2715:
1376:
4648:
98:
1959:
2196:
2701:
5867:
1718:
39:
27:
5718:
1866:
the occipital cortex but not in the primary visual cortex. The extrastriate regions included V3A, V4, V7, and the lateral occipital complex (LOC). Electrophysiological studies have found a late posterior negativity that originates from the same areas. In general, a large part of the visual system seems to be involved in processing visual symmetry, and these areas involve similar networks to those responsible for detecting and recognising objects.
2743:
2095:
1862:. The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry is faster when this is a property of a single object. Studies of human perception and psychophysics have shown that detection of symmetry is fast, efficient and robust to perturbations. For example, symmetry can be detected with presentations between 100 and 150 milliseconds.
2687:
2673:
59:
2659:
2729:
4087:"My starting point for this construction was a simple statement which I once read (and which does not necessarily reflect my personal views): 'Only a bad architect relies on symmetry; instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses.'
1849:
made this observation in his book "The analysis of sensations" (1897), and this implies that perception of symmetry is not a general response to all types of regularities. Both behavioural and neurophysiological studies have confirmed the special sensitivity to reflection symmetry in humans and also
1387:
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of
1865:
More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al. used functional magnetic resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots. A strong activity was present in extrastriate regions of
2037:
to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from
2624:
in faces physically attractive; it indicates health and genetic fitness. Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Rudolf
Arnheim suggested that people prefer shapes that have some symmetry, and enough complexity to make them interesting.
2186:
Symmetry is also used in designing logos. By creating a logo on a grid and using the theory of symmetry, designers can organize their work, create a symmetric or asymmetrical design, determine the space between letters, determine how much negative space is required in the design, and how to
1768:
In biology, the notion of symmetry is also used as in physics, that is to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics.
2022:
2055:
4385:
Jones, B. C., Little, A. C., Tiddeman, B. P., Burt, D. M., & Perrett, D. I. (2001). Facial symmetry and judgements of apparent health
Support for a ââ good genes ââ explanation of the attractiveness â symmetry relationship, 22,
1672:(which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language); and also,
6061:
2041:
Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient
1737:
with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.
3430:
2113:
are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.
2046:, for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.
1733:
which divides the body into left and right halves. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The
5625:
1845:
For a human observer, some symmetry types are more salient than others, in particular the most salient is a reflection with a vertical axis, like that present in the human face.
3767:
1914:
message "we are all the same" while asymmetrical interactions may send the message "I am special; better than you." Peer relationships, such as can be governed by the
4866:
3434:
2852:
ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.
1572:, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a
1874:
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of
4647:
1263: 'agreement in dimensions, due proportion, arrangement') in everyday life refers to a sense of harmonious and beautiful proportion and balance. In
2777:
1291:. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
5657:
1918:, are based on symmetry, whereas power relationships are based on asymmetry. Symmetrical relationships can to some degree be maintained by simple (
5652:
4826:
4527:
2074:
have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a
4310:
Grammer, K.; Thornhill, R. (1994). "Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness".
5756:
4806:
1250:
2641:
where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern of
4144:
3865:
1660:, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate
1226:
2330:⊠the other kind of identity. ⊠has to do with axes of symmetry. CâE belongs to a family of symmetrically related dyads as follows:"
3457:
3716:
Barlow, H.B.; Reeves, B.C. (1979). "The versatility and absolute efficiency of detecting mirror symmetry in random dot displays".
3822:
Makin, A.D.J.; Rampone, G.; Pecchinenda, A.; Bertamini, M. (2013). "Electrophysiological responses to visuospatial regularity".
1499:. Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.
4469:
4294:
4173:
4051:
3519:
3350:
3124:
2943:
326:
4101:
2530:
Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are
5437:
6042:
5597:
4486:
3933:
4217:
4206:
5611:
5509:
4416:
2611:
2010:
1616:
1970:
Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic
4520:
4448:
4370:
3591:
3041:
2998:
2411:
Thus in addition to being part of the interval-4 family, CâE is also a part of the sum-4 family (with C equal to 0).
292:
6080:
5749:
4961:
3342:
1676:, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.
3275:
2714:
5096:
4117:
1906:
is the balance that may be attained through deliberative mutual adjustment among general principles and specific
1875:
5672:
5642:
5082:
3178:
1219:
1173:
779:
238:
2002:
are ornamented with complex patterns made using translational and reflection symmetries as well as rotations.
1998:
make elaborate use of symmetry both in their structure and in their ornamentation. Moorish buildings like the
2803:
2798:
1944:
There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.
2038:
ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives.
1749:, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the
1407:
if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.
6131:
6121:
6022:
5722:
5535:
5197:
5047:
5016:
4911:
4858:
4773:
4672:
4513:
3628:
2561:
The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's
2005:
It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";
1953:
1721:
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
5742:
2911:
1375:
1579:
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include
4714:
4699:
3334:
2935:
2164:
2122:
1859:
1673:
1194:
804:
3609:
Symmetries and Group Theory in
Particle Physics: An Introduction to Space-Time and Internal Symmetries
3261:
Symmetries and Group Theory in
Particle Physics: An Introduction to Space-Time and Internal Symmetries
1425:
if it can be simultaneously translated and rotated in three-dimensional space along a line known as a
5846:
5054:
4615:
3015:
1272:
1212:
20:
5647:
5068:
4189:
4027:
1303:
1280:
181:
5856:
3140:
1668:
that "it is only slightly overstating the case to say that physics is the study of symmetry." See
5371:
5061:
4704:
4585:
4433:
The
Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
2767:
1569:
1555:
1397:
1268:
607:
287:
144:
79:
2550:
form a link with the cyclic pitch successions in the atonal music of
Modernists such as BartĂłk,
1400:
a plane) going through it which divides it into two pieces that are mirror images of each other.
5662:
5503:
5399:
4780:
2772:
2295:
2006:
1903:
1855:
1791:
interactions between molecules in nature (i.e., via the interaction of natural and human-made
1661:
1580:
1415:
1411:
1284:
1276:
1243:
683:
394:
272:
157:
5561:
4286:
4041:
3671:
Bertamini, M. (2010). "Sensitivity to reflection and translation is modulated by objectness".
2286:(diminished-diminished seventh), are said to lack direction or a sense of forward motion, are
1388:
symmetry is determined by the way the pieces are organized, or by the type of transformation:
5543:
5517:
5365:
5344:
5186:
5103:
4931:
4886:
4812:
4763:
4570:
3924:
3861:"The neural basis of visual symmetry and its role in middle and high-level visual processing"
3543:. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches.
3371:
2931:
2882:
2562:
1441:
1393:
455:
416:
375:
370:
223:
67:
4278:
4141:
2083:
5468:
5339:
5220:
4833:
4071:
3983:
3874:
3776:
3673:
3408:
3380:
3292:
3231:
2793:
2195:
1669:
1418:(moving every point of the object by the same distance) without changing its overall shape.
1267:, the term has a more precise definition and is usually used to refer to an object that is
1123:
1046:
894:
799:
321:
216:
130:
3468:
2558:, and the Vienna school. At the same time, these progressions signal the end of tonality.
8:
6116:
6090:
6050:
5834:
5693:
5618:
5357:
4988:
4607:
4536:
2959:
2762:
2692:
2591:
2148:
2079:
1958:
1939:
1746:
1708:
1680:
1657:
1649:
1643:
1573:
1561:
1514:
1508:
1404:
1370:
1128:
1072:
985:
839:
819:
744:
634:
505:
495:
358:
233:
228:
211:
186:
174:
126:
121:
102:
71:
47:
5009:
3878:
3780:
3384:
3296:
3235:
6028:
5928:
5765:
5698:
5555:
5385:
5181:
5171:
5024:
4896:
4751:
4335:
3743:
3698:
3653:
3564:
3412:
3396:
3316:
3059:
2808:
2706:
2621:
2583:
2551:
1851:
1800:
1796:
1792:
1778:
1653:
1632:
1503:
1288:
1087:
814:
654:
282:
206:
196:
167:
152:
51:
6062:
How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
4891:
3799:
3762:
3641:
1725:
In biology, the notion of symmetry is mostly used explicitly to describe body shapes.
6126:
6111:
6055:
5955:
5950:
5891:
5881:
5819:
5419:
5393:
5275:
4936:
4444:
4366:
4327:
4290:
4279:
4169:
4098:
4047:
3902:
3841:
3804:
3735:
3731:
3690:
3645:
3623:
3587:
3556:
3552:
3515:
3416:
3346:
3308:
3283:
3120:
3037:
2994:
2939:
2700:
2664:
2599:
2595:
2587:
2535:
2271:
2238:. In classical music, Bach used the symmetry concepts of permutation and invariance.
1854:
tradition suggested that bilateral symmetry was one of the key factors in perceptual
1840:
1812:
1692:
1684:
1452:
1158:
946:
924:
849:
708:
434:
363:
255:
201:
162:
5407:
4950:
3821:
3747:
3702:
3320:
2326:. George Perle explains that "CâE, DâFâŻ, EbâG, are different instances of the same
1148:
1077:
874:
784:
5906:
5569:
5260:
5210:
5156:
5111:
5040:
4840:
4620:
4398:
4354:
4339:
4319:
4253:
3995:
3892:
3882:
3833:
3824:
3794:
3784:
3727:
3682:
3657:
3637:
3568:
3548:
3455:
3388:
3300:
3239:
2990:
2971:
2963:
2678:
2327:
2303:
2275:
2231:
1995:
1824:
1596:
1592:
1560:
Generalizing from geometrical symmetry in the previous section, one can say that a
1448:
1422:
1311:
1138:
879:
589:
467:
402:
260:
245:
110:
5265:
5090:
5076:
4466:
3947:
3858:
2813:
2555:
5829:
5703:
5431:
5320:
5235:
5215:
5166:
4756:
4662:
4637:
4575:
4490:
4478:
4473:
4221:
4210:
4148:
4105:
3961:
3928:
3718:
3509:
3404:
3304:
2878:
2828:
2748:
2279:
2199:
1975:
1923:
1820:
1712:
1588:
1465:
1440:
also exhibit a form of scale symmetry, where smaller portions of the fractal are
1307:
561:
424:
267:
250:
191:
97:
75:
5225:
4323:
3165:
The Basic
Writings of Josiah Royce: Logic, loyalty, and community (Google eBook)
1133:
1102:
1036:
884:
829:
764:
6017:
6012:
5945:
5270:
5245:
5191:
5176:
5132:
5003:
4995:
4876:
4595:
4440:
3920:
3859:
Bertamini, M.; Silvanto, J.; Norcia, A.M.; Makin, A.D.J.; Wagemans, J. (2018).
3339:
The
Noether theorems: Invariance and conservation laws in the twentieth century
2823:
2818:
2720:
2547:
2539:
2534:
with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal
2311:
2263:
2043:
1730:
1600:
1520:
1433:
1189:
1097:
1041:
1006:
914:
824:
794:
754:
659:
43:
4498:, BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (
4483:
4257:
4000:
3763:"Symmetry activates extrastriate visual cortex in human and nonhuman primates"
1163:
774:
6105:
6007:
6002:
5688:
5667:
5604:
5456:
5250:
5161:
4982:
4942:
4926:
4847:
4819:
4687:
4677:
4602:
4590:
4214:
4203:
4161:
3760:
3366:
2543:
2323:
2283:
2259:
2251:
2247:
2136:
2071:
2067:
2034:
2026:
1734:
1696:
1608:
1168:
1153:
1082:
899:
859:
809:
584:
547:
514:
352:
348:
5866:
4495:
3789:
3535:
Montévil, Maël; Mossio, Matteo; Pocheville, Arnaud; Longo, Giuseppe (2016).
2906:
1811:. A rigorous understanding of symmetry explains fundamental observations in
1717:
5918:
5841:
5824:
5480:
5379:
5230:
5141:
5136:
5034:
4955:
4871:
4790:
4694:
4667:
4632:
4565:
4274:
4241:
3906:
3845:
3808:
3694:
3649:
3560:
3341:. Sources and Studies in the History of Mathematics and Physical Sciences.
3312:
3029:
2757:
2307:
2235:
2219:
2156:
2099:
1828:
1816:
1808:
1604:
1530:
1342:
1107:
1056:
869:
724:
639:
429:
4331:
2206:
Symmetry is not restricted to the visual arts. Its role in the history of
2127:
Symmetries appear in the design of objects of all kinds. Examples include
38:
6036:
5994:
5913:
5809:
5804:
5529:
5302:
5240:
5151:
5146:
4746:
4709:
4436:
4015:
3897:
3761:
Sasaki, Y.; Vanduffel, W.; Knutsen, T.; Tyler, C.W.; Tootell, R. (2005).
3739:
3369:(1939), "On unitary representations of the inhomogeneous Lorentz group",
2734:
2580:
2315:
2291:
2267:
2227:
2152:
1966:
has bilateral symmetry; from the top (in plan), it has fourfold symmetry.
1927:
1919:
1915:
1795:
molecules with inherently chiral biological systems). The control of the
1750:
1742:
1542:
1326:
1306:; through other kinds of functional transformations; and as an aspect of
1264:
1143:
1016:
834:
769:
697:
669:
644:
3511:
Perspectives on
Organisms: Biological time, Symmetries and Singularities
5983:
5978:
5896:
5799:
5350:
5255:
5205:
4785:
4580:
4359:
Facial Attractiveness: Evolutionary, Cognitive, and Social Perspectives
3887:
3860:
3837:
3400:
2788:
2638:
2634:
2617:
2531:
2299:
2180:
1979:
1846:
1762:
1758:
1652:âthat is, lack of changeâunder any kind of transformation, for example
1612:
1426:
1380:
1001:
980:
970:
960:
919:
864:
759:
749:
649:
500:
4505:
6085:
5988:
5734:
5314:
4901:
4881:
4550:
3244:
3219:
2849:
2287:
2223:
2160:
2132:
2082:
that are reflected across both the horizontal and vertical axes (see
2075:
1991:
1971:
1963:
1784:
1726:
1688:
1353:
1294:
Mathematical symmetry may be observed with respect to the passage of
1256:
1011:
729:
692:
556:
528:
31:
4129:
3536:
3392:
5779:
5127:
4729:
4555:
2987:
Symmetry and Complexity: The Spirit and Beauty of Nonlinear Science
2874:
2783:
2576:
2319:
2255:
2128:
1999:
1911:
1907:
1891:
1883:
1754:
1729:, including humans, are more or less symmetric with respect to the
1620:
1584:
1437:
1330:
1315:
1092:
1051:
1021:
909:
904:
854:
579:
538:
486:
343:
89:
26:
3686:
1528:(â) is not symmetric. Other symmetric logical connectives include
5940:
5935:
5923:
5901:
5886:
5794:
5029:
4768:
4682:
4560:
2870:
2862:
2742:
2643:
2176:
2094:
1982:, and down to the design of individual building elements such as
1899:
1895:
1887:
1879:
1334:
1026:
739:
533:
477:
277:
63:
5626:
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
4166:
Manifold Mirrors: The Crossing Paths of the Arts and Mathematics
3988:
Journal of Interdisciplinary Methodologies and Issues in Science
3456:
Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002).
2246:
Symmetry is also an important consideration in the formation of
2021:
4734:
2063:
1983:
1436:
if it does not change shape when it is expanded or contracted.
1338:
975:
965:
844:
789:
664:
627:
615:
570:
523:
441:
106:
2686:
2210:
touches many aspects of the creation and perception of music.
58:
5973:
4719:
4362:
3534:
3220:"Chirality and Symmetry Measures: A Transdisciplinary Review"
2866:
2207:
2168:
2144:
2110:
2102:
1987:
1319:
1299:
1031:
955:
889:
734:
338:
333:
19:
This article is about the broad concept. For other uses, see
1656:. This concept has become one of the most powerful tools of
1325:
This article describes symmetry from three perspectives: in
5851:
5789:
5784:
4741:
4724:
4168:. Cambridge University Press. pp. 77â78, 83, 89, 103.
4018:. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.
2172:
2140:
1823:. The theory and application of symmetry to these areas of
1804:
1295:
622:
472:
4461:
4132:. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.
3768:
Proceedings of the National Academy of Sciences of the USA
3581:
2728:
4244:(1992). "Symmetry, the twelve-tone scale, and tonality".
3981:
2070:
Indians used bold diagonals and rectangular motifs. Many
2054:
1536:
1346:
1333:, the most familiar type of symmetry for many people; in
3715:
3624:"Characteristics and models of human symmetry detection"
3079:. Prometheus Books. Especially chapter 12. Nontechnical.
2202:
triads on the white piano keys are symmetrical to the D.
1455:
symmetry (a combination of a rotation and a reflection).
2066:
and rug patterns spans a variety of cultures. American
78:, three forms of symmetry. This shape is obtained by a
1451:
symmetry (a reflection followed by a translation) and
4190:"How to Design a Perfect Logo with Grid and Symmetry"
3263:. Springer Science & Business Media. p. 112.
2861:
Symmetric objects can be material, such as a person,
3495:
What Shape is a Snowflake? Magical Numbers in Nature
2654:
1396:(line or mirror symmetry) if there is a line (or in
3333:
3206:
Academic Press Dictionary of Science and Technology
3186:
University of Waterloo â School of Computer Science
2187:accentuate parts of the logo to make it stand out.
2013:, relies instead on "wings and balance of masses".
5523:The Drawing of Geometric Patterns in Saracenic Art
4397:
4151:. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.
3075:Stenger, Victor J. (2000) and Mahou Shiro (2007).
3058:
2778:Fixed points of isometry groups in Euclidean space
1803:contributes to the ability of scientists to offer
5658:Goudreau Museum of Mathematics in Art and Science
4118:Marla Mallett Textiles & Tribal Oriental Rugs
4108:. Chinavoc (2007-11-19). Retrieved on 2013-04-16.
4039:
2258:music being made up of non-symmetrical groups of
1834:
1648:Symmetry in physics has been generalized to mean
6103:
4414:
4309:
3056:
4352:
3537:"Theoretical principles for biology: Variation"
2016:
3179:"Propositional Logic: Introduction and Syntax"
3103:Conceptual Foundations of Quantum Field Theory
1741:Plants and sessile (attached) animals such as
5750:
5653:European Society for Mathematics and the Arts
4827:Mathematica: A World of Numbers... and Beyond
4521:
4273:
4046:. Princeton University Press. pp. 269â.
4028:Aslaksen: Mathematics in Art and Architecture
3670:
3507:
3217:
1549:
1220:
16:Mathematical invariance under transformations
4807:List of works designed with the golden ratio
4269:
4267:
4236:
4234:
4232:
4230:
3621:
3541:Progress in Biophysics and Molecular Biology
3497:. Weidenfeld & Nicolson. pp. 64â65.
3492:
2958:
4467:Dutch: Symmetry Around a Point in the Plane
4303:
4120:. The Metropolitan Museum of Art, New York.
4030:. Math.nus.edu.sg. Retrieved on 2013-04-16.
2116:
2062:A long tradition of the use of symmetry in
1850:in other animals. Early studies within the
1356:, which refers to the absence of symmetry.
5757:
5743:
4528:
4514:
4285:. University of California Press. p.
3866:Annals of the New York Academy of Sciences
3467:. McGraw-Hill. p. 139. Archived from
3259:Costa, Giovanni; Fogli, Gianluigi (2012).
3258:
3163:Josiah Royce, Ignas K. Skrupskelis (2005)
2633:Symmetry can be found in various forms in
2294:or tonal center, and have a less specific
2222:constraint by many composers, such as the
1827:draws heavily on the mathematical area of
1227:
1213:
96:
4867:Cathedral of Saint Mary of the Assumption
4379:
4264:
4227:
4072:"Behind the Scenes: Edgar Martins Speaks"
4065:
4063:
3999:
3896:
3886:
3798:
3788:
3243:
3036:. Princeton: Princeton University Press.
2978:
1869:
1664:to write in his widely read 1972 article
1502:In propositional logic, symmetric binary
4462:International Symmetry Association (ISA)
4142:Quate: Exploring Geometry Through Quilts
3508:Longo, Giuseppe; Montévil, Maël (2016).
3273:
3065:. Springer Science & Business Media.
2194:
2093:
2053:
2020:
1957:
1716:
1679:Important symmetries in physics include
1626:
1374:
57:
37:
25:
5413:Vier BĂŒcher von Menschlicher Proportion
4535:
4395:
3919:
3465:Chapter 8: Acoelomate Bilateral Animals
3327:
2984:
2151:. Symmetries are central to the art of
1978:, through the layout of the individual
6104:
5764:
4160:
4069:
4060:
3365:
3114:
3108:
2049:
1787:because it undergirds essentially all
327:Straightedge and compass constructions
5738:
4509:
4419:. University of Tennessee, Knoxville.
4240:
3959:
3582:Lowe, John P; Peterson, Kirk (2005).
3428:
2058:Persian rug with rectangular symmetry
3982:Bouissou, C.; Petitjean, M. (2018).
3606:
3105:Cambridge University Press p.154-155
2835:
2594:are horizontally symmetrical, under
1910:. Symmetrical interactions send the
1654:arbitrary coordinate transformations
6043:The Chemical Basis of Morphogenesis
5598:Journal of Mathematics and the Arts
4417:"Symmetrical Aesthetics of Beowulf"
3934:Stanford Encyclopedia of Philosophy
3176:
3119:. Paris/New York: Masson Springer.
3092:Dover Publications (September 1990)
2968:Symmetry and the Beautiful Universe
2925:
2241:
2175:, carpet-making, and many kinds of
1617:symmetric probability distributions
13:
5612:Making Mathematics with Needlework
4426:
4016:Williams: Symmetry in Architecture
3586:(Third ed.). Academic Press.
3458:"Animal Diversity (Third Edition)"
2904:
2612:Symmetry (physical attractiveness)
2310:have used axes of symmetry and/or
1947:
14:
6143:
5438:I quattro libri dell'architettura
4455:
4404:. University of California Press.
4312:Journal of Comparative Psychology
4070:Dunlap, David W. (31 July 2009).
3013:E. H. Lockwood, R. H. Macmillan,
1623:âthe asymmetry of distributions.
1479:is symmetric if for all elements
1359:
293:Noncommutative algebraic geometry
5865:
5717:
5716:
4962:Self-portrait in a Convex Mirror
4646:
4043:What Science Is and How It Works
3553:10.1016/j.pbiomolbio.2016.08.005
3230:(3): 271â312 (see section 2.9).
2741:
2727:
2713:
2699:
2685:
2671:
2657:
2628:
2616:The relationship of symmetry to
2605:
2084:Klein four-group § Geometry
1799:of molecules produced in modern
4479:Chapman: Aesthetics of Symmetry
4408:
4389:
4346:
4318:(3). Washington, D.C.: 233â42.
4196:
4182:
4154:
4135:
4123:
4111:
4092:
4033:
4021:
4009:
3975:
3953:
3941:
3913:
3852:
3815:
3754:
3709:
3664:
3615:
3600:
3575:
3528:
3501:
3486:
3449:
3433:. AccessScience. Archived from
3422:
3359:
3267:
3252:
3211:
3198:
3170:
3157:
3133:
3095:
3082:
3069:
3019:, London: Cambridge Press, 1978
2213:
2159:in art and craft forms such as
1772:
1540:(not-biconditional, or â»), and
1383:has 3-fold rotational symmetry.
5673:National Museum of Mathematics
5425:Regole generali d'architettura
3611:. Open Court Publishing House.
3204:Christopher G. Morris (1992)
3117:Physics and fractal structures
3115:Gouyet, Jean-François (1996).
3050:
3028:
3022:
3007:
2952:
2919:
2898:
2855:
2842:
2571:
2163:, ceramic tilework such as in
1933:
1835:In psychology and neuroscience
1815:, and in the applied areas of
1364:
686:- / other-dimensional
50:. The yellow region shows the
1:
3642:10.1016/S1364-6613(97)01105-4
2892:
2885:or a series of tones (music).
2804:Symmetry-breaking constraints
2799:Spontaneous symmetry breaking
2637:, a simple example being the
2298:. However, composers such as
2155:and the many applications of
1702:
1637:
1615:, symmetry also manifests as
5198:Garden of Cosmic Speculation
3732:10.1016/0042-6989(79)90154-8
3629:Trends in Cognitive Sciences
3305:10.1126/science.177.4047.393
3208:Gulf Professional Publishing
2218:Symmetry has been used as a
2165:Islamic geometric decoration
2089:
2029:acquire rotational symmetry.
2017:In pottery and metal vessels
1954:Mathematics and architecture
1444:in shape to larger portions.
1352:The opposite of symmetry is
1341:; and in the arts, covering
7:
4324:10.1037/0735-7036.108.3.233
3335:Kosmann-Schwarzbach, Yvette
2912:Online Etymology Dictionary
2650:
2190:
2033:Since the earliest uses of
1807:interventions with minimal
1491:, whenever it is true that
1459:
10:
6148:
4715:Islamic geometric patterns
4099:The Art of Chinese Bronzes
4040:Derry, Gregory N. (2002).
3167:Fordham Univ Press, p. 790
2936:Princeton University Press
2609:
2123:Islamic geometric patterns
2120:
1951:
1937:
1838:
1776:
1753:, the group that includes
1706:
1641:
1630:
1553:
1550:Other areas of mathematics
1524:(â), while the connective
1368:
1249:
18:
6073:
6023:D'Arcy Wentworth Thompson
5966:
5874:
5863:
5772:
5712:
5681:
5635:
5589:
5536:A Mathematician's Apology
5496:
5449:
5332:
5295:
5288:
5120:
4973:
4919:
4910:
4857:
4799:
4655:
4644:
4543:
4415:Jenny Lea Bowman (2009).
4258:10.1080/07494469200640151
4246:Contemporary Music Review
4001:10.18713/JIMIS-230718-4-1
3088:Bottema, O, and B. Roth,
3061:Geometry: Plane and Fancy
3057:Singer, David A. (1998).
2819:Symmetries of polyominoes
2814:Symmetries of polyiamonds
2416:
1783:Symmetry is important to
1447:Other symmetries include
1304:geometric transformations
21:Symmetry (disambiguation)
5648:The Bridges Organization
4396:Arnheim, Rudolf (1969).
3962:"The Symmetry Principle"
3925:"Reflective Equilibrium"
2620:is complex. Humans find
2117:In other arts and crafts
1962:Seen from the side, the
1735:head becomes specialized
1568:with respect to a given
182:Non-Archimedean geometry
5510:The Grammar of Ornament
5462:Nature's Harmonic Unity
5372:De prospectiva pingendi
3790:10.1073/pnas.0500319102
3274:Anderson, P.W. (1972).
3090:Theoretical Kinematics,
2985:Mainzer, Klaus (2005).
2314:in an analogous way to
2274:or chords, such as the
1858:. This is known as the
1674:Wigner's classification
1556:Symmetry in mathematics
1495:, it is also true that
288:Noncommutative geometry
80:finite subdivision rule
5663:Institute For Figuring
5575:The 'Life' of a Carpet
5400:A Treatise on Painting
4281:The Listening Composer
4130:Dilucchio: Navajo Rugs
3984:"Asymmetric Exchanges"
3218:Petitjean, M. (2003).
2773:Even and odd functions
2296:diatonic functionality
2203:
2106:
2059:
2030:
2025:Clay pots thrown on a
2007:Modernist architecture
1990:buildings such as the
1967:
1904:Reflective equilibrium
1870:In social interactions
1722:
1699:of physical theories.
1581:even and odd functions
1570:mathematical operation
1412:translational symmetry
1384:
256:Discrete/Combinatorial
83:
55:
35:
5857:WidmanstÀtten pattern
5544:George David Birkhoff
5518:Ernest Hanbury Hankin
5386:De divina proportione
5366:Piero della Francesca
5345:Leon Battista Alberti
4932:Piero della Francesca
4571:Hyperboloid structure
4202:see ("Fugue No. 21,"
3622:Wagemans, J. (1997).
3493:Stewart, Ian (2001).
3372:Annals of Mathematics
3145:TheFreeDictionary.com
3141:"Rotoreflection Axis"
2932:Princeton, New Jersey
2883:mathematical equation
2598:vertically. See also
2198:
2121:Further information:
2097:
2057:
2024:
1961:
1952:Further information:
1938:Further information:
1922:) strategies seen in
1839:Further information:
1745:often have radial or
1720:
1707:Further information:
1681:continuous symmetries
1631:Further information:
1627:In science and nature
1394:reflectional symmetry
1378:
239:Discrete differential
68:reflectional symmetry
66:-like shape that has
61:
41:
29:
5469:Frederik Macody Lund
5340:Filippo Brunelleschi
5221:Hamid Naderi Yeganeh
5083:La condition humaine
4355:Zebrowitz, Leslie A.
3948:Emotional Competency
3607:Mach, Ernst (1897).
3429:Valentine, James W.
2881:structure such as a
2794:Spacetime symmetries
2780:â center of symmetry
1300:spatial relationship
30:Symmetry (left) and
6132:Theoretical physics
6122:Artistic techniques
6091:Mathematics and art
6081:Pattern recognition
6051:Aristid Lindenmayer
5694:Mathematical beauty
5619:Rhythm of Structure
5562:Gödel, Escher, Bach
5358:De re aedificatoria
4989:The Ancient of Days
4608:Projective geometry
4537:Mathematics and art
3879:2018NYASA1426..111B
3781:2005PNAS..102.3159S
3385:1939AnMat..40..149W
3297:1972Sci...177..393A
3276:"More is Different"
3236:2003Entrp...5..271P
3177:Gao, Alice (2019).
2693:Architecture portal
2149:musical instruments
2050:In carpets and rugs
2011:International style
1940:Mathematics and art
1747:rotational symmetry
1709:symmetry in biology
1693:internal symmetries
1685:discrete symmetries
1658:theoretical physics
1644:Symmetry in physics
1562:mathematical object
1504:logical connectives
1405:rotational symmetry
1371:Symmetry (geometry)
506:Pythagorean theorem
72:rotational symmetry
48:octahedral symmetry
6029:On Growth and Form
5929:Logarithmic spiral
5766:Patterns in nature
5699:Patterns in nature
5556:Douglas Hofstadter
5182:Desmond Paul Henry
5172:Bathsheba Grossman
5104:The Swallow's Tail
5025:Giorgio de Chirico
4897:Sydney Opera House
4752:Croatian interlace
4489:2009-09-22 at the
4472:2004-01-02 at the
4220:2005-10-26 at the
4209:2005-09-13 at the
4147:2003-12-31 at the
4104:2003-12-11 at the
3960:Lutus, P. (2008).
3888:10.1111/nyas.13667
3838:10.1111/psyp.12082
3437:on 18 January 2008
3016:Geometric Symmetry
2877:, or it can be an
2809:Symmetric relation
2707:Mathematics portal
2622:bilateral symmetry
2552:Alexander Scriabin
2542:composers such as
2272:Symmetrical scales
2204:
2107:
2060:
2031:
1968:
1801:chemical synthesis
1779:Molecular symmetry
1723:
1695:of particles; and
1633:Patterns in nature
1597:symmetric matrices
1385:
84:
56:
52:fundamental domain
36:
6099:
6098:
6056:BenoĂźt Mandelbrot
5956:Self-organization
5892:Natural selection
5882:Pattern formation
5732:
5731:
5585:
5584:
5549:Aesthetic Measure
5420:Sebastiano Serlio
5394:Leonardo da Vinci
5284:
5283:
5276:Margaret Wertheim
4937:Leonardo da Vinci
4353:Rhodes, Gillian;
4296:978-0-520-06991-6
4175:978-0-521-72876-8
4053:978-1-4008-2311-6
4006:(see appendix 1)
3832:(10): 1045â1055.
3584:Quantum Chemistry
3521:978-3-662-51229-6
3352:978-0-387-87867-6
3291:(4047): 393â396.
3126:978-0-387-94153-0
2945:978-0-691-13482-6
2905:Harper, Douglas.
2836:Explanatory notes
2665:Philosophy portal
2600:Asymmetric rhythm
2528:
2527:
2409:
2408:
2254:, traditional or
2224:arch (swell) form
1841:Visual perception
1813:quantum chemistry
1727:Bilateral animals
1670:Noether's theorem
1666:More is Different
1534:(not-and, or âŒ),
1237:
1236:
1202:
1201:
925:List of geometers
608:Three-dimensional
597:
596:
6139:
5907:Sexual selection
5869:
5759:
5752:
5745:
5736:
5735:
5720:
5719:
5570:Nikos Salingaros
5293:
5292:
5261:Hiroshi Sugimoto
5211:Robert Longhurst
5157:Helaman Ferguson
5112:Crockett Johnson
5041:Circle Limit III
5010:Danseuse au café
4917:
4916:
4887:Pyramid of Khufu
4650:
4530:
4523:
4516:
4507:
4506:
4502:, Apr. 19, 2007)
4421:
4420:
4412:
4406:
4405:
4403:
4393:
4387:
4383:
4377:
4376:
4350:
4344:
4343:
4307:
4301:
4300:
4284:
4271:
4262:
4261:
4238:
4225:
4200:
4194:
4193:
4186:
4180:
4179:
4158:
4152:
4139:
4133:
4127:
4121:
4115:
4109:
4096:
4090:
4089:
4084:
4082:
4067:
4058:
4057:
4037:
4031:
4025:
4019:
4013:
4007:
4005:
4003:
3979:
3973:
3972:
3970:
3968:
3957:
3951:
3945:
3939:
3938:
3929:Zalta, Edward N.
3917:
3911:
3910:
3900:
3890:
3856:
3850:
3849:
3825:Psychophysiology
3819:
3813:
3812:
3802:
3792:
3775:(8): 3159â3163.
3758:
3752:
3751:
3713:
3707:
3706:
3668:
3662:
3661:
3619:
3613:
3612:
3604:
3598:
3597:
3579:
3573:
3572:
3532:
3526:
3525:
3505:
3499:
3498:
3490:
3484:
3483:
3481:
3479:
3473:
3462:
3453:
3447:
3446:
3444:
3442:
3426:
3420:
3419:
3363:
3357:
3356:
3331:
3325:
3324:
3280:
3271:
3265:
3264:
3256:
3250:
3249:
3247:
3245:10.3390/e5030271
3215:
3209:
3202:
3196:
3195:
3193:
3192:
3183:
3174:
3168:
3161:
3155:
3154:
3152:
3151:
3137:
3131:
3130:
3112:
3106:
3099:
3093:
3086:
3080:
3077:Timeless Reality
3073:
3067:
3066:
3064:
3054:
3048:
3047:
3026:
3020:
3011:
3005:
3004:
2991:World Scientific
2982:
2976:
2975:
2972:Prometheus Books
2956:
2950:
2949:
2928:Fearful Symmetry
2926:Zee, A. (2007).
2923:
2917:
2916:
2902:
2886:
2859:
2853:
2846:
2763:Burnside's lemma
2751:
2746:
2745:
2737:
2732:
2731:
2723:
2718:
2717:
2709:
2704:
2703:
2695:
2690:
2689:
2681:
2679:Astronomy portal
2676:
2675:
2674:
2667:
2662:
2661:
2660:
2538:in the works of
2414:
2413:
2333:
2332:
2282:, or diminished
2276:whole tone scale
2242:Pitch structures
2226:(ABCBA) used by
2009:, starting with
1996:Lotfollah mosque
1825:physical science
1593:abstract algebra
1589:symmetric groups
1546:(not-or, or âœ).
1449:glide reflection
1423:helical symmetry
1312:theoretic models
1308:abstract objects
1260:
1253:
1229:
1222:
1215:
943:
942:
462:
461:
395:Zero-dimensional
100:
86:
85:
6147:
6146:
6142:
6141:
6140:
6138:
6137:
6136:
6102:
6101:
6100:
6095:
6069:
5962:
5870:
5861:
5768:
5763:
5733:
5728:
5708:
5704:Sacred geometry
5677:
5643:Ars Mathematica
5631:
5581:
5492:
5445:
5432:Andrea Palladio
5328:
5321:De architectura
5280:
5236:Antoine Pevsner
5216:Jeanette McLeod
5167:Susan Goldstine
5116:
4975:
4969:
4906:
4892:Sagrada FamĂlia
4853:
4795:
4663:Algorithmic art
4651:
4642:
4638:Wallpaper group
4576:Minimal surface
4539:
4534:
4491:Wayback Machine
4474:Wayback Machine
4458:
4429:
4427:Further reading
4424:
4413:
4409:
4400:Visual Thinking
4394:
4390:
4384:
4380:
4373:
4351:
4347:
4308:
4304:
4297:
4272:
4265:
4239:
4228:
4222:Wayback Machine
4211:Wayback Machine
4201:
4197:
4188:
4187:
4183:
4176:
4159:
4155:
4149:Wayback Machine
4140:
4136:
4128:
4124:
4116:
4112:
4106:Wayback Machine
4097:
4093:
4080:
4078:
4068:
4061:
4054:
4038:
4034:
4026:
4022:
4014:
4010:
3980:
3976:
3966:
3964:
3958:
3954:
3946:
3942:
3921:Daniels, Norman
3918:
3914:
3857:
3853:
3820:
3816:
3759:
3755:
3719:Vision Research
3714:
3710:
3669:
3665:
3620:
3616:
3605:
3601:
3594:
3580:
3576:
3533:
3529:
3522:
3506:
3502:
3491:
3487:
3477:
3475:
3474:on May 17, 2016
3471:
3460:
3454:
3450:
3440:
3438:
3427:
3423:
3393:10.2307/1968551
3364:
3360:
3353:
3343:Springer-Verlag
3332:
3328:
3278:
3272:
3268:
3257:
3253:
3216:
3212:
3203:
3199:
3190:
3188:
3181:
3175:
3171:
3162:
3158:
3149:
3147:
3139:
3138:
3134:
3127:
3113:
3109:
3100:
3096:
3087:
3083:
3074:
3070:
3055:
3051:
3044:
3027:
3023:
3012:
3008:
3001:
2983:
2979:
2964:Lederman, L. M.
2957:
2953:
2946:
2924:
2920:
2903:
2899:
2895:
2890:
2889:
2860:
2856:
2847:
2843:
2838:
2833:
2829:Wallpaper group
2749:Clothing portal
2747:
2740:
2733:
2726:
2719:
2712:
2705:
2698:
2691:
2684:
2677:
2672:
2670:
2663:
2658:
2656:
2653:
2631:
2614:
2608:
2574:
2312:interval cycles
2280:augmented chord
2244:
2216:
2200:Major and minor
2193:
2125:
2119:
2092:
2052:
2019:
1976:The White House
1956:
1950:
1948:In architecture
1942:
1936:
1924:symmetric games
1872:
1860:Law of Symmetry
1843:
1837:
1821:crystallography
1781:
1775:
1715:
1713:facial symmetry
1705:
1646:
1640:
1635:
1629:
1558:
1552:
1512:(â§, or &),
1466:dyadic relation
1462:
1373:
1367:
1362:
1273:transformations
1233:
1204:
1203:
940:
939:
930:
929:
720:
719:
703:
702:
688:
687:
675:
674:
611:
610:
599:
598:
459:
458:
456:Two-dimensional
447:
446:
420:
419:
417:One-dimensional
408:
407:
398:
397:
386:
385:
319:
318:
317:
300:
299:
148:
147:
136:
113:
76:self-similarity
24:
17:
12:
11:
5:
6145:
6135:
6134:
6129:
6124:
6119:
6114:
6097:
6096:
6094:
6093:
6088:
6083:
6077:
6075:
6071:
6070:
6068:
6067:
6066:
6065:
6053:
6048:
6047:
6046:
6034:
6033:
6032:
6020:
6018:Wilson Bentley
6015:
6013:Joseph Plateau
6010:
6005:
6000:
5999:
5998:
5986:
5981:
5976:
5970:
5968:
5964:
5963:
5961:
5960:
5959:
5958:
5953:
5951:Plateau's laws
5948:
5946:Fluid dynamics
5943:
5933:
5932:
5931:
5926:
5921:
5911:
5910:
5909:
5904:
5899:
5894:
5884:
5878:
5876:
5872:
5871:
5864:
5862:
5860:
5859:
5854:
5849:
5844:
5839:
5838:
5837:
5832:
5827:
5822:
5812:
5807:
5802:
5797:
5792:
5787:
5782:
5776:
5774:
5770:
5769:
5762:
5761:
5754:
5747:
5739:
5730:
5729:
5727:
5726:
5713:
5710:
5709:
5707:
5706:
5701:
5696:
5691:
5685:
5683:
5679:
5678:
5676:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5639:
5637:
5633:
5632:
5630:
5629:
5622:
5615:
5608:
5601:
5593:
5591:
5587:
5586:
5583:
5582:
5580:
5579:
5578:
5577:
5567:
5566:
5565:
5553:
5552:
5551:
5541:
5540:
5539:
5527:
5526:
5525:
5515:
5514:
5513:
5500:
5498:
5494:
5493:
5491:
5490:
5489:
5488:
5486:The Greek Vase
5478:
5477:
5476:
5466:
5465:
5464:
5453:
5451:
5447:
5446:
5444:
5443:
5442:
5441:
5429:
5428:
5427:
5417:
5416:
5415:
5408:Albrecht DĂŒrer
5405:
5404:
5403:
5391:
5390:
5389:
5377:
5376:
5375:
5363:
5362:
5361:
5354:
5342:
5336:
5334:
5330:
5329:
5327:
5326:
5325:
5324:
5312:
5311:
5310:
5299:
5297:
5290:
5286:
5285:
5282:
5281:
5279:
5278:
5273:
5271:Roman Verostko
5268:
5263:
5258:
5253:
5248:
5246:Alba Rojo Cama
5243:
5238:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5202:
5201:
5192:Charles Jencks
5189:
5184:
5179:
5177:George W. Hart
5174:
5169:
5164:
5159:
5154:
5149:
5144:
5139:
5130:
5124:
5122:
5118:
5117:
5115:
5114:
5109:
5108:
5107:
5100:
5088:
5087:
5086:
5074:
5073:
5072:
5065:
5058:
5051:
5044:
5032:
5027:
5022:
5021:
5020:
5013:
5004:Jean Metzinger
5001:
5000:
4999:
4992:
4979:
4977:
4971:
4970:
4968:
4967:
4966:
4965:
4953:
4951:Albrecht DĂŒrer
4948:
4947:
4946:
4934:
4929:
4923:
4921:
4914:
4908:
4907:
4905:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4863:
4861:
4855:
4854:
4852:
4851:
4844:
4837:
4830:
4823:
4816:
4809:
4803:
4801:
4797:
4796:
4794:
4793:
4788:
4783:
4778:
4777:
4776:
4766:
4761:
4760:
4759:
4754:
4749:
4739:
4738:
4737:
4732:
4727:
4722:
4712:
4707:
4702:
4697:
4692:
4691:
4690:
4685:
4680:
4670:
4668:Anamorphic art
4665:
4659:
4657:
4653:
4652:
4645:
4643:
4641:
4640:
4635:
4630:
4625:
4624:
4623:
4618:
4610:
4605:
4600:
4599:
4598:
4596:Camera obscura
4593:
4583:
4578:
4573:
4568:
4563:
4558:
4553:
4547:
4545:
4541:
4540:
4533:
4532:
4525:
4518:
4510:
4504:
4503:
4493:
4481:
4476:
4464:
4457:
4456:External links
4454:
4453:
4452:
4441:Souvenir Press
4428:
4425:
4423:
4422:
4407:
4388:
4378:
4371:
4345:
4302:
4295:
4263:
4226:
4195:
4181:
4174:
4162:Cucker, Felipe
4153:
4134:
4122:
4110:
4091:
4076:New York Times
4059:
4052:
4032:
4020:
4008:
3974:
3952:
3940:
3923:(2003-04-28).
3912:
3873:(1): 280â293.
3851:
3814:
3753:
3726:(7): 783â793.
3708:
3663:
3636:(9): 346â352.
3614:
3599:
3592:
3574:
3527:
3520:
3500:
3485:
3448:
3421:
3379:(1): 149â204,
3358:
3351:
3326:
3266:
3251:
3210:
3197:
3169:
3156:
3132:
3125:
3107:
3094:
3081:
3068:
3049:
3042:
3021:
3006:
2999:
2977:
2951:
2944:
2918:
2896:
2894:
2891:
2888:
2887:
2854:
2840:
2839:
2837:
2834:
2832:
2831:
2826:
2824:Symmetry group
2821:
2816:
2811:
2806:
2801:
2796:
2791:
2786:
2781:
2775:
2770:
2765:
2760:
2754:
2753:
2752:
2738:
2724:
2721:Physics portal
2710:
2696:
2682:
2668:
2652:
2649:
2630:
2627:
2610:Main article:
2607:
2604:
2573:
2570:
2548:Richard Wagner
2526:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2495:
2491:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2465:
2463:
2460:
2458:
2454:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2428:
2426:
2423:
2421:
2418:
2407:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2381:
2379:
2376:
2374:
2370:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2344:
2342:
2339:
2337:
2264:diatonic scale
2262:, such as the
2243:
2240:
2215:
2212:
2192:
2189:
2137:sand paintings
2118:
2115:
2091:
2088:
2051:
2048:
2035:pottery wheels
2018:
2015:
1949:
1946:
1935:
1932:
1871:
1868:
1836:
1833:
1777:Main article:
1774:
1771:
1731:sagittal plane
1704:
1701:
1642:Main article:
1639:
1636:
1628:
1625:
1601:linear algebra
1554:Main article:
1551:
1548:
1521:if and only if
1518:(âš, or |) and
1461:
1458:
1457:
1456:
1453:rotoreflection
1445:
1434:scale symmetry
1432:An object has
1430:
1421:An object has
1419:
1410:An object has
1408:
1403:An object has
1401:
1392:An object has
1369:Main article:
1366:
1363:
1361:
1360:In mathematics
1358:
1235:
1234:
1232:
1231:
1224:
1217:
1209:
1206:
1205:
1200:
1199:
1198:
1197:
1192:
1184:
1183:
1179:
1178:
1177:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1118:
1117:
1113:
1112:
1111:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1067:
1066:
1062:
1061:
1060:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
996:
995:
991:
990:
989:
988:
983:
978:
973:
968:
963:
958:
950:
949:
941:
937:
936:
935:
932:
931:
928:
927:
922:
917:
912:
907:
902:
897:
892:
887:
882:
877:
872:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
721:
717:
716:
715:
712:
711:
705:
704:
701:
700:
695:
689:
682:
681:
680:
677:
676:
673:
672:
667:
662:
660:Platonic Solid
657:
652:
647:
642:
637:
632:
631:
630:
619:
618:
612:
606:
605:
604:
601:
600:
595:
594:
593:
592:
587:
582:
574:
573:
567:
566:
565:
564:
559:
551:
550:
544:
543:
542:
541:
536:
531:
526:
518:
517:
511:
510:
509:
508:
503:
498:
490:
489:
483:
482:
481:
480:
475:
470:
460:
454:
453:
452:
449:
448:
445:
444:
439:
438:
437:
432:
421:
415:
414:
413:
410:
409:
406:
405:
399:
393:
392:
391:
388:
387:
384:
383:
378:
373:
367:
366:
361:
356:
346:
341:
336:
330:
329:
320:
316:
315:
312:
308:
307:
306:
305:
302:
301:
298:
297:
296:
295:
285:
280:
275:
270:
265:
264:
263:
253:
248:
243:
242:
241:
236:
231:
221:
220:
219:
214:
204:
199:
194:
189:
184:
179:
178:
177:
172:
171:
170:
155:
149:
143:
142:
141:
138:
137:
135:
134:
124:
118:
115:
114:
101:
93:
92:
44:symmetry group
15:
9:
6:
4:
3:
2:
6144:
6133:
6130:
6128:
6125:
6123:
6120:
6118:
6115:
6113:
6110:
6109:
6107:
6092:
6089:
6087:
6084:
6082:
6079:
6078:
6076:
6072:
6064:
6063:
6059:
6058:
6057:
6054:
6052:
6049:
6045:
6044:
6040:
6039:
6038:
6035:
6031:
6030:
6026:
6025:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6008:Ernst Haeckel
6006:
6004:
6003:Adolf Zeising
6001:
5997:
5996:
5992:
5991:
5990:
5987:
5985:
5982:
5980:
5977:
5975:
5972:
5971:
5969:
5965:
5957:
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5938:
5937:
5934:
5930:
5927:
5925:
5922:
5920:
5917:
5916:
5915:
5912:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5889:
5888:
5885:
5883:
5880:
5879:
5877:
5873:
5868:
5858:
5855:
5853:
5850:
5848:
5847:Vortex street
5845:
5843:
5840:
5836:
5833:
5831:
5828:
5826:
5825:Quasicrystals
5823:
5821:
5818:
5817:
5816:
5813:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5777:
5775:
5771:
5767:
5760:
5755:
5753:
5748:
5746:
5741:
5740:
5737:
5725:
5724:
5715:
5714:
5711:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5689:Droste effect
5687:
5686:
5684:
5680:
5674:
5671:
5669:
5668:Mathemalchemy
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5640:
5638:
5636:Organizations
5634:
5628:
5627:
5623:
5621:
5620:
5616:
5614:
5613:
5609:
5607:
5606:
5605:Lumen Naturae
5602:
5600:
5599:
5595:
5594:
5592:
5588:
5576:
5573:
5572:
5571:
5568:
5564:
5563:
5559:
5558:
5557:
5554:
5550:
5547:
5546:
5545:
5542:
5538:
5537:
5533:
5532:
5531:
5528:
5524:
5521:
5520:
5519:
5516:
5512:
5511:
5507:
5506:
5505:
5502:
5501:
5499:
5495:
5487:
5484:
5483:
5482:
5479:
5475:
5472:
5471:
5470:
5467:
5463:
5460:
5459:
5458:
5457:Samuel Colman
5455:
5454:
5452:
5448:
5440:
5439:
5435:
5434:
5433:
5430:
5426:
5423:
5422:
5421:
5418:
5414:
5411:
5410:
5409:
5406:
5402:
5401:
5397:
5396:
5395:
5392:
5388:
5387:
5383:
5382:
5381:
5378:
5374:
5373:
5369:
5368:
5367:
5364:
5360:
5359:
5355:
5353:
5352:
5348:
5347:
5346:
5343:
5341:
5338:
5337:
5335:
5331:
5323:
5322:
5318:
5317:
5316:
5313:
5309:
5306:
5305:
5304:
5301:
5300:
5298:
5294:
5291:
5287:
5277:
5274:
5272:
5269:
5267:
5266:Daina TaimiĆa
5264:
5262:
5259:
5257:
5254:
5252:
5251:Reza Sarhangi
5249:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5200:
5199:
5195:
5194:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5162:Peter Forakis
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5134:
5131:
5129:
5126:
5125:
5123:
5119:
5113:
5110:
5106:
5105:
5101:
5099:
5098:
5094:
5093:
5092:
5091:Salvador DalĂ
5089:
5085:
5084:
5080:
5079:
5078:
5077:René Magritte
5075:
5071:
5070:
5066:
5064:
5063:
5059:
5057:
5056:
5052:
5050:
5049:
5048:Print Gallery
5045:
5043:
5042:
5038:
5037:
5036:
5033:
5031:
5028:
5026:
5023:
5019:
5018:
5017:L'Oiseau bleu
5014:
5012:
5011:
5007:
5006:
5005:
5002:
4998:
4997:
4993:
4991:
4990:
4986:
4985:
4984:
4983:William Blake
4981:
4980:
4978:
4972:
4964:
4963:
4959:
4958:
4957:
4954:
4952:
4949:
4945:
4944:
4943:Vitruvian Man
4940:
4939:
4938:
4935:
4933:
4930:
4928:
4927:Paolo Uccello
4925:
4924:
4922:
4918:
4915:
4913:
4909:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4864:
4862:
4860:
4856:
4850:
4849:
4848:Pi in the Sky
4845:
4843:
4842:
4838:
4836:
4835:
4831:
4829:
4828:
4824:
4822:
4821:
4820:Mathemalchemy
4817:
4815:
4814:
4810:
4808:
4805:
4804:
4802:
4798:
4792:
4789:
4787:
4784:
4782:
4779:
4775:
4772:
4771:
4770:
4767:
4765:
4762:
4758:
4755:
4753:
4750:
4748:
4745:
4744:
4743:
4740:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4717:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4689:
4688:Vastu shastra
4686:
4684:
4681:
4679:
4678:Geodesic dome
4676:
4675:
4674:
4671:
4669:
4666:
4664:
4661:
4660:
4658:
4654:
4649:
4639:
4636:
4634:
4631:
4629:
4626:
4622:
4619:
4617:
4614:
4613:
4611:
4609:
4606:
4604:
4603:Plastic ratio
4601:
4597:
4594:
4592:
4591:Camera lucida
4589:
4588:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4548:
4546:
4542:
4538:
4531:
4526:
4524:
4519:
4517:
4512:
4511:
4508:
4501:
4497:
4494:
4492:
4488:
4485:
4484:ISIS Symmetry
4482:
4480:
4477:
4475:
4471:
4468:
4465:
4463:
4460:
4459:
4450:
4449:0-285-63743-6
4446:
4442:
4438:
4434:
4431:
4430:
4418:
4411:
4402:
4401:
4392:
4382:
4374:
4372:1-56750-636-4
4368:
4364:
4360:
4356:
4349:
4341:
4337:
4333:
4329:
4325:
4321:
4317:
4313:
4306:
4298:
4292:
4288:
4283:
4282:
4276:
4275:Perle, George
4270:
4268:
4259:
4255:
4251:
4247:
4243:
4242:Perle, George
4237:
4235:
4233:
4231:
4223:
4219:
4216:
4212:
4208:
4205:
4199:
4191:
4185:
4177:
4171:
4167:
4163:
4157:
4150:
4146:
4143:
4138:
4131:
4126:
4119:
4114:
4107:
4103:
4100:
4095:
4088:
4077:
4073:
4066:
4064:
4055:
4049:
4045:
4044:
4036:
4029:
4024:
4017:
4012:
4002:
3997:
3993:
3989:
3985:
3978:
3963:
3956:
3949:
3944:
3936:
3935:
3930:
3926:
3922:
3916:
3908:
3904:
3899:
3898:11577/3289328
3894:
3889:
3884:
3880:
3876:
3872:
3868:
3867:
3862:
3855:
3847:
3843:
3839:
3835:
3831:
3827:
3826:
3818:
3810:
3806:
3801:
3796:
3791:
3786:
3782:
3778:
3774:
3770:
3769:
3764:
3757:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3720:
3712:
3704:
3700:
3696:
3692:
3688:
3687:10.1068/p6393
3684:
3680:
3676:
3675:
3667:
3659:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3630:
3625:
3618:
3610:
3603:
3595:
3593:0-12-457551-X
3589:
3585:
3578:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3531:
3523:
3517:
3513:
3512:
3504:
3496:
3489:
3470:
3466:
3459:
3452:
3436:
3432:
3425:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3373:
3368:
3367:Wigner, E. P.
3362:
3354:
3348:
3344:
3340:
3336:
3330:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3285:
3277:
3270:
3262:
3255:
3246:
3241:
3237:
3233:
3229:
3225:
3221:
3214:
3207:
3201:
3187:
3180:
3173:
3166:
3160:
3146:
3142:
3136:
3128:
3122:
3118:
3111:
3104:
3098:
3091:
3085:
3078:
3072:
3063:
3062:
3053:
3045:
3043:0-691-02374-3
3039:
3035:
3031:
3030:Weyl, Hermann
3025:
3018:
3017:
3010:
3002:
3000:981-256-192-7
2996:
2992:
2988:
2981:
2973:
2969:
2965:
2961:
2955:
2947:
2941:
2937:
2933:
2929:
2922:
2914:
2913:
2908:
2901:
2897:
2884:
2880:
2876:
2872:
2868:
2864:
2858:
2851:
2848:For example,
2845:
2841:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2750:
2744:
2739:
2736:
2730:
2725:
2722:
2716:
2711:
2708:
2702:
2697:
2694:
2688:
2683:
2680:
2669:
2666:
2655:
2648:
2646:
2645:
2640:
2636:
2629:In literature
2626:
2623:
2619:
2613:
2606:In aesthetics
2603:
2601:
2597:
2593:
2589:
2585:
2582:
2578:
2569:
2567:
2565:
2559:
2557:
2556:Edgard VarĂšse
2553:
2549:
2545:
2544:Gustav Mahler
2541:
2537:
2533:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2492:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2455:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2415:
2412:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2371:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2334:
2331:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2289:
2285:
2284:seventh chord
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2239:
2237:
2233:
2229:
2225:
2221:
2211:
2209:
2201:
2197:
2188:
2184:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2124:
2114:
2112:
2104:
2101:
2096:
2087:
2085:
2081:
2077:
2073:
2072:Oriental rugs
2069:
2065:
2056:
2047:
2045:
2039:
2036:
2028:
2027:pottery wheel
2023:
2014:
2012:
2008:
2003:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1965:
1960:
1955:
1945:
1941:
1931:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1867:
1863:
1861:
1857:
1853:
1848:
1842:
1832:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1780:
1770:
1766:
1764:
1760:
1756:
1752:
1748:
1744:
1739:
1736:
1732:
1728:
1719:
1714:
1710:
1700:
1698:
1697:supersymmetry
1694:
1690:
1686:
1682:
1677:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1645:
1634:
1624:
1622:
1618:
1614:
1610:
1609:Galois theory
1606:
1605:Galois groups
1602:
1598:
1594:
1590:
1586:
1582:
1577:
1575:
1571:
1567:
1563:
1557:
1547:
1545:
1544:
1539:
1538:
1533:
1532:
1527:
1523:
1522:
1517:
1516:
1511:
1510:
1505:
1500:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1467:
1454:
1450:
1446:
1443:
1439:
1435:
1431:
1428:
1424:
1420:
1417:
1414:if it can be
1413:
1409:
1406:
1402:
1399:
1395:
1391:
1390:
1389:
1382:
1377:
1372:
1357:
1355:
1350:
1349:, and music.
1348:
1344:
1340:
1336:
1332:
1328:
1323:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1292:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1259:
1252:
1248:
1245:
1244:Ancient Greek
1241:
1230:
1225:
1223:
1218:
1216:
1211:
1210:
1208:
1207:
1196:
1193:
1191:
1188:
1187:
1186:
1185:
1181:
1180:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1121:
1120:
1119:
1115:
1114:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1070:
1069:
1068:
1064:
1063:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
999:
998:
997:
993:
992:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
953:
952:
951:
948:
945:
944:
934:
933:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
722:
714:
713:
710:
707:
706:
699:
696:
694:
691:
690:
685:
679:
678:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
629:
626:
625:
624:
621:
620:
617:
614:
613:
609:
603:
602:
591:
588:
586:
585:Circumference
583:
581:
578:
577:
576:
575:
572:
569:
568:
563:
560:
558:
555:
554:
553:
552:
549:
548:Quadrilateral
546:
545:
540:
537:
535:
532:
530:
527:
525:
522:
521:
520:
519:
516:
515:Parallelogram
513:
512:
507:
504:
502:
499:
497:
494:
493:
492:
491:
488:
485:
484:
479:
476:
474:
471:
469:
466:
465:
464:
463:
457:
451:
450:
443:
440:
436:
433:
431:
428:
427:
426:
423:
422:
418:
412:
411:
404:
401:
400:
396:
390:
389:
382:
379:
377:
374:
372:
369:
368:
365:
362:
360:
357:
354:
353:Perpendicular
350:
349:Orthogonality
347:
345:
342:
340:
337:
335:
332:
331:
328:
325:
324:
323:
313:
310:
309:
304:
303:
294:
291:
290:
289:
286:
284:
281:
279:
276:
274:
273:Computational
271:
269:
266:
262:
259:
258:
257:
254:
252:
249:
247:
244:
240:
237:
235:
232:
230:
227:
226:
225:
222:
218:
215:
213:
210:
209:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
176:
173:
169:
166:
165:
164:
161:
160:
159:
158:Non-Euclidean
156:
154:
151:
150:
146:
140:
139:
132:
128:
125:
123:
120:
119:
117:
116:
112:
108:
104:
99:
95:
94:
91:
88:
87:
81:
77:
73:
69:
65:
60:
53:
49:
45:
40:
33:
28:
22:
6060:
6041:
6027:
5993:
5919:Chaos theory
5842:Tessellation
5814:
5721:
5624:
5617:
5610:
5603:
5596:
5590:Publications
5574:
5560:
5548:
5534:
5522:
5508:
5485:
5481:Jay Hambidge
5474:Ad Quadratum
5473:
5461:
5436:
5424:
5412:
5398:
5384:
5380:Luca Pacioli
5370:
5356:
5349:
5319:
5307:
5231:Hinke Osinga
5226:IstvĂĄn Orosz
5196:
5187:Anthony Hill
5142:Scott Draves
5137:Erik Demaine
5121:Contemporary
5102:
5095:
5081:
5067:
5060:
5053:
5046:
5039:
5035:M. C. Escher
5015:
5008:
4994:
4987:
4960:
4956:Parmigianino
4941:
4872:Hagia Sophia
4846:
4839:
4832:
4825:
4818:
4811:
4695:Computer art
4673:Architecture
4633:Tessellation
4627:
4616:Architecture
4566:Golden ratio
4499:
4432:
4410:
4399:
4391:
4381:
4358:
4348:
4315:
4311:
4305:
4280:
4252:(2): 81â96.
4249:
4245:
4198:
4184:
4165:
4156:
4137:
4125:
4113:
4094:
4086:
4079:. Retrieved
4075:
4042:
4035:
4023:
4011:
3991:
3987:
3977:
3967:28 September
3965:. Retrieved
3955:
3943:
3932:
3915:
3870:
3864:
3854:
3829:
3823:
3817:
3772:
3766:
3756:
3723:
3717:
3711:
3681:(1): 27â40.
3678:
3672:
3666:
3633:
3627:
3617:
3608:
3602:
3583:
3577:
3547:(1): 36â50.
3544:
3540:
3530:
3514:. Springer.
3510:
3503:
3494:
3488:
3476:. Retrieved
3469:the original
3464:
3451:
3439:. Retrieved
3435:the original
3424:
3376:
3370:
3361:
3338:
3329:
3288:
3282:
3269:
3260:
3254:
3227:
3223:
3213:
3205:
3200:
3189:. Retrieved
3185:
3172:
3164:
3159:
3148:. Retrieved
3144:
3135:
3116:
3110:
3102:
3101:Tian Yu Cao
3097:
3089:
3084:
3076:
3071:
3060:
3052:
3033:
3024:
3014:
3009:
2986:
2980:
2967:
2954:
2927:
2921:
2910:
2900:
2857:
2844:
2758:Automorphism
2642:
2632:
2615:
2575:
2563:
2560:
2536:progressions
2529:
2467:
2430:
2410:
2383:
2346:
2308:George Perle
2245:
2236:James Tenney
2217:
2214:Musical form
2205:
2185:
2157:tessellation
2126:
2108:
2100:kaleidoscope
2061:
2040:
2032:
2004:
1984:tile mosaics
1969:
1943:
1873:
1864:
1844:
1829:group theory
1817:spectroscopy
1809:side effects
1788:
1782:
1773:In chemistry
1767:
1743:sea anemones
1740:
1724:
1678:
1665:
1647:
1578:
1565:
1559:
1541:
1535:
1529:
1525:
1519:
1513:
1507:
1501:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1468:
1463:
1386:
1351:
1343:architecture
1329:, including
1324:
1310:, including
1293:
1257:
1254:
1247:
1239:
1238:
1057:Parameshvara
870:Parameshvara
640:Dodecahedron
380:
224:Differential
42:A spherical
6037:Alan Turing
5995:Liber Abaci
5914:Mathematics
5820:in crystals
5810:Soap bubble
5805:Phyllotaxis
5530:G. H. Hardy
5333:Renaissance
5303:Polykleitos
5241:Tony Robbin
5152:John Ernest
5147:Jan Dibbets
5097:Crucifixion
4920:Renaissance
4774:Mathematics
4747:Celtic knot
4710:Fractal art
4612:Proportion
4586:Perspective
4500:In Our Time
4437:Mario Livio
4081:11 November
3478:October 25,
3431:"Bilateria"
2960:Hill, C. T.
2871:floor tiles
2735:Star portal
2581:pitch class
2572:Equivalency
2304:BĂ©la BartĂłk
2268:major chord
2232:BĂ©la BartĂłk
2228:Steve Reich
2153:M.C. Escher
1980:floor plans
1934:In the arts
1928:tit for tat
1920:game theory
1916:Golden Rule
1894:, respect,
1876:reciprocity
1805:therapeutic
1759:sea urchins
1751:echinoderms
1662:PW Anderson
1365:In geometry
1327:mathematics
1277:translation
1271:under some
1265:mathematics
1182:Present day
1129:Lobachevsky
1116:1700sâ1900s
1073:JyeáčŁáčhadeva
1065:1400sâ1700s
1017:Brahmagupta
840:Lobachevsky
820:JyeáčŁáčhadeva
770:Brahmagupta
698:Hypersphere
670:Tetrahedron
645:Icosahedron
217:Diophantine
6117:Aesthetics
6106:Categories
5984:Empedocles
5979:Pythagoras
5897:Camouflage
5835:in biology
5830:in flowers
5800:Parastichy
5504:Owen Jones
5351:De pictura
5256:Oliver Sin
5206:Andy Lomas
5055:Relativity
4786:String art
4700:Fiber arts
4581:Paraboloid
3950:: Symmetry
3674:Perception
3191:2019-11-12
3150:2019-11-12
2907:"symmetry"
2893:References
2789:Palindrome
2639:palindrome
2635:literature
2618:aesthetics
2592:retrograde
2586:which are
2532:enharmonic
2300:Alban Berg
2290:as to the
2183:patterns.
2181:embroidery
2078:âthat is,
1972:cathedrals
1847:Ernst Mach
1763:sea lilies
1703:In biology
1650:invariance
1638:In physics
1613:statistics
1427:screw axis
1416:translated
1381:triskelion
1302:; through
1281:reflection
1275:, such as
1242:(from
1042:al-Yasamin
986:Apollonius
981:Archimedes
971:Pythagoras
961:Baudhayana
915:al-Yasamin
865:Pythagoras
760:Baudhayana
750:Archimedes
745:Apollonius
650:Octahedron
501:Hypotenuse
376:Similarity
371:Congruence
283:Incidence
234:Symplectic
229:Riemannian
212:Arithmetic
187:Projective
175:Hyperbolic
103:Projecting
6086:Emergence
5989:Fibonacci
5315:Vitruvius
5289:Theorists
5069:Waterfall
4974:19thâ20th
4902:Taj Mahal
4882:Parthenon
4859:Buildings
4813:Continuum
4781:Sculpture
4757:Interlace
4551:Algorithm
4215:Shockwave
3417:121773411
3032:(1982) .
2850:Aristotle
2768:Chirality
2596:inversion
2588:invariant
2577:Tone rows
2288:ambiguous
2161:wallpaper
2133:furniture
2090:In quilts
2076:rectangle
1992:Taj Mahal
1964:Taj Mahal
1908:judgments
1785:chemistry
1689:spacetime
1619:, and as
1566:symmetric
1354:asymmetry
1269:invariant
1258:summetrĂa
1251:ÏÏ
ΌΌΔÏÏία
1159:Minkowski
1078:Descartes
1012:Aryabhata
1007:KÄtyÄyana
938:by period
850:Minkowski
825:KÄtyÄyana
785:Descartes
730:Aryabhata
709:Geometers
693:Tesseract
557:Trapezoid
529:Rectangle
322:Dimension
207:Algebraic
197:Synthetic
168:Spherical
153:Euclidean
32:asymmetry
6127:Geometry
6112:Symmetry
5815:Symmetry
5773:Patterns
5723:Category
5450:Romantic
5128:Max Bill
5062:Reptiles
4877:Pantheon
4834:Octacube
4800:Artworks
4742:Knotting
4730:Muqarnas
4628:Symmetry
4556:Catenary
4544:Concepts
4496:Symmetry
4487:Archived
4470:Archived
4443:, 2006,
4386:417â429.
4357:(2002).
4277:(1990).
4218:Archived
4207:Archived
4164:(2013).
4145:Archived
4102:Archived
3994:: 1â18.
3907:29604083
3846:23941638
3809:15710884
3748:41530752
3703:22451173
3695:20301844
3650:21223945
3561:27530930
3337:(2010).
3321:34548824
3313:17796623
3034:Symmetry
2966:(2005).
2879:abstract
2875:molecule
2784:Isotropy
2651:See also
2568:(1910).
2540:Romantic
2328:interval
2191:In music
2141:knotwork
2129:beadwork
2098:Kitchen
2000:Alhambra
1994:and the
1926:such as
1892:dialogue
1884:sympathy
1856:grouping
1797:symmetry
1789:specific
1755:starfish
1621:skewness
1585:calculus
1506:include
1460:In logic
1438:Fractals
1331:geometry
1316:language
1285:rotation
1240:Symmetry
1149:Poincaré
1093:Minggatu
1052:Yang Hui
1022:Virasena
910:Yang Hui
905:Virasena
875:Poincaré
855:Minggatu
635:Cylinder
580:Diameter
539:Rhomboid
496:Altitude
487:Triangle
381:Symmetry
359:Parallel
344:Diagonal
314:Features
311:Concepts
202:Analytic
163:Elliptic
145:Branches
131:Timeline
90:Geometry
6074:Related
5941:Crystal
5936:Physics
5924:Fractal
5902:Mimicry
5887:Biology
5795:Meander
5682:Related
5296:Ancient
5030:Man Ray
4976:Century
4912:Artists
4769:Origami
4683:Pyramid
4561:Fractal
4340:1205083
4332:7924253
3931:(ed.).
3875:Bibcode
3777:Bibcode
3658:2143353
3569:3671068
3409:1503456
3401:1968551
3381:Bibcode
3293:Bibcode
3284:Science
3232:Bibcode
3224:Entropy
2863:crystal
2644:Beowulf
2566:, Op. 3
2564:Quartet
2324:centers
2318:or non-
2266:or the
2260:pitches
2177:textile
2044:Chinese
1988:Islamic
1900:revenge
1896:justice
1888:apology
1880:empathy
1852:Gestalt
1442:similar
1335:science
1298:; as a
1289:scaling
1174:Coxeter
1154:Hilbert
1139:Riemann
1088:Huygens
1047:al-Tusi
1037:KhayyĂĄm
1027:Alhazen
994:1â1400s
895:al-Tusi
880:Riemann
830:KhayyĂĄm
815:Huygens
810:Hilbert
780:Coxeter
740:Alhazen
718:by name
655:Pyramid
534:Rhombus
478:Polygon
430:segment
278:Fractal
261:Digital
246:Complex
127:History
122:Outline
64:fractal
34:(right)
5967:People
5875:Causes
5497:Modern
5133:Martin
4996:Newton
4791:Tiling
4735:Zellij
4705:4D art
4447:
4369:
4338:
4330:
4293:
4172:
4050:
3905:
3844:
3807:
3800:549500
3797:
3746:
3740:483597
3738:
3701:
3693:
3656:
3648:
3590:
3567:
3559:
3518:
3441:29 May
3415:
3407:
3399:
3349:
3319:
3311:
3123:
3040:
2997:
2942:
2590:under
2322:tonal
2306:, and
2252:chords
2248:scales
2234:, and
2220:formal
2147:, and
2111:quilts
2080:motifs
2068:Navajo
2064:carpet
1898:, and
1793:chiral
1761:, and
1603:, and
1339:nature
1318:, and
1195:Gromov
1190:Atiyah
1169:Veblen
1164:Cartan
1134:Bolyai
1103:Sakabe
1083:Pascal
976:Euclid
966:Manava
900:Veblen
885:Sakabe
860:Pascal
845:Manava
805:Gromov
790:Euclid
775:Cartan
765:Bolyai
755:Atiyah
665:Sphere
628:cuboid
616:Volume
571:Circle
524:Square
442:Length
364:Vertex
268:Convex
251:Finite
192:Affine
107:sphere
5974:Plato
5780:Crack
5308:Canon
4764:Music
4720:Girih
4656:Forms
4621:Human
4363:Ablex
4336:S2CID
3927:. In
3744:S2CID
3699:S2CID
3654:S2CID
3565:S2CID
3472:(PDF)
3461:(PDF)
3413:S2CID
3397:JSTOR
3317:S2CID
3279:(PDF)
3182:(PDF)
2873:, or
2867:quilt
2320:tonal
2256:tonal
2208:music
2169:batik
2145:masks
2105:block
2103:quilt
1912:moral
1611:. In
1574:group
1320:music
1287:, or
1246:
1144:Klein
1124:Gauss
1098:Euler
1032:Sijzi
1002:Zhang
956:Ahmes
920:Zhang
890:Sijzi
835:Klein
800:Gauss
795:Euler
735:Ahmes
468:Plane
403:Point
339:Curve
334:Angle
111:plane
109:to a
46:with
5852:Wave
5790:Foam
5785:Dune
5135:and
4725:Jali
4445:ISBN
4367:ISBN
4328:PMID
4291:ISBN
4170:ISBN
4083:2014
4048:ISBN
3969:2015
3903:PMID
3842:PMID
3805:PMID
3736:PMID
3691:PMID
3646:PMID
3588:ISBN
3557:PMID
3516:ISBN
3480:2012
3443:2013
3347:ISBN
3309:PMID
3121:ISBN
3038:ISBN
2995:ISBN
2940:ISBN
2584:sets
2546:and
2316:keys
2250:and
2179:and
2173:ikat
1974:and
1819:and
1711:and
1683:and
1531:nand
1379:The
1337:and
1296:time
1108:Aida
725:Aida
684:Four
623:Cube
590:Area
562:Kite
473:Area
425:Line
74:and
4320:doi
4316:108
4254:doi
4213:or
4204:pdf
3996:doi
3893:hdl
3883:doi
3871:132
3834:doi
3795:PMC
3785:doi
3773:102
3728:doi
3683:doi
3638:doi
3549:doi
3545:122
3389:doi
3301:doi
3289:177
3240:doi
2579:or
2479:10
2474:11
2405:GâŻ
2395:AâŻ
2378:CâŻ
2368:GâŻ
2358:FâŻ
2341:DâŻ
2292:key
2109:As
2086:).
1687:of
1607:in
1599:in
1591:in
1583:in
1564:is
1543:nor
1537:xor
1509:and
1497:Rba
1493:Rab
1487:in
1347:art
947:BCE
435:ray
6108::
4841:Pi
4439:,
4435:,
4365:.
4361:.
4334:.
4326:.
4314:.
4289:.
4287:21
4266:^
4248:.
4229:^
4085:.
4074:.
4062:^
3990:.
3986:.
3901:.
3891:.
3881:.
3869:.
3863:.
3840:.
3830:50
3828:.
3803:.
3793:.
3783:.
3771:.
3765:.
3742:.
3734:.
3724:19
3722:.
3697:.
3689:.
3679:39
3677:.
3652:.
3644:.
3632:.
3626:.
3563:.
3555:.
3539:.
3463:.
3411:,
3405:MR
3403:,
3395:,
3387:,
3377:40
3375:,
3345:.
3315:.
3307:.
3299:.
3287:.
3281:.
3238:.
3226:.
3222:.
3184:.
3143:.
2993:.
2989:.
2970:.
2962:;
2938:.
2934::
2930:.
2909:.
2869:,
2865:,
2647:.
2602:.
2554:,
2524:4
2519:4
2514:4
2509:4
2504:4
2499:4
2494:4
2489:8
2484:9
2462:1
2457:2
2452:8
2447:7
2442:6
2437:5
2425:3
2420:2
2417:+
2400:A
2390:B
2373:D
2363:G
2353:F
2336:D
2302:,
2278:,
2270:.
2230:,
2171:,
2167:,
2143:,
2139:,
2135:,
2131:,
1986:.
1930:.
1902:.
1890:,
1886:,
1882:,
1878:,
1831:.
1765:.
1757:,
1691:;
1595:,
1587:,
1576:.
1526:if
1515:or
1483:,
1475:Ă
1471:=
1464:A
1398:3D
1345:,
1322:.
1314:,
1283:,
1279:,
105:a
70:,
62:A
5758:e
5751:t
5744:v
4529:e
4522:t
4515:v
4451:.
4375:.
4342:.
4322::
4299:.
4260:.
4256::
4250:6
4224:)
4192:.
4178:.
4056:.
4004:.
3998::
3992:4
3971:.
3937:.
3909:.
3895::
3885::
3877::
3848:.
3836::
3811:.
3787::
3779::
3750:.
3730::
3705:.
3685::
3660:.
3640::
3634:1
3596:.
3571:.
3551::
3524:.
3482:.
3445:.
3391::
3383::
3355:.
3323:.
3303::
3295::
3248:.
3242::
3234::
3228:5
3194:.
3153:.
3129:.
3046:.
3003:.
2974:.
2948:.
2915:.
2468:0
2431:4
2384:C
2347:E
1489:S
1485:b
1481:a
1477:S
1473:S
1469:R
1429:.
1261:)
1255:(
1228:e
1221:t
1214:v
355:)
351:(
133:)
129:(
82:.
54:.
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
.svg)
