597:
2274:
983:
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1048:
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941:
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439:
764:
757:
904:
1100:
40:
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1128:
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1114:
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65:
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1135:
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948:
1142:
421:
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1390:
See in particular
Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2.9.1, p. 103 (classification of colored tilings), Figure 2.9.2, p. 105 (illustration of colored tilings), Figure 2.5.3(d), p. 83 (topologically equivalent star
523:. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an
413:
The kagome pattern is common in bamboo weaving in East Asia. In 2022, archaeologists found bamboo weaving remains at the
Dongsunba ruins in Chongqing, China, 200 BC. After 2200 years, the kagome pattern is still clear.
488:, and first appeared in a 1951 paper by his assistant IchirĆ ShĆji. The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling. Despite the name, these crossing points do not form a
120:
919:
can be geometrically distorted into topologically equivalent tilings of lower symmetry. In these variants of the tiling, the edges do not necessarily line up to form straight lines.
1835:
Yin, Jia-Xin; Zhang, Songtian S.; Chang, Guoqing; Wang, Qi; Tsirkin, Stepan S.; Guguchia, Zurab; Lian, Biao; Zhou, Huibin; Jiang, Kun; Belopolski, Ilya; Shumiya, Nana (2019).
1199:, sharing the vertices of the trihexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons
1522:
2190:
1680:
Yen, F.; Chaudhury, R. P.; Galstyan, E.; Lorenz, B.; Wang, Y. Q.; Sun, Y. Y.; Chu, C. W. (2008). "Magnetic phase diagrams of the Kagome staircase compound Co
460:
composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The
3214:
581:
The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice.
2479:
2412:
1464:; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings; Euclidean plane tessellations".
3219:
2434:
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2137:
2104:
2028:
2003:
1473:
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3179:
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3069:
3019:
2999:
2814:
2699:
1607:
Lawler, Michael J.; Kee, Hae-Young; Kim, Yong Baek; Vishwanath, Ashvin (2008). "Topological spin liquid on the hyperkagome lattice of Na
1333:
3189:
3184:
3124:
3119:
3074:
3024:
3009:
1770:
Yin, Jia-Xin; Zhang, Songtian S.; Li, Hang; Jiang, Kun; Chang, Guoqing; Zhang, Bingjing; Lian, Biao; Xiang, Cheng; Belopolski (2018).
821:
180:
316:, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular
3209:
2994:
2242:
2058:
1413:
695:
of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232. The second is called a
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1268:
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198:
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894:, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (
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rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures. Quantum magnets realized on
82:
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The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.
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1439:. Memoirs of the American Philosophical Society. Vol. 209. American Philosophical Society. pp. 104â105.
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1499:"[News Live Room] Bamboo weaving products of Ba culture first appeared in Chongqing about 2200 years ago"
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have been discovered to exhibit many unexpected electronic and magnetic phenomena. It is also proposed that
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1939:
Wei, Chenan; Sedrakyan, Tigran (2021-01-29). "Optical lattice platform for the
Sachdev-Ye-Kitaev model".
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tiling), and
Exercise 4.1.3, p. 171 (topological equivalence of trihexagonal and two-triangle tilings).
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1008:), progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With
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373:, combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille).
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1772:"Giant and anisotropic many-body spinâorbit tunability in a strongly correlated kagome magnet"
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1536:
Yin, Jia-Xin (March 2023). "Exploring hitherto unknown quantum phases in kagome crystals".
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This pattern, and its place in the classification of uniform tilings, was already known to
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361:. The Japanese term for this pattern has been taken up in physics, where it is called a
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903:
895:
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771:
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402:) is a traditional Japanese woven bamboo pattern; its name is composed from the words
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546:, contain two-dimensional layers or three-dimensional kagome lattice arrangement of
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954:
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1970:
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1837:"Negative flat band magnetism in a spinâorbit-coupled correlated kagome magnet"
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1805:
1372:
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A related three dimensional structure formed by the vertices and edges of the
1998:. Springer Series in Materials Science. Vol. 126. Springer. p. 20.
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2462:
1024:
of symmetry, with generator points at the right angle corner of the domain.
17:
1813:
1658:
2354:
652:
539:
429:
410:, meaning "eye(s)", referring to the pattern of holes in a woven basket.
354:
290:
1406:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1549:
535:
461:
450:
2280:
1589:
1564:
1953:
1853:
1788:
578:
can be observed in two dimensional kagome lattice with impurities.
381:
1995:
Crystallography of
Quasicrystals: Concepts, Methods and Structures
1706:
1633:
554:. These minerals display novel physical properties connected with
426:
380:
2099:(2nd ed.). Cambridge University Press. pp. 111â2, 136.
1000:
exists in a sequence of symmetries of quasiregular tilings with
547:
457:
365:. It occurs also in the crystal structures of certain minerals.
2542:
2392:
2292:
2188:
2150:
2146:
558:. For instance, the spin arrangement of the magnetic ions in Co
27:
Tiling of a plane by regular hexagons and equilateral triangles
1497:
China
Central Television, CCTV-13 News Channel (2022-03-25).
2047:(1973). "V. The Kaleidoscope, §5.7 Wythoff's construction".
683:, one of eight derived from the regular hexagonal tiling.
2074:
324:. Two hexagons and two triangles alternate around each
91:
511:. It is represented by the vertices and edges of the
85:
921:
29:
2698:
2625:
2594:
2556:
1371:
115:{\displaystyle {\begin{Bmatrix}6\\3\end{Bmatrix}}}
114:
706:{6,3}, with two colors of triangles, existing in
659:p6mm, (*632), and the tiling can be derived as a
1468:. Wellesley, MA: A K Peters, Ltd. p. 288.
1565:"Kagome: The story of the basketweave lattice"
1035:32 orbifold symmetries of quasiregular tilings
2162:
353:. The pattern has long been used in Japanese
8:
1746:"A quantum magnet with a topological twist"
1521:: CS1 maint: numeric names: authors list (
1436:The Harmony of the World by Johannes Kepler
2553:
2539:
2389:
2289:
2185:
2169:
2155:
2147:
1026:
890:The trihexagonal tiling can be used as a
675:, alternating two types of polygons, with
385:Japanese basket showing the kagome pattern
2480:Dividing a square into similar rectangles
2079:
1952:
1852:
1787:
1705:
1632:
1588:
1429:Aiton, E. J.; Duncan, Alistair Matheson;
86:
84:
1992:Steurer, Walter; Deloudi, Sofia (2009).
1602:
1600:
1244:
712:
595:
2023:. Thames & Hudson. pp. 74â75.
2019:Critchlow, Keith (2000) . "pattern G".
1408:. Dover Publications, Inc. p. 38.
1350:
416:
1514:
1358:
1356:
1354:
7:
2021:Order in Space: A design source book
644:, symbolizing the fact that it is a
1334:Cyclotruncated simplectic honeycomb
531:represents its edges and vertices.
1191:Related regular complex apeirogons
556:geometrically frustrated magnetism
464:process gives the Kagome a chiral
25:
484:was coined by Japanese physicist
328:, and its edges form an infinite
2279:
2272:
1750:Discovery: Research at Princeton
1563:Mekata, Mamoru (February 2003).
1329:Trihexagonal prismatic honeycomb
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911:Topologically equivalent tilings
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837:
829:
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671:. The trihexagonal tiling is a
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529:trihexagonal prismatic honeycomb
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1651:10.1103/physrevlett.100.227201
1:
2505:Regular Division of the Plane
2130:Introduction to Tessellations
1016:32 all of these tilings are
992:Related quasiregular tilings
865:
782:
769:
746:
608:The trihexagonal tiling has
2413:Architectonic and catoptric
2311:Aperiodic set of prototiles
1971:10.1103/PhysRevA.103.013323
1724:10.1016/j.physb.2007.10.334
1694:Physica B: Condensed Matter
525:orthorhombic-kagome lattice
515:, filling space by regular
499:, filling space by regular
3295:
1197:regular complex apeirogons
2552:
2538:
2399:
2388:
2301:
2288:
2270:
2197:
2184:
2097:Regular Complex Polytopes
1918:10.1038/s41567-019-0451-6
1871:10.1038/s41567-019-0426-7
1806:10.1038/s41586-018-0502-7
1227:vertices arranged like a
1064:
1056:
1045:
1029:
930:
924:
397:
37:
32:
2095:Coxeter, H.S.M. (1991).
1892:Yazyev, Oleg V. (2019).
1466:The Symmetries of Things
663:within the reflectional
655:can be described by the
444:Kagome pattern in detail
406:, meaning "basket", and
2053:(3rd ed.). Dover.
1894:"An upside-down magnet"
1621:Physical Review Letters
1339:List of uniform tilings
691:There are two distinct
513:quarter cubic honeycomb
497:quarter cubic honeycomb
3269:Quasiregular polyhedra
1431:Field, Judith Veronica
1211:are constrained by: 1/
605:
604:of p6m (*632) symmetry
386:
116:
1314:Percolation threshold
1002:vertex configurations
599:
384:
357:, where it is called
310:equilateral triangles
117:
1374:Tilings and Patterns
1018:wythoff construction
679:(3.6). It is also a
677:vertex configuration
661:Wythoff construction
521:truncated tetrahedra
509:hyper-kagome lattice
507:, has been called a
505:truncated tetrahedra
490:mathematical lattice
330:arrangement of lines
83:
59:Vertex configuration
33:Trihexagonal tiling
3274:Japanese bamboowork
3264:Semiregular tilings
1963:2021PhRvA.103a3323W
1910:2019NatPh..15..424Y
1863:2019NatPh..15..443Y
1798:2018Natur.562...91Y
1716:2008PhyB..403.1487Y
1643:2008PhRvL.100v7201L
1581:2003PhT....56b..12M
998:trihexagonal tiling
917:trihexagonal tiling
673:quasiregular tiling
665:fundamental domains
602:fundamental domains
306:by regular polygons
295:trihexagonal tiling
2132:. pp. 50â56.
1700:(5â9): 1487â1489.
1550:10.7693/wl20230301
1022:fundamental domain
606:
600:30-60-90 triangle
387:
112:
106:
52:Semiregular tiling
3249:Euclidean tilings
3236:
3235:
3232:
3231:
3228:
3227:
2534:
2533:
2425:Computer graphics
2384:
2383:
2268:
2267:
2139:978-0-86651-461-3
2106:978-0-521-39490-1
2050:Regular Polytopes
2030:978-0-500-34033-2
2005:978-3-642-01899-2
1590:10.1063/1.1564329
1475:978-1-56881-220-5
1446:978-0-87169-209-2
1385:978-0-7167-1193-3
1378:. W. H. Freeman.
1305:
1304:
1188:
1187:
1010:orbifold notation
989:
988:
883:
882:
710:(*333) symmetry.
693:uniform colorings
687:Uniform colorings
552:crystal structure
347:in his 1619 book
322:triangular tiling
308:. It consists of
287:
286:
279:Vertex-transitive
239:Rotation symmetry
16:(Redirected from
3286:
3259:Isotoxal tilings
3254:Isogonal tilings
2554:
2540:
2492:Conway criterion
2419:Circle Limit III
2390:
2323:Einstein problem
2290:
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2212:Schwarz triangle
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1402:Williams, Robert
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1364:GrĂŒnbaum, Branko
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1223:= 1. Edges have
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699:hexagonal tiling
649:hexagonal tiling
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350:Harmonices Mundi
338:rhombille tiling
318:hexagonal tiling
314:regular hexagons
269:Rhombille tiling
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3279:Crystallography
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2255:Wallpaper group
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2124:Seymour, Dale;
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2118:Further reading
2115:
2114:
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2045:Coxeter, H.S.M.
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2018:
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1991:
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1782:(7725): 91â95.
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1462:Conway, John H.
1460:
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1447:
1433:, eds. (1997).
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1368:Shephard, G. C.
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1229:regular polygon
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614:Coxeter diagram
610:SchlÀfli symbol
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303:Euclidean plane
299:uniform tilings
282:Edge-transitive
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2081:10.1.1.30.8536
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2011:
2004:
1984:
1931:
1898:Nature Physics
1884:
1841:Nature Physics
1827:
1762:
1737:
1689:
1685:
1681:
1672:
1627:(22): 227201.
1616:
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1596:
1555:
1544:(3): 157â165.
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1233:vertex figures
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896:kissing number
892:circle packing
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886:Circle packing
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703:
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681:uniform tiling
612:of r{6,3}, or
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482:kagome lattice
477:
476:Kagome lattice
474:
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363:kagome lattice
320:and a regular
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255:Bowers acronym
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2126:Britton, Jill
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2060:0-486-61480-8
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2015:
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1947:(1): 013323.
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1569:Physics Today
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1324:Star of David
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1019:
1015:
1012:symmetry of *
1011:
1007:
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980:
977:
973:
970:
966:
963:
959:
956:
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937:
933:
927:
925:p3m1, (*333)
923:
920:
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905:
901:
900:
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893:
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874:
871:
869:
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779:3 3 | 3
778:
776:2 | 6 3
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651:, {6,3}. Its
650:
647:
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611:
603:
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591:
589:
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586:Kagome crests
582:
579:
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573:
572:Kagome metals
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541:
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300:
297:is one of 11
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139:3 3 | 3
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78:
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72:
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62:
60:
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47:
46:
41:
36:
31:
19:
2669:
2516:Substitution
2511:Regular grid
2503:
2417:
2350:Quaquaversal
2248:Kisrhombille
2178:Tessellation
2129:
2096:
2090:
2069:
2049:
2039:
2020:
2014:
1994:
1987:
1944:
1941:Phys. Rev. A
1940:
1934:
1904:(5): 424â5.
1901:
1897:
1887:
1847:(5): 443â8.
1844:
1840:
1830:
1779:
1775:
1765:
1754:. Retrieved
1752:. 2019-02-22
1749:
1740:
1697:
1693:
1675:
1624:
1620:
1575:(2): 12â13.
1572:
1568:
1558:
1541:
1538:Physics (ç©ç)
1537:
1531:
1506:. Retrieved
1502:
1492:
1465:
1456:
1435:
1424:
1405:
1396:
1373:
1319:Kagome crest
1241:
1236:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1195:There are 2
1194:
1184:(3.∞)
1169:
1093:Quasiregular
1053:Construction
1038:
1032:
1013:
1005:
997:
995:
934:cmm, (2*22)
931:p31m, (3*3)
916:
914:
889:
724:p3m, (*333)
721:p6m, (*632)
696:
690:
607:
583:
580:
576:SYK behavior
533:
524:
508:
494:
481:
479:
448:
412:
407:
403:
389:
388:
371:hexadeltille
370:
362:
358:
348:
342:
294:
288:
137:2 | 6 3
18:Kagome crest
2546:vertex type
2404:Anisohedral
2359:Self-tiling
2202:Pythagorean
1503:tv.cctv.com
1088:*∞32
1065:Hyperbolic
749:fundamental
584:See also:
486:KĂŽdi Husimi
454:arrangement
369:calls it a
234:, , (*632)
3243:Categories
2450:Pentagonal
1954:2005.07640
1854:1901.04822
1789:1810.00218
1756:2020-04-26
1508:2023-03-20
1345:References
1264:3{12}2 or
1062:Euclidean
928:p3, (333)
669:this group
653:symmetries
517:tetrahedra
501:tetrahedra
468:symmetry,
275:Properties
250:, , (333)
79:r{6,3} or
2558:Spherical
2526:Voderberg
2487:Prototile
2454:Problems
2430:Honeycomb
2408:Isohedral
2295:Aperiodic
2233:honeycomb
2217:Rectangle
2207:Rhombille
2076:CiteSeerX
1979:234363891
1926:128299874
1879:119363372
1822:205570556
1707:0710.1009
1634:0705.0990
1284:6{6}2 or
1058:Spherical
1020:within a
729:Coloring
646:rectified
550:in their
540:jarosites
538:, namely
480:The term
245:, , (632)
2640:V3.4.3.4
2475:Squaring
2470:Heesch's
2435:Isotoxal
2355:Rep-tile
2345:Pinwheel
2238:Coloring
2191:Periodic
2128:(1989).
1814:30209398
1732:14958188
1667:31984687
1659:18643453
1517:cite web
1404:(1979).
1370:(1987).
1308:See also
1239:-gonal.
1095:figures
1085:*832...
875:r{3} = h
868:SchlÀfli
717:Symmetry
592:Symmetry
536:minerals
449:It is a
430:snowshoe
394:Japanese
355:basketry
291:geometry
227:Symmetry
3100:6.4.8.4
3055:5.4.6.4
3015:4.12.16
3005:4.10.12
2975:V4.8.10
2950:V4.6.16
2940:V4.6.14
2840:3.6.4.6
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2720:3.4.3.7
2715:3.4.3.6
2710:3.4.3.5
2665:3.4.6.4
2635:3.4.3.4
2628:regular
2595:Regular
2521:Voronoi
2445:Packing
2376:Truchet
2371:Socolar
2340:Penrose
2335:Gilbert
2260:Wythoff
1959:Bibcode
1906:Bibcode
1859:Bibcode
1794:Bibcode
1712:Bibcode
1639:Bibcode
1577:Bibcode
1484:2410150
872:r{6,3}
785:Coxeter
772:Wythoff
697:cantic
472:(632).
336:is the
301:of the
2990:4.8.16
2985:4.8.14
2980:4.8.12
2970:4.8.10
2945:4.6.16
2935:4.6.14
2930:4.6.12
2700:Hyper-
2685:4.6.12
2458:Domino
2364:Sphinx
2243:Convex
2222:Domino
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432:Tagluk
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377:Kagome
367:Conway
359:kagome
332:. Its
326:vertex
293:, the
127:{6,3}
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2702:bolic
2670:(3.6)
2626:Semi-
2497:Girih
2394:Other
1975:S2CID
1949:arXiv
1922:S2CID
1875:S2CID
1849:arXiv
1818:S2CID
1784:arXiv
1728:S2CID
1702:arXiv
1663:S2CID
1629:arXiv
1180:(3.8)
1175:(3.7)
1170:(3.6)
1165:(3.5)
1160:(3.4)
1155:(3.3)
1082:*732
1079:*632
1076:*532
1073:*432
1070:*332
1037:: (3.
548:atoms
534:Some
462:woven
458:laths
451:woven
427:Inuit
258:That
3190:8.16
3185:8.12
3155:7.14
3125:6.16
3120:6.12
3115:6.10
3075:5.12
3070:5.10
3025:4.16
3020:4.14
3010:4.12
3000:4.10
2860:3.16
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2675:3.12
2660:V3.6
2586:V4.n
2576:V3.n
2463:Wang
2440:List
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2357:and
2316:List
2231:and
2134:ISBN
2101:ISBN
2055:ISBN
2025:ISBN
2000:ISBN
1810:PMID
1655:PMID
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