tiling
760:German
756:sphinx
401:square
351:, are
349:sphinx
339:, are
333:square
271:, the
248:. The
239:sphinx
219:square
217:Every
118:sphinx
40:sphinx
2303:(6.8)
2258:(5.6)
2193:4.8.∞
2163:(4.8)
2158:(4.7)
2153:4.6.∞
2123:(4.6)
2118:(4.5)
2088:4.∞.4
2083:4.8.4
2078:4.7.4
2073:4.6.4
2068:4.5.4
2048:(3.8)
2043:(3.7)
2013:(3.4)
2008:(3.4)
1900:bolic
1868:(3.6)
1824:Semi-
1695:Girih
1592:Other
1201:JSTOR
1054:Notes
919:is a
371:like
267:. By
233:, or
2388:8.16
2383:8.12
2353:7.14
2323:6.16
2318:6.12
2313:6.10
2273:5.12
2268:5.10
2223:4.16
2218:4.14
2208:4.12
2198:4.10
2058:3.16
2053:3.14
1873:3.12
1858:V3.6
1784:V4.n
1774:V3.n
1661:Wang
1638:List
1604:and
1555:and
1514:List
1429:and
692:rep-
444:and
425:and
383:and
375:and
335:and
54:, a
2418:∞.8
2413:∞.6
2378:8.6
2348:7.8
2343:7.6
2308:6.8
2263:5.8
2228:4.∞
2063:3.∞
1988:3.4
1983:3.∞
1978:3.8
1973:3.7
1888:4.8
1878:4.∞
1853:3.6
1848:3.∞
1843:3.4
1779:4.n
1769:3.n
1742:By
1237:doi
1193:doi
310:is
103:in
74:by
68:pun
58:or
50:of
2443::
1304:.
1245:MR
1243:,
1233:85
1231:,
1217:MR
1199:,
1189:24
1187:,
1140:.
843:.
786:.
746:.
591:.
550:.
506:A
282:.
259:A
229:,
225:,
221:,
175:kn
109:.
2408:∞
2403:∞
2398:∞
2393:∞
2373:8
2368:8
2363:8
2358:8
2338:7
2333:7
2328:7
2293:6
2288:6
2283:6
2278:6
2248:5
2243:5
2238:5
2233:5
2113:4
2108:4
2103:4
2098:4
2093:4
2003:3
1998:3
1993:3
1815:6
1810:4
1805:3
1800:2
1764:2
1368:e
1361:t
1354:v
1310:.
1293:.
1239::
1195::
1144:.
1115:.
1092:.
1080:.
762:/
694:n
685:n
680:n
579:3
530:2
315:n
304:n
294:2
279:5
195:n
191:n
187:k
183:n
179:k
171:n
167:n
163:n
159:n
147:n
143:n
20:)
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