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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.
2754:
2106:
1873:. 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.
2698:
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4098:"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.'
1860:
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
1398:
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
1876:
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
2048:
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
2635:
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.
2197:
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
1779:
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.
2033:
2066:
4396:
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,
1683:(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,
6072:
2052:
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
1748:
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.
3441:
2124:
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.
2057:, 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.
1744:
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
5636:
1856:
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.
3778:
1925:
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
4877:
3445:
2863:
ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.
1583:, 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
1885:
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of
4658:
1274: 'agreement in dimensions, due proportion, arrangement') in everyday life refers to a sense of harmonious and beautiful proportion and balance. In
2788:
1302:. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
5668:
1929:, are based on symmetry, whereas power relationships are based on asymmetry. Symmetrical relationships can to some degree be maintained by simple (
5663:
4837:
4538:
2085:
have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a
4321:
Grammer, K.; Thornhill, R. (1994). "Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness".
5767:
4817:
1261:
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where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern of
4155:
3876:
1671:, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate
1237:
2341:⊠the other kind of identity. ⊠has to do with axes of symmetry. CâE belongs to a family of symmetrically related dyads as follows:"
3468:
3727:
Barlow, H.B.; Reeves, B.C. (1979). "The versatility and absolute efficiency of detecting mirror symmetry in random dot displays".
3833:
Makin, A.D.J.; Rampone, G.; Pecchinenda, A.; Bertamini, M. (2013). "Electrophysiological responses to visuospatial regularity".
1510:. 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.
4480:
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Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are
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Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic
4531:
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2422:
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).
303:
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1687:, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.
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1917:
is the balance that may be attained through deliberative mutual adjustment among general principles and specific
1886:
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2013:
are ornamented with complex patterns made using translational and reflection symmetries as well as rotations.
2009:
make elaborate use of symmetry both in their structure and in their ornamentation. Moorish buildings like the
2814:
2809:
1955:
There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.
2049:
ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives.
1760:, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the
1418:
if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.
6142:
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The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's
2016:
It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";
1964:
1732:
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
5753:
2922:
1386:
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In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include
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2946:
2175:
2133:
1870:
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1205:
815:
17:
3620:
Symmetries and Group Theory in
Particle Physics: An Introduction to Space-Time and Internal Symmetries
3272:
Symmetries and Group Theory in
Particle Physics: An Introduction to Space-Time and Internal Symmetries
1436:
if it can be simultaneously translated and rotated in three-dimensional space along a line known as a
5857:
5065:
4626:
3026:
1283:
1223:
31:
5658:
5079:
4200:
4038:
1314:
1291:
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that "it is only slightly overstating the case to say that physics is the study of symmetry." See
5382:
5072:
4715:
4596:
4444:
The
Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
2778:
1580:
1566:
1408:
1279:
618:
298:
155:
90:
2561:
form a link with the cyclic pitch successions in the atonal music of
Modernists such as BartĂłk,
1411:
a plane) going through it which divides it into two pieces that are mirror images of each other.
5673:
5514:
5410:
4791:
2783:
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2017:
1914:
1866:
1802:
interactions between molecules in nature (i.e., via the interaction of natural and human-made
1672:
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1422:
1295:
1287:
1254:
694:
405:
283:
168:
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4052:
3682:
Bertamini, M. (2010). "Sensitivity to reflection and translation is modulated by objectness".
2297:(diminished-diminished seventh), are said to lack direction or a sense of forward motion, are
1399:
symmetry is determined by the way the pieces are organized, or by the type of transformation:
5554:
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5376:
5355:
5197:
5114:
4942:
4897:
4823:
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3935:
3872:"The neural basis of visual symmetry and its role in middle and high-level visual processing"
3554:. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches.
3382:
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2804:
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1680:
1429:(moving every point of the object by the same distance) without changing its overall shape.
1278:, the term has a more precise definition and is usually used to refer to an object that is
1134:
1057:
905:
810:
332:
227:
141:
3479:
2569:, and the Vienna school. At the same time, these progressions signal the end of tonality.
8:
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62:
6073:
How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
4902:
3810:
3773:
3652:
1736:
In biology, the notion of symmetry is mostly used explicitly to describe body shapes.
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6122:
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2249:. In classical music, Bach used the symmetry concepts of permutation and invariance.
1865:
tradition suggested that bilateral symmetry was one of the key factors in perceptual
1851:
1823:
1703:
1695:
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1169:
957:
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860:
719:
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266:
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173:
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2337:. George Perle explains that "CâE, DâFâŻ, EbâG, are different instances of the same
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1088:
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3311:
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3001:
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2974:
2689:
2338:
2314:
2286:
2242:
2006:
1835:
1607:
1603:
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Generalizing from geometrical symmetry in the previous section, one can say that a
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600:
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121:
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5331:
5246:
5226:
5177:
4767:
4673:
4648:
4586:
4501:
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4484:
4232:
4221:
4159:
4116:
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3939:
3729:
3520:
3415:
3315:
2889:
2839:
2759:
2290:
2210:
1986:
1934:
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also exhibit a form of scale symmetry, where smaller portions of the fractal are
1318:
572:
435:
278:
261:
202:
108:
86:
5236:
4334:
3176:
The Basic
Writings of Josiah Royce: Logic, loyalty, and community (Google eBook)
1144:
1113:
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775:
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6023:
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5202:
5187:
5143:
5014:
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4887:
4606:
4451:
3931:
3870:
Bertamini, M.; Silvanto, J.; Norcia, A.M.; Makin, A.D.J.; Wagemans, J. (2018).
3350:
The
Noether theorems: Invariance and conservation laws in the twentieth century
2834:
2829:
2731:
2558:
2550:
2545:
with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal
2322:
2274:
2054:
1741:
1611:
1531:
1444:
1200:
1108:
1052:
1017:
925:
835:
805:
765:
670:
54:
4509:, BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (
4494:
4268:
4011:
3774:"Symmetry activates extrastriate visual cortex in human and nonhuman primates"
1174:
785:
6116:
6018:
6013:
5699:
5678:
5615:
5467:
5261:
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4993:
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4858:
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4601:
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4172:
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2334:
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2045:
2037:
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1707:
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870:
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525:
363:
359:
5877:
4506:
3800:
3546:
Montévil, Maël; Mossio, Matteo; Pocheville, Arnaud; Longo, Giuseppe (2016).
2917:
1822:. A rigorous understanding of symmetry explains fundamental observations in
1728:
5929:
5852:
5835:
5491:
5390:
5241:
5152:
5147:
5045:
4966:
4882:
4801:
4705:
4678:
4643:
4576:
4285:
4252:
3917:
3856:
3819:
3705:
3660:
3571:
3352:. Sources and Studies in the History of Mathematics and Physical Sciences.
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2110:
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1827:
1819:
1615:
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880:
735:
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Symmetry is not restricted to the visual arts. Its role in the history of
2138:
Symmetries appear in the design of objects of all kinds. Examples include
49:
6047:
6005:
5924:
5820:
5815:
5540:
5313:
5251:
5162:
5157:
4757:
4720:
4447:
4026:
3908:
3772:
Sasaki, Y.; Vanduffel, W.; Knutsen, T.; Tyler, C.W.; Tootell, R. (2005).
3750:
3380:(1939), "On unitary representations of the inhomogeneous Lorentz group",
2745:
2591:
2326:
2302:
2278:
2238:
2163:
1977:
has bilateral symmetry; from the top (in plan), it has fourfold symmetry.
1938:
1930:
1926:
1806:
molecules with inherently chiral biological systems). The control of the
1761:
1753:
1553:
1337:
1317:; through other kinds of functional transformations; and as an aspect of
1275:
1154:
1027:
845:
780:
708:
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655:
3522:
Perspectives on
Organisms: Biological time, Symmetries and Singularities
5994:
5989:
5907:
5810:
5361:
5266:
5216:
4796:
4591:
4370:
Facial Attractiveness: Evolutionary, Cognitive, and Social Perspectives
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3848:
3411:
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2649:
2645:
2628:
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2310:
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1990:
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1663:âthat is, lack of changeâunder any kind of transformation, for example
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that are reflected across both the horizontal and vertical axes (see
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2002:
1982:
1974:
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1305:
Mathematical symmetry may be observed with respect to the passage of
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703:
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539:
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Symmetry and Complexity: The Spirit and Beauty of Nonlinear Science
2885:
2794:
2587:
2330:
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2139:
2010:
1922:
1918:
1902:
1894:
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1740:, including humans, are more or less symmetric with respect to the
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1539:(â) is not symmetric. Other symmetric logical connectives include
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1993:, and down to the design of individual building elements such as
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1906:
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1890:
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544:
488:
288:
74:
5637:
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
4177:
Manifold Mirrors: The Crossing Paths of the Arts and Mathematics
3999:
Journal of Interdisciplinary Methodologies and Issues in Science
3467:
Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002).
2257:
Symmetry is also an important consideration in the formation of
2032:
4745:
2074:
1994:
1447:
if it does not change shape when it is expanded or contracted.
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986:
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touches many aspects of the creation and perception of music.
69:
5984:
4730:
4373:
3545:
3231:"Chirality and Symmetry Measures: A Transdisciplinary Review"
2877:
2218:
2179:
2155:
2121:
2113:
1998:
1330:
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900:
745:
349:
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30:
This article is about the broad concept. For other uses, see
1667:. This concept has become one of the most powerful tools of
1336:
This article describes symmetry from three perspectives: in
5862:
5800:
5795:
4752:
4735:
4179:. Cambridge University Press. pp. 77â78, 83, 89, 103.
4029:. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.
2183:
2151:
1834:. The theory and application of symmetry to these areas of
1815:
1306:
633:
483:
4472:
4143:. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.
3779:
Proceedings of the National Academy of Sciences of the USA
3592:
2739:
4255:(1992). "Symmetry, the twelve-tone scale, and tonality".
3992:
2081:
Indians used bold diagonals and rectangular motifs. Many
2065:
1547:
1357:
1344:, the most familiar type of symmetry for many people; in
3726:
3635:"Characteristics and models of human symmetry detection"
3090:. Prometheus Books. Especially chapter 12. Nontechnical.
2213:
triads on the white piano keys are symmetrical to the D.
1466:
symmetry (a combination of a rotation and a reflection).
2077:
and rug patterns spans a variety of cultures. American
89:, three forms of symmetry. This shape is obtained by a
1462:
symmetry (a reflection followed by a translation) and
4201:"How to Design a Perfect Logo with Grid and Symmetry"
3274:. Springer Science & Business Media. p. 112.
2872:
Symmetric objects can be material, such as a person,
3506:
What Shape is a Snowflake? Magical Numbers in Nature
2665:
1407:(line or mirror symmetry) if there is a line (or in
3344:
3217:
Academic Press Dictionary of Science and Technology
3197:
University of Waterloo â School of Computer Science
2198:accentuate parts of the logo to make it stand out.
2024:, relies instead on "wings and balance of masses".
5534:The Drawing of Geometric Patterns in Saracenic Art
4408:
4162:. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.
3086:Stenger, Victor J. (2000) and Mahou Shiro (2007).
3069:
2789:Fixed points of isometry groups in Euclidean space
1814:contributes to the ability of scientists to offer
5669:Goudreau Museum of Mathematics in Art and Science
4129:Marla Mallett Textiles & Tribal Oriental Rugs
4119:. Chinavoc (2007-11-19). Retrieved on 2013-04-16.
4050:
2269:music being made up of non-symmetrical groups of
1845:
1659:Symmetry in physics has been generalized to mean
6114:
4425:
4320:
3067:
4363:
3548:"Theoretical principles for biology: Variation"
2027:
3190:"Propositional Logic: Introduction and Syntax"
3114:Conceptual Foundations of Quantum Field Theory
1752:Plants and sessile (attached) animals such as
5761:
5664:European Society for Mathematics and the Arts
4838:Mathematica: A World of Numbers... and Beyond
4532:
4284:
4057:. Princeton University Press. pp. 269â.
4039:Aslaksen: Mathematics in Art and Architecture
3681:
3518:
3228:
1560:
1231:
27:Mathematical invariance under transformations
4818:List of works designed with the golden ratio
4280:
4278:
4247:
4245:
4243:
4241:
3632:
3552:Progress in Biophysics and Molecular Biology
3508:. Weidenfeld & Nicolson. pp. 64â65.
3503:
2969:
4478:Dutch: Symmetry Around a Point in the Plane
4314:
4131:. The Metropolitan Museum of Art, New York.
4041:. Math.nus.edu.sg. Retrieved on 2013-04-16.
2127:
2073:A long tradition of the use of symmetry in
1861:in other animals. Early studies within the
1367:, which refers to the absence of symmetry.
5768:
5754:
4539:
4525:
4296:. University of California Press. p.
3877:Annals of the New York Academy of Sciences
3478:. McGraw-Hill. p. 139. Archived from
3270:Costa, Giovanni; Fogli, Gianluigi (2012).
3269:
3174:Josiah Royce, Ignas K. Skrupskelis (2005)
2644:Symmetry can be found in various forms in
2305:or tonal center, and have a less specific
2233:constraint by many composers, such as the
1838:draws heavily on the mathematical area of
1238:
1224:
107:
4878:Cathedral of Saint Mary of the Assumption
4390:
4275:
4238:
4083:"Behind the Scenes: Edgar Martins Speaks"
4076:
4074:
4010:
3907:
3897:
3809:
3799:
3254:
3047:. Princeton: Princeton University Press.
2989:
1880:
1675:to write in his widely read 1972 article
1513:In propositional logic, symmetric binary
4473:International Symmetry Association (ISA)
4153:Quate: Exploring Geometry Through Quilts
3519:Longo, Giuseppe; Montévil, Maël (2016).
3284:
3076:. Springer Science & Business Media.
2205:
2104:
2064:
2031:
1968:
1727:
1690:Important symmetries in physics include
1637:
1385:
68:
48:
36:
5424:Vier BĂŒcher von Menschlicher Proportion
4546:
4406:
3930:
3476:Chapter 8: Acoelomate Bilateral Animals
3338:
2995:
2162:. Symmetries are central to the art of
1989:, through the layout of the individual
14:
6115:
5775:
4171:
4080:
4071:
3376:
3125:
3119:
2060:
1798:because it undergirds essentially all
338:Straightedge and compass constructions
5749:
4520:
4430:. University of Tennessee, Knoxville.
4251:
3970:
3593:Lowe, John P; Peterson, Kirk (2005).
3439:
2069:Persian rug with rectangular symmetry
3993:Bouissou, C.; Petitjean, M. (2018).
3617:
3116:Cambridge University Press p.154-155
2846:
2605:are horizontally symmetrical, under
1921:. Symmetrical interactions send the
1665:arbitrary coordinate transformations
6054:The Chemical Basis of Morphogenesis
5609:Journal of Mathematics and the Arts
4428:"Symmetrical Aesthetics of Beowulf"
3945:Stanford Encyclopedia of Philosophy
3187:
3130:. Paris/New York: Masson Springer.
3103:Dover Publications (September 1990)
2979:Symmetry and the Beautiful Universe
2936:
2252:
2186:, carpet-making, and many kinds of
1628:symmetric probability distributions
24:
5623:Making Mathematics with Needlework
4437:
4027:Williams: Symmetry in Architecture
3597:(Third ed.). Academic Press.
3469:"Animal Diversity (Third Edition)"
2915:
2623:Symmetry (physical attractiveness)
2321:have used axes of symmetry and/or
1958:
25:
6154:
5449:I quattro libri dell'architettura
4466:
4415:. University of California Press.
4323:Journal of Comparative Psychology
4081:Dunlap, David W. (31 July 2009).
3024:E. H. Lockwood, R. H. Macmillan,
1634:âthe asymmetry of distributions.
1490:is symmetric if for all elements
1370:
304:Noncommutative algebraic geometry
5876:
5728:
5727:
4973:Self-portrait in a Convex Mirror
4657:
4054:What Science Is and How It Works
3564:10.1016/j.pbiomolbio.2016.08.005
3241:(3): 271â312 (see section 2.9).
2752:
2738:
2724:
2710:
2696:
2682:
2668:
2639:
2627:The relationship of symmetry to
2616:
2095:Klein four-group § Geometry
1810:of molecules produced in modern
4490:Chapman: Aesthetics of Symmetry
4419:
4400:
4357:
4329:(3). Washington, D.C.: 233â42.
4207:
4193:
4165:
4146:
4134:
4122:
4103:
4044:
4032:
4020:
3986:
3964:
3952:
3924:
3863:
3826:
3765:
3720:
3675:
3626:
3611:
3586:
3539:
3512:
3497:
3460:
3444:. AccessScience. Archived from
3433:
3370:
3278:
3263:
3222:
3209:
3181:
3168:
3144:
3106:
3093:
3080:
3030:, London: Cambridge Press, 1978
2224:
2170:in art and craft forms such as
1783:
1551:(not-biconditional, or â»), and
1394:has 3-fold rotational symmetry.
5684:National Museum of Mathematics
5436:Regole generali d'architettura
3622:. Open Court Publishing House.
3215:Christopher G. Morris (1992)
3128:Physics and fractal structures
3126:Gouyet, Jean-François (1996).
3061:
3039:
3033:
3018:
2963:
2930:
2909:
2866:
2853:
2582:
2174:, ceramic tilework such as in
1944:
1846:In psychology and neuroscience
1826:, and in the applied areas of
1375:
697:- / other-dimensional
61:. The yellow region shows the
13:
1:
3653:10.1016/S1364-6613(97)01105-4
2903:
2896:or a series of tones (music).
2815:Symmetry-breaking constraints
2810:Spontaneous symmetry breaking
2648:, a simple example being the
2309:. However, composers such as
2166:and the many applications of
1713:
1648:
1626:, symmetry also manifests as
5209:Garden of Cosmic Speculation
3743:10.1016/0042-6989(79)90154-8
3640:Trends in Cognitive Sciences
3316:10.1126/science.177.4047.393
3219:Gulf Professional Publishing
2229:Symmetry has been used as a
2176:Islamic geometric decoration
2100:
2040:acquire rotational symmetry.
2028:In pottery and metal vessels
1965:Mathematics and architecture
1455:in shape to larger portions.
1363:The opposite of symmetry is
1352:; and in the arts, covering
7:
4335:10.1037/0735-7036.108.3.233
3346:Kosmann-Schwarzbach, Yvette
2923:Online Etymology Dictionary
2661:
2201:
2044:Since the earliest uses of
1818:interventions with minimal
1502:, whenever it is true that
1470:
10:
6159:
4726:Islamic geometric patterns
4110:The Art of Chinese Bronzes
4051:Derry, Gregory N. (2002).
3178:Fordham Univ Press, p. 790
2947:Princeton University Press
2620:
2134:Islamic geometric patterns
2131:
1962:
1948:
1849:
1787:
1764:, the group that includes
1717:
1652:
1641:
1564:
1561:Other areas of mathematics
1535:(â), while the connective
1379:
1260:
29:
6084:
6034:D'Arcy Wentworth Thompson
5977:
5885:
5874:
5783:
5723:
5692:
5646:
5600:
5547:A Mathematician's Apology
5507:
5460:
5343:
5306:
5299:
5131:
4984:
4930:
4921:
4868:
4810:
4666:
4655:
4554:
4426:Jenny Lea Bowman (2009).
4269:10.1080/07494469200640151
4257:Contemporary Music Review
4012:10.18713/JIMIS-230718-4-1
3099:Bottema, O, and B. Roth,
3072:Geometry: Plane and Fancy
3068:Singer, David A. (1998).
2830:Symmetries of polyominoes
2825:Symmetries of polyiamonds
2427:
1794:Symmetry is important to
1458:Other symmetries include
1315:geometric transformations
32:Symmetry (disambiguation)
5659:The Bridges Organization
4407:Arnheim, Rudolf (1969).
3973:"The Symmetry Principle"
3936:"Reflective Equilibrium"
2631:is complex. Humans find
2128:In other arts and crafts
1973:Seen from the side, the
1746:head becomes specialized
1579:with respect to a given
193:Non-Archimedean geometry
5521:The Grammar of Ornament
5473:Nature's Harmonic Unity
5383:De prospectiva pingendi
3801:10.1073/pnas.0500319102
3285:Anderson, P.W. (1972).
3101:Theoretical Kinematics,
2996:Mainzer, Klaus (2005).
2325:in an analogous way to
2285:or chords, such as the
1869:. This is known as the
1685:Wigner's classification
1567:Symmetry in mathematics
1506:, it is also true that
299:Noncommutative geometry
91:finite subdivision rule
5674:Institute For Figuring
5586:The 'Life' of a Carpet
5411:A Treatise on Painting
4292:The Listening Composer
4141:Dilucchio: Navajo Rugs
3995:"Asymmetric Exchanges"
3229:Petitjean, M. (2003).
2784:Even and odd functions
2307:diatonic functionality
2214:
2117:
2070:
2041:
2036:Clay pots thrown on a
2018:Modernist architecture
2001:buildings such as the
1978:
1915:Reflective equilibrium
1881:In social interactions
1733:
1710:of physical theories.
1592:even and odd functions
1581:mathematical operation
1423:translational symmetry
1395:
267:Discrete/Combinatorial
94:
66:
46:
5868:WidmanstÀtten pattern
5555:George David Birkhoff
5529:Ernest Hanbury Hankin
5397:De divina proportione
5377:Piero della Francesca
5356:Leon Battista Alberti
4943:Piero della Francesca
4582:Hyperboloid structure
4213:see ("Fugue No. 21,"
3633:Wagemans, J. (1997).
3504:Stewart, Ian (2001).
3383:Annals of Mathematics
3156:TheFreeDictionary.com
3152:"Rotoreflection Axis"
2943:Princeton, New Jersey
2894:mathematical equation
2609:vertically. See also
2209:
2132:Further information:
2108:
2068:
2035:
1972:
1963:Further information:
1949:Further information:
1933:) strategies seen in
1850:Further information:
1756:often have radial or
1731:
1718:Further information:
1692:continuous symmetries
1642:Further information:
1638:In science and nature
1405:reflectional symmetry
1389:
250:Discrete differential
79:reflectional symmetry
77:-like shape that has
72:
52:
40:
5480:Frederik Macody Lund
5351:Filippo Brunelleschi
5232:Hamid Naderi Yeganeh
5094:La condition humaine
4366:Zebrowitz, Leslie A.
3959:Emotional Competency
3618:Mach, Ernst (1897).
3440:Valentine, James W.
2892:structure such as a
2805:Spacetime symmetries
2791:â center of symmetry
1311:spatial relationship
41:Symmetry (left) and
6143:Theoretical physics
6133:Artistic techniques
6102:Mathematics and art
6092:Pattern recognition
6062:Aristid Lindenmayer
5705:Mathematical beauty
5630:Rhythm of Structure
5573:Gödel, Escher, Bach
5369:De re aedificatoria
5000:The Ancient of Days
4619:Projective geometry
4548:Mathematics and art
3890:2018NYASA1426..111B
3792:2005PNAS..102.3159S
3396:1939AnMat..40..149W
3308:1972Sci...177..393A
3287:"More is Different"
3247:2003Entrp...5..271P
3188:Gao, Alice (2019).
2704:Architecture portal
2160:musical instruments
2061:In carpets and rugs
2022:International style
1951:Mathematics and art
1758:rotational symmetry
1720:symmetry in biology
1704:internal symmetries
1696:discrete symmetries
1669:theoretical physics
1655:Symmetry in physics
1573:mathematical object
1515:logical connectives
1416:rotational symmetry
1382:Symmetry (geometry)
517:Pythagorean theorem
83:rotational symmetry
59:octahedral symmetry
6040:On Growth and Form
5940:Logarithmic spiral
5777:Patterns in nature
5710:Patterns in nature
5567:Douglas Hofstadter
5193:Desmond Paul Henry
5183:Bathsheba Grossman
5115:The Swallow's Tail
5036:Giorgio de Chirico
4908:Sydney Opera House
4763:Croatian interlace
4500:2009-09-22 at the
4483:2004-01-02 at the
4231:2005-10-26 at the
4220:2005-09-13 at the
4158:2003-12-31 at the
4115:2003-12-11 at the
3971:Lutus, P. (2008).
3899:10.1111/nyas.13667
3849:10.1111/psyp.12082
3448:on 18 January 2008
3027:Geometric Symmetry
2888:, or it can be an
2820:Symmetric relation
2718:Mathematics portal
2633:bilateral symmetry
2563:Alexander Scriabin
2553:composers such as
2283:Symmetrical scales
2215:
2118:
2071:
2042:
1979:
1812:chemical synthesis
1790:Molecular symmetry
1734:
1706:of particles; and
1644:Patterns in nature
1608:symmetric matrices
1396:
95:
67:
63:fundamental domain
47:
6110:
6109:
6067:BenoĂźt Mandelbrot
5967:Self-organization
5903:Natural selection
5893:Pattern formation
5743:
5742:
5596:
5595:
5560:Aesthetic Measure
5431:Sebastiano Serlio
5405:Leonardo da Vinci
5295:
5294:
5287:Margaret Wertheim
4948:Leonardo da Vinci
4364:Rhodes, Gillian;
4307:978-0-520-06991-6
4186:978-0-521-72876-8
4064:978-1-4008-2311-6
4017:(see appendix 1)
3843:(10): 1045â1055.
3595:Quantum Chemistry
3532:978-3-662-51229-6
3363:978-0-387-87867-6
3302:(4047): 393â396.
3137:978-0-387-94153-0
2956:978-0-691-13482-6
2916:Harper, Douglas.
2847:Explanatory notes
2676:Philosophy portal
2611:Asymmetric rhythm
2539:
2538:
2420:
2419:
2265:, traditional or
2235:arch (swell) form
1852:Visual perception
1824:quantum chemistry
1738:Bilateral animals
1681:Noether's theorem
1677:More is Different
1545:(not-and, or âŒ),
1248:
1247:
1213:
1212:
936:List of geometers
619:Three-dimensional
608:
607:
16:(Redirected from
6150:
5918:Sexual selection
5880:
5770:
5763:
5756:
5747:
5746:
5731:
5730:
5581:Nikos Salingaros
5304:
5303:
5272:Hiroshi Sugimoto
5222:Robert Longhurst
5168:Helaman Ferguson
5123:Crockett Johnson
5052:Circle Limit III
5021:Danseuse au café
4928:
4927:
4898:Pyramid of Khufu
4661:
4541:
4534:
4527:
4518:
4517:
4513:, Apr. 19, 2007)
4432:
4431:
4423:
4417:
4416:
4414:
4404:
4398:
4394:
4388:
4387:
4361:
4355:
4354:
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4295:
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4069:
4068:
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4018:
4016:
4014:
3990:
3984:
3983:
3981:
3979:
3968:
3962:
3956:
3950:
3949:
3940:Zalta, Edward N.
3928:
3922:
3921:
3911:
3901:
3867:
3861:
3860:
3836:Psychophysiology
3830:
3824:
3823:
3813:
3803:
3786:(8): 3159â3163.
3769:
3763:
3762:
3724:
3718:
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3275:
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3261:
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3256:10.3390/e5030271
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3110:
3104:
3097:
3091:
3088:Timeless Reality
3084:
3078:
3077:
3075:
3065:
3059:
3058:
3037:
3031:
3022:
3016:
3015:
3002:World Scientific
2993:
2987:
2986:
2983:Prometheus Books
2967:
2961:
2960:
2939:Fearful Symmetry
2937:Zee, A. (2007).
2934:
2928:
2927:
2913:
2897:
2870:
2864:
2857:
2774:Burnside's lemma
2762:
2757:
2756:
2748:
2743:
2742:
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2720:
2715:
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2701:
2700:
2692:
2690:Astronomy portal
2687:
2686:
2685:
2678:
2673:
2672:
2671:
2549:in the works of
2425:
2424:
2344:
2343:
2293:, or diminished
2287:whole tone scale
2253:Pitch structures
2237:(ABCBA) used by
2020:, starting with
2007:Lotfollah mosque
1836:physical science
1604:abstract algebra
1600:symmetric groups
1557:(not-or, or âœ).
1460:glide reflection
1434:helical symmetry
1323:theoretic models
1319:abstract objects
1271:
1264:
1240:
1233:
1226:
954:
953:
473:
472:
406:Zero-dimensional
111:
97:
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5774:
5744:
5739:
5719:
5715:Sacred geometry
5688:
5654:Ars Mathematica
5642:
5592:
5503:
5456:
5443:Andrea Palladio
5339:
5332:De architectura
5291:
5247:Antoine Pevsner
5227:Jeanette McLeod
5178:Susan Goldstine
5127:
4986:
4980:
4917:
4903:Sagrada FamĂlia
4864:
4806:
4674:Algorithmic art
4662:
4653:
4649:Wallpaper group
4587:Minimal surface
4550:
4545:
4502:Wayback Machine
4485:Wayback Machine
4469:
4440:
4438:Further reading
4435:
4424:
4420:
4411:Visual Thinking
4405:
4401:
4395:
4391:
4384:
4362:
4358:
4319:
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4308:
4283:
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4250:
4239:
4233:Wayback Machine
4222:Wayback Machine
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4160:Wayback Machine
4151:
4147:
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4117:Wayback Machine
4108:
4104:
4091:
4089:
4079:
4072:
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4037:
4033:
4025:
4021:
3991:
3987:
3977:
3975:
3969:
3965:
3957:
3953:
3932:Daniels, Norman
3929:
3925:
3868:
3864:
3831:
3827:
3770:
3766:
3730:Vision Research
3725:
3721:
3680:
3676:
3631:
3627:
3616:
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3605:
3591:
3587:
3544:
3540:
3533:
3517:
3513:
3502:
3498:
3488:
3486:
3485:on May 17, 2016
3482:
3471:
3465:
3461:
3451:
3449:
3438:
3434:
3404:10.2307/1968551
3375:
3371:
3364:
3354:Springer-Verlag
3343:
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3081:
3066:
3062:
3055:
3038:
3034:
3023:
3019:
3012:
2994:
2990:
2975:Lederman, L. M.
2968:
2964:
2957:
2935:
2931:
2914:
2910:
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2901:
2900:
2871:
2867:
2858:
2854:
2849:
2844:
2840:Wallpaper group
2760:Clothing portal
2758:
2751:
2744:
2737:
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2723:
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2695:
2688:
2683:
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2667:
2664:
2642:
2625:
2619:
2585:
2323:interval cycles
2291:augmented chord
2255:
2227:
2211:Major and minor
2204:
2136:
2130:
2103:
2063:
2030:
1987:The White House
1967:
1961:
1959:In architecture
1953:
1947:
1935:symmetric games
1883:
1871:Law of Symmetry
1854:
1848:
1832:crystallography
1792:
1786:
1726:
1724:facial symmetry
1716:
1657:
1651:
1646:
1640:
1569:
1563:
1523:(â§, or &),
1477:dyadic relation
1473:
1384:
1378:
1373:
1284:transformations
1244:
1215:
1214:
951:
950:
941:
940:
731:
730:
714:
713:
699:
698:
686:
685:
622:
621:
610:
609:
470:
469:
467:Two-dimensional
458:
457:
431:
430:
428:One-dimensional
419:
418:
409:
408:
397:
396:
330:
329:
328:
311:
310:
159:
158:
147:
124:
87:self-similarity
35:
28:
23:
22:
15:
12:
11:
5:
6156:
6146:
6145:
6140:
6135:
6130:
6125:
6108:
6107:
6105:
6104:
6099:
6094:
6088:
6086:
6082:
6081:
6079:
6078:
6077:
6076:
6064:
6059:
6058:
6057:
6045:
6044:
6043:
6031:
6029:Wilson Bentley
6026:
6024:Joseph Plateau
6021:
6016:
6011:
6010:
6009:
5997:
5992:
5987:
5981:
5979:
5975:
5974:
5972:
5971:
5970:
5969:
5964:
5962:Plateau's laws
5959:
5957:Fluid dynamics
5954:
5944:
5943:
5942:
5937:
5932:
5922:
5921:
5920:
5915:
5910:
5905:
5895:
5889:
5887:
5883:
5882:
5875:
5873:
5871:
5870:
5865:
5860:
5855:
5850:
5849:
5848:
5843:
5838:
5833:
5823:
5818:
5813:
5808:
5803:
5798:
5793:
5787:
5785:
5781:
5780:
5773:
5772:
5765:
5758:
5750:
5741:
5740:
5738:
5737:
5724:
5721:
5720:
5718:
5717:
5712:
5707:
5702:
5696:
5694:
5690:
5689:
5687:
5686:
5681:
5676:
5671:
5666:
5661:
5656:
5650:
5648:
5644:
5643:
5641:
5640:
5633:
5626:
5619:
5612:
5604:
5602:
5598:
5597:
5594:
5593:
5591:
5590:
5589:
5588:
5578:
5577:
5576:
5564:
5563:
5562:
5552:
5551:
5550:
5538:
5537:
5536:
5526:
5525:
5524:
5511:
5509:
5505:
5504:
5502:
5501:
5500:
5499:
5497:The Greek Vase
5489:
5488:
5487:
5477:
5476:
5475:
5464:
5462:
5458:
5457:
5455:
5454:
5453:
5452:
5440:
5439:
5438:
5428:
5427:
5426:
5419:Albrecht DĂŒrer
5416:
5415:
5414:
5402:
5401:
5400:
5388:
5387:
5386:
5374:
5373:
5372:
5365:
5353:
5347:
5345:
5341:
5340:
5338:
5337:
5336:
5335:
5323:
5322:
5321:
5310:
5308:
5301:
5297:
5296:
5293:
5292:
5290:
5289:
5284:
5282:Roman Verostko
5279:
5274:
5269:
5264:
5259:
5257:Alba Rojo Cama
5254:
5249:
5244:
5239:
5234:
5229:
5224:
5219:
5214:
5213:
5212:
5203:Charles Jencks
5200:
5195:
5190:
5188:George W. Hart
5185:
5180:
5175:
5170:
5165:
5160:
5155:
5150:
5141:
5135:
5133:
5129:
5128:
5126:
5125:
5120:
5119:
5118:
5111:
5099:
5098:
5097:
5085:
5084:
5083:
5076:
5069:
5062:
5055:
5043:
5038:
5033:
5032:
5031:
5024:
5015:Jean Metzinger
5012:
5011:
5010:
5003:
4990:
4988:
4982:
4981:
4979:
4978:
4977:
4976:
4964:
4962:Albrecht DĂŒrer
4959:
4958:
4957:
4945:
4940:
4934:
4932:
4925:
4919:
4918:
4916:
4915:
4910:
4905:
4900:
4895:
4890:
4885:
4880:
4874:
4872:
4866:
4865:
4863:
4862:
4855:
4848:
4841:
4834:
4827:
4820:
4814:
4812:
4808:
4807:
4805:
4804:
4799:
4794:
4789:
4788:
4787:
4777:
4772:
4771:
4770:
4765:
4760:
4750:
4749:
4748:
4743:
4738:
4733:
4723:
4718:
4713:
4708:
4703:
4702:
4701:
4696:
4691:
4681:
4679:Anamorphic art
4676:
4670:
4668:
4664:
4663:
4656:
4654:
4652:
4651:
4646:
4641:
4636:
4635:
4634:
4629:
4621:
4616:
4611:
4610:
4609:
4607:Camera obscura
4604:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4558:
4556:
4552:
4551:
4544:
4543:
4536:
4529:
4521:
4515:
4514:
4504:
4492:
4487:
4475:
4468:
4467:External links
4465:
4464:
4463:
4452:Souvenir Press
4439:
4436:
4434:
4433:
4418:
4399:
4389:
4382:
4356:
4313:
4306:
4274:
4237:
4206:
4192:
4185:
4173:Cucker, Felipe
4164:
4145:
4133:
4121:
4102:
4087:New York Times
4070:
4063:
4043:
4031:
4019:
3985:
3963:
3951:
3934:(2003-04-28).
3923:
3884:(1): 280â293.
3862:
3825:
3764:
3737:(7): 783â793.
3719:
3674:
3647:(9): 346â352.
3625:
3610:
3603:
3585:
3538:
3531:
3511:
3496:
3459:
3432:
3390:(1): 149â204,
3369:
3362:
3337:
3277:
3262:
3221:
3208:
3180:
3167:
3143:
3136:
3118:
3105:
3092:
3079:
3060:
3053:
3032:
3017:
3010:
2988:
2962:
2955:
2929:
2907:
2905:
2902:
2899:
2898:
2865:
2851:
2850:
2848:
2845:
2843:
2842:
2837:
2835:Symmetry group
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2786:
2781:
2776:
2771:
2765:
2764:
2763:
2749:
2735:
2732:Physics portal
2721:
2707:
2693:
2679:
2663:
2660:
2641:
2638:
2621:Main article:
2618:
2615:
2584:
2581:
2559:Richard Wagner
2537:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2502:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2476:
2474:
2471:
2469:
2465:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2439:
2437:
2434:
2432:
2429:
2418:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2392:
2390:
2387:
2385:
2381:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2355:
2353:
2350:
2348:
2275:diatonic scale
2273:, such as the
2254:
2251:
2226:
2223:
2203:
2200:
2148:sand paintings
2129:
2126:
2102:
2099:
2062:
2059:
2046:pottery wheels
2029:
2026:
1960:
1957:
1946:
1943:
1882:
1879:
1847:
1844:
1788:Main article:
1785:
1782:
1742:sagittal plane
1715:
1712:
1653:Main article:
1650:
1647:
1639:
1636:
1612:linear algebra
1565:Main article:
1562:
1559:
1532:if and only if
1529:(âš, or |) and
1472:
1469:
1468:
1467:
1464:rotoreflection
1456:
1445:scale symmetry
1443:An object has
1441:
1432:An object has
1430:
1421:An object has
1419:
1414:An object has
1412:
1403:An object has
1380:Main article:
1377:
1374:
1372:
1371:In mathematics
1369:
1246:
1245:
1243:
1242:
1235:
1228:
1220:
1217:
1216:
1211:
1210:
1209:
1208:
1203:
1195:
1194:
1190:
1189:
1188:
1187:
1182:
1177:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1129:
1128:
1124:
1123:
1122:
1121:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1078:
1077:
1073:
1072:
1071:
1070:
1065:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1007:
1006:
1002:
1001:
1000:
999:
994:
989:
984:
979:
974:
969:
961:
960:
952:
948:
947:
946:
943:
942:
939:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
732:
728:
727:
726:
723:
722:
716:
715:
712:
711:
706:
700:
693:
692:
691:
688:
687:
684:
683:
678:
673:
671:Platonic Solid
668:
663:
658:
653:
648:
643:
642:
641:
630:
629:
623:
617:
616:
615:
612:
611:
606:
605:
604:
603:
598:
593:
585:
584:
578:
577:
576:
575:
570:
562:
561:
555:
554:
553:
552:
547:
542:
537:
529:
528:
522:
521:
520:
519:
514:
509:
501:
500:
494:
493:
492:
491:
486:
481:
471:
465:
464:
463:
460:
459:
456:
455:
450:
449:
448:
443:
432:
426:
425:
424:
421:
420:
417:
416:
410:
404:
403:
402:
399:
398:
395:
394:
389:
384:
378:
377:
372:
367:
357:
352:
347:
341:
340:
331:
327:
326:
323:
319:
318:
317:
316:
313:
312:
309:
308:
307:
306:
296:
291:
286:
281:
276:
275:
274:
264:
259:
254:
253:
252:
247:
242:
232:
231:
230:
225:
215:
210:
205:
200:
195:
190:
189:
188:
183:
182:
181:
166:
160:
154:
153:
152:
149:
148:
146:
145:
135:
129:
126:
125:
112:
104:
103:
55:symmetry group
26:
9:
6:
4:
3:
2:
6155:
6144:
6141:
6139:
6136:
6134:
6131:
6129:
6126:
6124:
6121:
6120:
6118:
6103:
6100:
6098:
6095:
6093:
6090:
6089:
6087:
6083:
6075:
6074:
6070:
6069:
6068:
6065:
6063:
6060:
6056:
6055:
6051:
6050:
6049:
6046:
6042:
6041:
6037:
6036:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6019:Ernst Haeckel
6017:
6015:
6014:Adolf Zeising
6012:
6008:
6007:
6003:
6002:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5982:
5980:
5976:
5968:
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5949:
5948:
5945:
5941:
5938:
5936:
5933:
5931:
5928:
5927:
5926:
5923:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5900:
5899:
5896:
5894:
5891:
5890:
5888:
5884:
5879:
5869:
5866:
5864:
5861:
5859:
5858:Vortex street
5856:
5854:
5851:
5847:
5844:
5842:
5839:
5837:
5836:Quasicrystals
5834:
5832:
5829:
5828:
5827:
5824:
5822:
5819:
5817:
5814:
5812:
5809:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5788:
5786:
5782:
5778:
5771:
5766:
5764:
5759:
5757:
5752:
5751:
5748:
5736:
5735:
5726:
5725:
5722:
5716:
5713:
5711:
5708:
5706:
5703:
5701:
5700:Droste effect
5698:
5697:
5695:
5691:
5685:
5682:
5680:
5679:Mathemalchemy
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5651:
5649:
5647:Organizations
5645:
5639:
5638:
5634:
5632:
5631:
5627:
5625:
5624:
5620:
5618:
5617:
5616:Lumen Naturae
5613:
5611:
5610:
5606:
5605:
5603:
5599:
5587:
5584:
5583:
5582:
5579:
5575:
5574:
5570:
5569:
5568:
5565:
5561:
5558:
5557:
5556:
5553:
5549:
5548:
5544:
5543:
5542:
5539:
5535:
5532:
5531:
5530:
5527:
5523:
5522:
5518:
5517:
5516:
5513:
5512:
5510:
5506:
5498:
5495:
5494:
5493:
5490:
5486:
5483:
5482:
5481:
5478:
5474:
5471:
5470:
5469:
5468:Samuel Colman
5466:
5465:
5463:
5459:
5451:
5450:
5446:
5445:
5444:
5441:
5437:
5434:
5433:
5432:
5429:
5425:
5422:
5421:
5420:
5417:
5413:
5412:
5408:
5407:
5406:
5403:
5399:
5398:
5394:
5393:
5392:
5389:
5385:
5384:
5380:
5379:
5378:
5375:
5371:
5370:
5366:
5364:
5363:
5359:
5358:
5357:
5354:
5352:
5349:
5348:
5346:
5342:
5334:
5333:
5329:
5328:
5327:
5324:
5320:
5317:
5316:
5315:
5312:
5311:
5309:
5305:
5302:
5298:
5288:
5285:
5283:
5280:
5278:
5277:Daina TaimiĆa
5275:
5273:
5270:
5268:
5265:
5263:
5262:Reza Sarhangi
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5211:
5210:
5206:
5205:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5173:Peter Forakis
5171:
5169:
5166:
5164:
5161:
5159:
5156:
5154:
5151:
5149:
5145:
5142:
5140:
5137:
5136:
5134:
5130:
5124:
5121:
5117:
5116:
5112:
5110:
5109:
5105:
5104:
5103:
5102:Salvador DalĂ
5100:
5096:
5095:
5091:
5090:
5089:
5088:René Magritte
5086:
5082:
5081:
5077:
5075:
5074:
5070:
5068:
5067:
5063:
5061:
5060:
5059:Print Gallery
5056:
5054:
5053:
5049:
5048:
5047:
5044:
5042:
5039:
5037:
5034:
5030:
5029:
5028:L'Oiseau bleu
5025:
5023:
5022:
5018:
5017:
5016:
5013:
5009:
5008:
5004:
5002:
5001:
4997:
4996:
4995:
4994:William Blake
4992:
4991:
4989:
4983:
4975:
4974:
4970:
4969:
4968:
4965:
4963:
4960:
4956:
4955:
4954:Vitruvian Man
4951:
4950:
4949:
4946:
4944:
4941:
4939:
4938:Paolo Uccello
4936:
4935:
4933:
4929:
4926:
4924:
4920:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4875:
4873:
4871:
4867:
4861:
4860:
4859:Pi in the Sky
4856:
4854:
4853:
4849:
4847:
4846:
4842:
4840:
4839:
4835:
4833:
4832:
4831:Mathemalchemy
4828:
4826:
4825:
4821:
4819:
4816:
4815:
4813:
4809:
4803:
4800:
4798:
4795:
4793:
4790:
4786:
4783:
4782:
4781:
4778:
4776:
4773:
4769:
4766:
4764:
4761:
4759:
4756:
4755:
4754:
4751:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4728:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4700:
4699:Vastu shastra
4697:
4695:
4692:
4690:
4689:Geodesic dome
4687:
4686:
4685:
4682:
4680:
4677:
4675:
4672:
4671:
4669:
4665:
4660:
4650:
4647:
4645:
4642:
4640:
4637:
4633:
4630:
4628:
4625:
4624:
4622:
4620:
4617:
4615:
4614:Plastic ratio
4612:
4608:
4605:
4603:
4602:Camera lucida
4600:
4599:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4559:
4557:
4553:
4549:
4542:
4537:
4535:
4530:
4528:
4523:
4522:
4519:
4512:
4508:
4505:
4503:
4499:
4496:
4495:ISIS Symmetry
4493:
4491:
4488:
4486:
4482:
4479:
4476:
4474:
4471:
4470:
4461:
4460:0-285-63743-6
4457:
4453:
4449:
4445:
4442:
4441:
4429:
4422:
4413:
4412:
4403:
4393:
4385:
4383:1-56750-636-4
4379:
4375:
4371:
4367:
4360:
4352:
4348:
4344:
4340:
4336:
4332:
4328:
4324:
4317:
4309:
4303:
4299:
4294:
4293:
4287:
4286:Perle, George
4281:
4279:
4270:
4266:
4262:
4258:
4254:
4253:Perle, George
4248:
4246:
4244:
4242:
4234:
4230:
4227:
4223:
4219:
4216:
4210:
4202:
4196:
4188:
4182:
4178:
4174:
4168:
4161:
4157:
4154:
4149:
4142:
4137:
4130:
4125:
4118:
4114:
4111:
4106:
4099:
4088:
4084:
4077:
4075:
4066:
4060:
4056:
4055:
4047:
4040:
4035:
4028:
4023:
4013:
4008:
4004:
4000:
3996:
3989:
3974:
3967:
3960:
3955:
3947:
3946:
3941:
3937:
3933:
3927:
3919:
3915:
3910:
3909:11577/3289328
3905:
3900:
3895:
3891:
3887:
3883:
3879:
3878:
3873:
3866:
3858:
3854:
3850:
3846:
3842:
3838:
3837:
3829:
3821:
3817:
3812:
3807:
3802:
3797:
3793:
3789:
3785:
3781:
3780:
3775:
3768:
3760:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3731:
3723:
3715:
3711:
3707:
3703:
3699:
3698:10.1068/p6393
3695:
3691:
3687:
3686:
3678:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3641:
3636:
3629:
3621:
3614:
3606:
3604:0-12-457551-X
3600:
3596:
3589:
3581:
3577:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3542:
3534:
3528:
3524:
3523:
3515:
3507:
3500:
3481:
3477:
3470:
3463:
3447:
3443:
3436:
3429:
3425:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3384:
3379:
3378:Wigner, E. P.
3373:
3365:
3359:
3355:
3351:
3347:
3341:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3297:
3296:
3288:
3281:
3273:
3266:
3257:
3252:
3248:
3244:
3240:
3236:
3232:
3225:
3218:
3212:
3198:
3191:
3184:
3177:
3171:
3157:
3153:
3147:
3139:
3133:
3129:
3122:
3115:
3109:
3102:
3096:
3089:
3083:
3074:
3073:
3064:
3056:
3054:0-691-02374-3
3050:
3046:
3042:
3041:Weyl, Hermann
3036:
3029:
3028:
3021:
3013:
3011:981-256-192-7
3007:
3003:
2999:
2992:
2984:
2980:
2976:
2972:
2966:
2958:
2952:
2948:
2944:
2940:
2933:
2925:
2924:
2919:
2912:
2908:
2895:
2891:
2887:
2883:
2879:
2875:
2869:
2862:
2859:For example,
2856:
2852:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2766:
2761:
2755:
2750:
2747:
2741:
2736:
2733:
2727:
2722:
2719:
2713:
2708:
2705:
2699:
2694:
2691:
2680:
2677:
2666:
2659:
2657:
2656:
2651:
2647:
2640:In literature
2637:
2634:
2630:
2624:
2617:In aesthetics
2614:
2612:
2608:
2604:
2600:
2596:
2593:
2589:
2580:
2578:
2576:
2570:
2568:
2567:Edgard VarĂšse
2564:
2560:
2556:
2555:Gustav Mahler
2552:
2548:
2544:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2503:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2466:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2426:
2423:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2382:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2345:
2342:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2296:
2295:seventh chord
2292:
2288:
2284:
2280:
2276:
2272:
2268:
2264:
2260:
2250:
2248:
2244:
2240:
2236:
2232:
2222:
2220:
2212:
2208:
2199:
2195:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2135:
2125:
2123:
2115:
2112:
2107:
2098:
2096:
2092:
2088:
2084:
2083:Oriental rugs
2080:
2076:
2067:
2058:
2056:
2050:
2047:
2039:
2038:pottery wheel
2034:
2025:
2023:
2019:
2014:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1976:
1971:
1966:
1956:
1952:
1942:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1878:
1874:
1872:
1868:
1864:
1859:
1853:
1843:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1791:
1781:
1777:
1775:
1771:
1767:
1763:
1759:
1755:
1750:
1747:
1743:
1739:
1730:
1725:
1721:
1711:
1709:
1708:supersymmetry
1705:
1701:
1697:
1693:
1688:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1656:
1645:
1635:
1633:
1629:
1625:
1621:
1620:Galois theory
1617:
1616:Galois groups
1613:
1609:
1605:
1601:
1597:
1593:
1588:
1586:
1582:
1578:
1574:
1568:
1558:
1556:
1555:
1550:
1549:
1544:
1543:
1538:
1534:
1533:
1528:
1527:
1522:
1521:
1516:
1511:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1478:
1465:
1461:
1457:
1454:
1450:
1446:
1442:
1439:
1435:
1431:
1428:
1425:if it can be
1424:
1420:
1417:
1413:
1410:
1406:
1402:
1401:
1400:
1393:
1388:
1383:
1368:
1366:
1361:
1360:, and music.
1359:
1355:
1351:
1347:
1343:
1339:
1334:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1303:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1270:
1263:
1259:
1256:
1255:Ancient Greek
1252:
1241:
1236:
1234:
1229:
1227:
1222:
1221:
1219:
1218:
1207:
1204:
1202:
1199:
1198:
1197:
1196:
1192:
1191:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1132:
1131:
1130:
1126:
1125:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1081:
1080:
1079:
1075:
1074:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1010:
1009:
1008:
1004:
1003:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
964:
963:
962:
959:
956:
955:
945:
944:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
733:
725:
724:
721:
718:
717:
710:
707:
705:
702:
701:
696:
690:
689:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
640:
637:
636:
635:
632:
631:
628:
625:
624:
620:
614:
613:
602:
599:
597:
596:Circumference
594:
592:
589:
588:
587:
586:
583:
580:
579:
574:
571:
569:
566:
565:
564:
563:
560:
559:Quadrilateral
557:
556:
551:
548:
546:
543:
541:
538:
536:
533:
532:
531:
530:
527:
526:Parallelogram
524:
523:
518:
515:
513:
510:
508:
505:
504:
503:
502:
499:
496:
495:
490:
487:
485:
482:
480:
477:
476:
475:
474:
468:
462:
461:
454:
451:
447:
444:
442:
439:
438:
437:
434:
433:
429:
423:
422:
415:
412:
411:
407:
401:
400:
393:
390:
388:
385:
383:
380:
379:
376:
373:
371:
368:
365:
364:Perpendicular
361:
360:Orthogonality
358:
356:
353:
351:
348:
346:
343:
342:
339:
336:
335:
334:
324:
321:
320:
315:
314:
305:
302:
301:
300:
297:
295:
292:
290:
287:
285:
284:Computational
282:
280:
277:
273:
270:
269:
268:
265:
263:
260:
258:
255:
251:
248:
246:
243:
241:
238:
237:
236:
233:
229:
226:
224:
221:
220:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
187:
184:
180:
177:
176:
175:
172:
171:
170:
169:Non-Euclidean
167:
165:
162:
161:
157:
151:
150:
143:
139:
136:
134:
131:
130:
128:
127:
123:
119:
115:
110:
106:
105:
102:
99:
98:
92:
88:
84:
80:
76:
71:
64:
60:
56:
51:
44:
39:
33:
19:
6071:
6052:
6038:
6004:
5930:Chaos theory
5853:Tessellation
5825:
5732:
5635:
5628:
5621:
5614:
5607:
5601:Publications
5585:
5571:
5559:
5545:
5533:
5519:
5496:
5492:Jay Hambidge
5485:Ad Quadratum
5484:
5472:
5447:
5435:
5423:
5409:
5395:
5391:Luca Pacioli
5381:
5367:
5360:
5330:
5318:
5242:Hinke Osinga
5237:IstvĂĄn Orosz
5207:
5198:Anthony Hill
5153:Scott Draves
5148:Erik Demaine
5132:Contemporary
5113:
5106:
5092:
5078:
5071:
5064:
5057:
5050:
5046:M. C. Escher
5026:
5019:
5005:
4998:
4971:
4967:Parmigianino
4952:
4883:Hagia Sophia
4857:
4850:
4843:
4836:
4829:
4822:
4706:Computer art
4684:Architecture
4644:Tessellation
4638:
4627:Architecture
4577:Golden ratio
4510:
4443:
4421:
4410:
4402:
4392:
4369:
4359:
4326:
4322:
4316:
4291:
4263:(2): 81â96.
4260:
4256:
4209:
4195:
4176:
4167:
4148:
4136:
4124:
4105:
4097:
4090:. Retrieved
4086:
4053:
4046:
4034:
4022:
4002:
3998:
3988:
3978:28 September
3976:. Retrieved
3966:
3954:
3943:
3926:
3881:
3875:
3865:
3840:
3834:
3828:
3783:
3777:
3767:
3734:
3728:
3722:
3692:(1): 27â40.
3689:
3683:
3677:
3644:
3638:
3628:
3619:
3613:
3594:
3588:
3558:(1): 36â50.
3555:
3551:
3541:
3525:. Springer.
3521:
3514:
3505:
3499:
3487:. Retrieved
3480:the original
3475:
3462:
3450:. Retrieved
3446:the original
3435:
3387:
3381:
3372:
3349:
3340:
3299:
3293:
3280:
3271:
3265:
3238:
3234:
3224:
3216:
3211:
3200:. Retrieved
3196:
3183:
3175:
3170:
3159:. Retrieved
3155:
3146:
3127:
3121:
3113:
3112:Tian Yu Cao
3108:
3100:
3095:
3087:
3082:
3071:
3063:
3044:
3035:
3025:
3020:
2997:
2991:
2978:
2965:
2938:
2932:
2921:
2911:
2868:
2855:
2769:Automorphism
2653:
2643:
2626:
2586:
2574:
2571:
2547:progressions
2540:
2478:
2441:
2421:
2394:
2357:
2319:George Perle
2256:
2247:James Tenney
2228:
2225:Musical form
2216:
2196:
2168:tessellation
2137:
2119:
2111:kaleidoscope
2072:
2051:
2043:
2015:
1995:tile mosaics
1980:
1954:
1884:
1875:
1855:
1840:group theory
1828:spectroscopy
1820:side effects
1799:
1793:
1784:In chemistry
1778:
1754:sea anemones
1751:
1735:
1689:
1676:
1658:
1589:
1576:
1570:
1552:
1546:
1540:
1536:
1530:
1524:
1518:
1512:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1474:
1397:
1362:
1354:architecture
1340:, including
1335:
1321:, including
1304:
1268:
1265:
1258:
1250:
1249:
1068:Parameshvara
881:Parameshvara
651:Dodecahedron
391:
235:Differential
53:A spherical
6048:Alan Turing
6006:Liber Abaci
5925:Mathematics
5831:in crystals
5821:Soap bubble
5816:Phyllotaxis
5541:G. H. Hardy
5344:Renaissance
5314:Polykleitos
5252:Tony Robbin
5163:John Ernest
5158:Jan Dibbets
5108:Crucifixion
4931:Renaissance
4785:Mathematics
4758:Celtic knot
4721:Fractal art
4623:Proportion
4597:Perspective
4511:In Our Time
4448:Mario Livio
4092:11 November
3489:October 25,
3442:"Bilateria"
2971:Hill, C. T.
2882:floor tiles
2746:Star portal
2592:pitch class
2583:Equivalency
2315:BĂ©la BartĂłk
2279:major chord
2243:BĂ©la BartĂłk
2239:Steve Reich
2164:M.C. Escher
1991:floor plans
1945:In the arts
1939:tit for tat
1931:game theory
1927:Golden Rule
1905:, respect,
1887:reciprocity
1816:therapeutic
1770:sea urchins
1762:echinoderms
1673:PW Anderson
1376:In geometry
1338:mathematics
1288:translation
1282:under some
1276:mathematics
1193:Present day
1140:Lobachevsky
1127:1700sâ1900s
1084:JyeáčŁáčhadeva
1076:1400sâ1700s
1028:Brahmagupta
851:Lobachevsky
831:JyeáčŁáčhadeva
781:Brahmagupta
709:Hypersphere
681:Tetrahedron
656:Icosahedron
228:Diophantine
6128:Aesthetics
6117:Categories
5995:Empedocles
5990:Pythagoras
5908:Camouflage
5846:in biology
5841:in flowers
5811:Parastichy
5515:Owen Jones
5362:De pictura
5267:Oliver Sin
5217:Andy Lomas
5066:Relativity
4797:String art
4711:Fiber arts
4592:Paraboloid
3961:: Symmetry
3685:Perception
3202:2019-11-12
3161:2019-11-12
2918:"symmetry"
2904:References
2800:Palindrome
2650:palindrome
2646:literature
2629:aesthetics
2603:retrograde
2597:which are
2543:enharmonic
2311:Alban Berg
2301:as to the
2194:patterns.
2192:embroidery
2089:âthat is,
1983:cathedrals
1858:Ernst Mach
1774:sea lilies
1714:In biology
1661:invariance
1649:In physics
1624:statistics
1438:screw axis
1427:translated
1392:triskelion
1313:; through
1292:reflection
1286:, such as
1253:(from
1053:al-Yasamin
997:Apollonius
992:Archimedes
982:Pythagoras
972:Baudhayana
926:al-Yasamin
876:Pythagoras
771:Baudhayana
761:Archimedes
756:Apollonius
661:Octahedron
512:Hypotenuse
387:Similarity
382:Congruence
294:Incidence
245:Symplectic
240:Riemannian
223:Arithmetic
198:Projective
186:Hyperbolic
114:Projecting
18:Symmetries
6097:Emergence
6000:Fibonacci
5326:Vitruvius
5300:Theorists
5080:Waterfall
4985:19thâ20th
4913:Taj Mahal
4893:Parthenon
4870:Buildings
4824:Continuum
4792:Sculpture
4768:Interlace
4562:Algorithm
4226:Shockwave
3428:121773411
3043:(1982) .
2861:Aristotle
2779:Chirality
2607:inversion
2599:invariant
2588:Tone rows
2299:ambiguous
2172:wallpaper
2144:furniture
2101:In quilts
2087:rectangle
2003:Taj Mahal
1975:Taj Mahal
1919:judgments
1796:chemistry
1700:spacetime
1630:, and as
1577:symmetric
1365:asymmetry
1280:invariant
1269:summetrĂa
1262:ÏÏ
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1170:Minkowski
1089:Descartes
1023:Aryabhata
1018:KÄtyÄyana
949:by period
861:Minkowski
836:KÄtyÄyana
796:Descartes
741:Aryabhata
720:Geometers
704:Tesseract
568:Trapezoid
540:Rectangle
333:Dimension
218:Algebraic
208:Synthetic
179:Spherical
164:Euclidean
43:asymmetry
6138:Geometry
6123:Symmetry
5826:Symmetry
5784:Patterns
5734:Category
5461:Romantic
5139:Max Bill
5073:Reptiles
4888:Pantheon
4845:Octacube
4811:Artworks
4753:Knotting
4741:Muqarnas
4639:Symmetry
4567:Catenary
4555:Concepts
4507:Symmetry
4498:Archived
4481:Archived
4454:, 2006,
4397:417â429.
4368:(2002).
4288:(1990).
4229:Archived
4218:Archived
4175:(2013).
4156:Archived
4113:Archived
4005:: 1â18.
3918:29604083
3857:23941638
3820:15710884
3759:41530752
3714:22451173
3706:20301844
3661:21223945
3572:27530930
3348:(2010).
3332:34548824
3324:17796623
3045:Symmetry
2977:(2005).
2890:abstract
2886:molecule
2795:Isotropy
2662:See also
2579:(1910).
2551:Romantic
2339:interval
2202:In music
2152:knotwork
2140:beadwork
2109:Kitchen
2011:Alhambra
2005:and the
1937:such as
1903:dialogue
1895:sympathy
1867:grouping
1808:symmetry
1800:specific
1766:starfish
1632:skewness
1596:calculus
1517:include
1471:In logic
1449:Fractals
1342:geometry
1327:language
1296:rotation
1251:Symmetry
1160:Poincaré
1104:Minggatu
1063:Yang Hui
1033:Virasena
921:Yang Hui
916:Virasena
886:Poincaré
866:Minggatu
646:Cylinder
591:Diameter
550:Rhomboid
507:Altitude
498:Triangle
392:Symmetry
370:Parallel
355:Diagonal
325:Features
322:Concepts
213:Analytic
174:Elliptic
156:Branches
142:Timeline
101:Geometry
6085:Related
5952:Crystal
5947:Physics
5935:Fractal
5913:Mimicry
5898:Biology
5806:Meander
5693:Related
5307:Ancient
5041:Man Ray
4987:Century
4923:Artists
4780:Origami
4694:Pyramid
4572:Fractal
4351:1205083
4343:7924253
3942:(ed.).
3886:Bibcode
3788:Bibcode
3669:2143353
3580:3671068
3420:1503456
3412:1968551
3392:Bibcode
3304:Bibcode
3295:Science
3243:Bibcode
3235:Entropy
2874:crystal
2655:Beowulf
2577:, Op. 3
2575:Quartet
2335:centers
2329:or non-
2277:or the
2271:pitches
2188:textile
2055:Chinese
1999:Islamic
1911:revenge
1907:justice
1899:apology
1891:empathy
1863:Gestalt
1453:similar
1346:science
1309:; as a
1300:scaling
1185:Coxeter
1165:Hilbert
1150:Riemann
1099:Huygens
1058:al-Tusi
1048:KhayyĂĄm
1038:Alhazen
1005:1â1400s
906:al-Tusi
891:Riemann
841:KhayyĂĄm
826:Huygens
821:Hilbert
791:Coxeter
751:Alhazen
729:by name
666:Pyramid
545:Rhombus
489:Polygon
441:segment
289:Fractal
272:Digital
257:Complex
138:History
133:Outline
75:fractal
45:(right)
5978:People
5886:Causes
5508:Modern
5144:Martin
5007:Newton
4802:Tiling
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4716:4D art
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2263:chords
2259:scales
2245:, and
2231:formal
2158:, and
2122:quilts
2091:motifs
2079:Navajo
2075:carpet
1909:, and
1804:chiral
1772:, and
1614:, and
1350:nature
1329:, and
1206:Gromov
1201:Atiyah
1180:Veblen
1175:Cartan
1145:Bolyai
1114:Sakabe
1094:Pascal
987:Euclid
977:Manava
911:Veblen
896:Sakabe
871:Pascal
856:Manava
816:Gromov
801:Euclid
786:Cartan
776:Bolyai
766:Atiyah
676:Sphere
639:cuboid
627:Volume
582:Circle
535:Square
453:Length
375:Vertex
279:Convex
262:Finite
203:Affine
118:sphere
5985:Plato
5791:Crack
5319:Canon
4775:Music
4731:Girih
4667:Forms
4632:Human
4374:Ablex
4347:S2CID
3938:. In
3755:S2CID
3710:S2CID
3665:S2CID
3576:S2CID
3483:(PDF)
3472:(PDF)
3424:S2CID
3408:JSTOR
3328:S2CID
3290:(PDF)
3193:(PDF)
2884:, or
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2267:tonal
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2180:batik
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1622:. In
1585:group
1331:music
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1155:Klein
1135:Gauss
1109:Euler
1043:Sijzi
1013:Zhang
967:Ahmes
931:Zhang
901:Sijzi
846:Klein
811:Gauss
806:Euler
746:Ahmes
479:Plane
414:Point
350:Curve
345:Angle
122:plane
120:to a
57:with
5863:Wave
5801:Foam
5796:Dune
5146:and
4736:Jali
4456:ISBN
4378:ISBN
4339:PMID
4302:ISBN
4181:ISBN
4094:2014
4059:ISBN
3980:2015
3914:PMID
3853:PMID
3816:PMID
3747:PMID
3702:PMID
3657:PMID
3599:ISBN
3568:PMID
3527:ISBN
3491:2012
3454:2013
3358:ISBN
3320:PMID
3132:ISBN
3049:ISBN
3006:ISBN
2951:ISBN
2595:sets
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2327:keys
2261:and
2190:and
2184:ikat
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1390:The
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1307:time
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695:Four
634:Cube
601:Area
573:Kite
484:Area
436:Line
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4331:doi
4327:108
4265:doi
4224:or
4215:pdf
4007:doi
3904:hdl
3894:doi
3882:132
3845:doi
3806:PMC
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