1136:: If the requirement that every cross section of the helical object be identical is relaxed, then additional lesser helical symmetries would become possible. For example, the cross section of the helical object may change, but may still repeat itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle of rotation at which the symmetry occurs divides evenly into a full circle (360°), then the result is the helical equivalent of a regular polygon. This case is called
667:
1127:
symmetries if for any small rotation of the object around its central axis, there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
909:
1102:
1209:
27:
1346:
1086:
1075:
859:
776:
850:. The illustration on the right shows four congruent footprints generated by translations along the arrow. If the line of footprints were to extend to infinity in both directions, then they would have a discrete translational symmetry; any translation that mapped one footprint onto another would leave the whole line unchanged.
888:
in general) means that a reflection in a line or plane combined with a translation along the line or in the plane, results in the same object (such as in the case of footprints). The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry
1126:
of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical
1121:
or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object—to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a
1393:
is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables small twigs to stand in for full
1357:
Scale symmetry means that if an object is expanded or reduced in size, the new object has the same properties as the original. This self-similarity is seen in many natural structures such as cumulus clouds, lightning, ferns and coastlines, over a wide range of scales. It is generally not found in
469:
373:
In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions, there is an axis of symmetry (a.k.a., line of symmetry), and in three dimensions there is a plane of symmetry. An object or figure for which every point has a one-to-one mapping onto another,
102:
available, and on what object properties should remain unchanged after a transformation. Because the composition of two transforms is also a transform and every transform has, by definition, an inverse transform that undoes it, the set of transforms under which an object is symmetric form a
385:
is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical as mirror images of each other. For example. a
150:, and combinations of these basic operations. Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation. A geometric object is typically symmetric only under a subset or "
74:
under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be
1070:
that helps define the properties of the traced helix. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the object rotates slowly and translates quickly, the coiling angle will approach 90°.
397:
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. Thus one can describe this phenomenon unambiguously by saying that "T has a vertical symmetry axis", or that "T has left-right symmetry".
2260:, 1872. "Vergleichende Betrachtungen über neuere geometrische Forschungen" ('A comparative review of recent researches in geometry'), Mathematische Annalen, 43 (1893) pp. 63–100 (Also: Gesammelte Abh. Vol. 1, Springer, 1921, pp. 460–497).
701:
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations (because translations are compositions of rotations about distinct points), and the symmetry group is the whole
1482:
meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
1329:, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines
1230:
In 4D, a double rotation symmetry can be generated as the composite of two orthogonal rotations. It is similar to 3D screw axis which is the composite of a rotation and an orthogonal translation.
848:
154:" of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups, and by the types of object invariance that are possible in geometry.
1171:
749:), the group of rigid motions; that is, the intersection of the full symmetry group and the group of rigid motions. For chiral objects, it is the same as the full symmetry group.
1238:
A wider definition of geometric symmetry allows operations from a larger group than the
Euclidean group of isometries. Examples of larger geometric symmetry groups are:
938:
is a rotation about an axis combined with reflection in a plane perpendicular to that axis. The symmetry groups associated with rotoreflections include:
1225:
1082:
Three main classes of helical symmetry can be distinguished, based on the interplay of the angle of coiling and translation symmetries along the axis:
1537:
with compact stabilizers, i.e. if it is the maximal group of symmetries. Sometimes this condition is included in the definition of a model geometry.
1578:
is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on
1474:
of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a
1066:, while simultaneously translating at a constant linear speed along its axis of rotation. At any point in time, these two motions combine to give a
390:
has four axes of symmetry, because there are four different ways to fold it and have the edges match each other. Another example would be that of a
1187:. The angle of rotation never repeats exactly, no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating
1062:. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant
2319:
1040:
In 3D geometry and higher, a screw axis (or rotary translation) is a combination of a rotation and a translation along the rotation axis.
1405:, they have a beauty and familiarity not typically seen with mathematically generated functions. Fractals have also found a place in
1370:). Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight.
633:), a point reflection changes the orientation of the space, like a mirror-image symmetry. That explains why in physics, the term
127:", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in
1704:
2301:
2061:
1243:
1582:. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called
2241:. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp.
2097:
799:
1664:
2246:
2189:
2105:
1967:
1929:
1833:
1800:
2053:
1739:
1023:
2224:, §5 Similarity in the Euclidean Plane, pp. 67–76, §7 Isometry and Similarity in Euclidean Space, pp 96–104,
709:
For symmetry with respect to rotations about a point, one can take that point as origin. These rotations form the
644:) is used for both point reflection and mirror symmetry. Since a point reflection in three dimensions changes a
158:
1879:
649:
1381:, fractals are a mathematical concept in which the structure of a complex form looks similar at any degree of
1177:– that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
1764:
645:
1151:
986:, because it does not depend on the axis and the plane. It is characterized by just the point of inversion.
706:). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
1109:, constructed by augmented regular tetrahedra, is an example of a screw axis symmetry that is nonperiodic.
2160:
Ursyn, Anna (2012). "Chapter 12. Visual tweet: Nature inspired visual statements". In Ursyn, Anna (ed.).
752:
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of
666:
2045:
1406:
1106:
760:
611:
1637:
1572:
1263:
687:
162:
99:
51:
20:
1123:
710:
491:
441:
374:
equidistant from and on opposite sides of a common plane is called mirror symmetric (for more, see
139:
67:
1825:
1297:
690:. Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean group
1439:
1409:, where their ability to create complex curves with fractal symmetries results in more realistic
1094:
623:
71:
47:
1921:
1568:. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
794:
788:
147:
63:
55:
2167:
2091:
1290:
1247:
1183:: This is the case in which the angle of rotation θ required to observe the symmetry is
764:
745:
Phrased slightly differently, the rotation group of an object is the symmetry group within E(
556:
457:
394:, which has infinitely many axes of symmetry passing through its center for the same reason.
361:
88:
1913:
1817:
2225:
1251:
1188:
983:
913:
753:
739:
1378:
942:
if the rotation angle has no common divisor with 360°, the symmetry group is not discrete.
793:
Translational symmetry leaves an object invariant under a discrete or continuous group of
8:
1471:
1467:
1459:
1455:
1447:
1435:
1434:. The hierarchy of geometries is thus mathematically represented as a hierarchy of these
1334:
1281:
661:
104:
1818:
1998:
1600:
1443:
1402:
1367:
1330:
1308:
636:
449:
367:
59:
31:
98:
The types of symmetries that are possible for a geometric object depend on the set of
2339:
2297:
2242:
2185:
2168:"Background information about the concept of symmetry as related to geometry", p. 209
2101:
2057:
1963:
1925:
1914:
1829:
1796:
1718:
1710:
1259:
921:
727:
387:
1478:
of the group of projective geometry, any notion invariant in projective geometry is
1498:
1490:
1326:
1184:
908:
871:
757:
641:
592:
584:
477:
366:
Reflectional symmetry, linear symmetry, mirror symmetry, mirror-image symmetry, or
2324:
95:; it is also possible for a figure/object to have more than one line of symmetry.
1501:
1463:
1267:
1174:
1047:
968:
600:
572:
414:
332:
135:
120:
1586:. (There are also uncountably many model geometries without compact quotients.)
2217:
2050:
The
Noether theorems: Invariance and conservation laws in the twentieth century
1431:
1386:
1213:
863:
780:
425:
306:
132:
128:
108:
2162:
Biologically-inspired
Computing for the Arts: Scientific Data Through Graphics
1743:
2333:
1410:
1382:
1063:
1059:
410:
382:
92:
2148:
The
Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces
2052:. Sources and Studies in the History of Mathematics and Physical Sciences.
1788:
1442:. For example, lengths, angles and areas are preserved with respect to the
1145:
890:
607:
is an alternative name for a point reflection symmetry through the origin.
381:
The axis of symmetry of a two-dimensional figure is a line such that, if a
375:
84:
1714:
1358:
gravitationally bound structures, for example the shape of the legs of an
652:, symmetry under a point reflection is also called a left-right symmetry.
2257:
1451:
1427:
1322:
1301:
1277:
with determinant 1 or −1; i.e., the transformations which preserve area.
1101:
881:
731:
678:
Rotational symmetry is symmetry with respect to some or all rotations in
619:
437:
1208:
1199:
per turn, is an example of this type of non-repeating helical symmetry.
1055:
1035:
916:
with marked edges shows rotoreflectional symmetry, with an order of 10.
671:
453:
429:
889:
group comprising glide reflections and associated translations is the
119:
The most common group of transforms applied to objects are termed the
1519:
1508:
1350:
1196:
1090:
418:
406:
26:
1004:, this is generated by a single symmetry, and the abstract group is
440:
in three dimensions for more), one of the three types of order two (
169:-dimensional space can be represented by the composition of at most
1732:
1475:
1359:
1085:
483:
468:
463:
402:
151:
143:
124:
70:) that maps the figure/object onto itself (i.e., the object has an
39:
1345:
1226:
Rotations in 4-dimensional
Euclidean space § Double rotations
1148:). This concept can be further generalized to include cases where
1046:
symmetry is the kind of symmetry seen in everyday objects such as
1595:
1395:
1374:
1273:
The group of affine transformations represented by a matrix
1117:: If there are no distinguishing features along the length of a
756:, rotational symmetry of a physical system is equivalent to the
1312:
1051:
391:
1690:
Mathematics for
Elementary School Teachers: A Process Approach
1638:"Symmetry | Thinking about Geometry | Underground Mathematics"
1522:
can be thought of as the group of symmetries of the geometry.
858:
1390:
1363:
1217:
1118:
1043:
971:, for which the same notation is used; the abstract group is
775:
691:
683:
472:
In 2 dimensions, a point reflection is a 180 degree rotation.
1533:
is maximal among groups acting smoothly and transitively on
19:"Geometric symmetry" redirects here. For the 1978 book, see
1851:
Beauty for Truth's Sake: On the Re-enchantment of
Education
1093:
has a discrete (3-fold here) screw-axis symmetry, drawn in
34:, with left and right sides as mirror images of each other.
1493:
introduced a similar version of symmetries in geometry. A
599: = 2), a point reflection is the same as a half-
1192:
2080:. Prometheus Books. Especially chapter 12. Nontechnical.
1946:
Continuous
Symmetry: From Euclid to Klein (Google eBook)
1373:
A more subtle form of scale symmetry is demonstrated by
1074:
1957:
1878:
1333:, whereas the group of Möbius transformations defines
1155:
803:
2263:
An
English translation by Mellen Haskell appeared in
1896:
1154:
802:
2093:
Transformation
Geometry: An Introduction to Symmetry
1848:
1621:
Transformation Geometry: An Introduction to Symmetry
1216:, stereographically projected into 3D, looks like a
2044:
896:, and is isomorphic with the infinite cyclic group
2076:Stenger, Victor J. (2000) and Mahou Shiro (2007).
1997:
1982:Vladimir G. Ivancevic, Tijana T. Ivancevic (2005)
1960:Symmetry principles in elementary particle physics
1220:. A double rotation can be seen as a helical path.
1165:
842:
114:
2029:
1018:. This is not a basic symmetry but a combination.
843:{\displaystyle \scriptstyle T_{a}(p)\;=\;p\,+\,a}
2331:
1995:
1962:. Cambridge University Press. pp. 120–122.
1401:Because fractals can generate the appearance of
482:Reflection symmetry can be generalized to other
464:Point reflection and other involutive isometries
2239:Three-dimensional geometry and topology. Vol. 1
2280:Conceptual Foundations of Quantum Field Theory
1863:
1665:"Symmetry - MathBitsNotebook(Geo - CCSS Math)"
1340:
87:about a line, then the figure is said to have
2184:. Gulf Professional Publishing. p. 101.
2120:Robert O. Gould, Steffen Borchardt-Ott (2011)
1687:
2014:
1899:Introduction to Möbius Differential Geometry
1868:(5 ed.). Cengage Learning. p. 499.
1816:Cowin, Stephen C.; Doty, Stephen B. (2007).
853:
717:), which can be represented by the group of
682:-dimensional Euclidean space. Rotations are
2325:Dutch: Symmetry Around a Point in the Plane
1233:
1203:
583:, then such a transformation is known as a
259:
253:
193:
183:
1951:
1920:. European Mathematical Society. pp.
1866:Mathematics for Elementary School Teachers
1618:
903:
827:
823:
424:For each line or plane of reflection, the
1815:
1795:. Princeton: Princeton University Press.
1144: = 360° (such as the case of a
835:
831:
770:
83:. If the isometry is the reflection of a
2034:. Springer Science & Business Media.
2019:. New Age International. pp. 111ff.
2004:. Springer Science & Business Media.
1344:
1207:
1100:
1084:
1073:
907:
857:
774:
665:
467:
370:is symmetry with respect to reflection.
355:
25:
2038:
2017:Elements of Group Theory for Physicists
1958:W.M. Gibson & B.R. Pollard (1980).
1507:together with a transitive action of a
2332:
2291:
2285:
2179:
2089:
1882:(2018). "11: Finite symmetry groups".
1706:Symmetry Groups and Their Applications
1702:
1470:. Then, by abstracting the underlying
655:
444:), hence algebraically isomorphic to C
2159:
2124:Springer Science & Business Media
1911:
1454:are preserved under the most general
1166:{\displaystyle \scriptstyle m\theta }
686:, which are isometries that preserve
2282:Cambridge University Press p.154-155
1659:
1657:
1632:
1630:
1560:, where Γ is a discrete subgroup of
1416:
949:-fold rotation angle (angle of 180°/
2296:. Paris/New York: Masson Springer.
2208:S. Sonnenschein & Company p.223
2206:The Fourth Dimension (Google eBook)
2137:Dover Publications (September 1990)
1029:
559:. This reflects the space along an
13:
2098:Undergraduate Texts in Mathematics
1485:
14:
2351:
2313:
1740:"Higher Dimensional Group Theory"
1654:
1627:
1122:helix, but which can also have a
2122:Crystallography: An Introduction
1692:. Cengage Learning. p. 721.
1407:computer-generated movie effects
1024:point groups in three dimensions
2272:
2251:
2231:
2211:
2198:
2173:
2164:. IGI Global. pp. 207–239.
2153:
2150:Cambridge University Press p.64
2140:
2127:
2114:
2083:
2070:
2023:
2008:
1989:
1976:
1938:
1905:
1890:
1872:
1857:
1842:
1421:
674:has 3-fold rotational symmetry.
629:(as well as for other odd
328:
326:
310:
305:
284:
281:
272:
270:
233:
228:
213:
208:
138:). These isometries consist of
115:Euclidean symmetries in general
2294:Physics and fractal structures
2292:Gouyet, Jean-François (1996).
2204:Charles Howard Hinton (1906)
1944:William H. Barker, Roger Howe
1897:Hertrich-Jeromin, Udo (2003).
1884:Geometries and Transformations
1809:
1787:
1781:
1757:
1696:
1681:
1612:
1518:with compact stabilizers. The
1446:of symmetries, while only the
1181:Non-repeating helical symmetry
866:with glide reflection symmetry
820:
814:
650:right-handed coordinate system
622:number. This implies that for
610:Such a "reflection" preserves
177:Basic isometries by dimension
30:A drawing of a butterfly with
1:
1901:. Cambridge University Press.
1886:. Cambridge University Press.
1849:Caldecott, Stratford (2009).
1606:
1189:point group in two dimensions
945:if the rotoreflection has a 2
646:left-handed coordinate system
490:-dimensional space which are
405:with reflection symmetry are
2320:Calotta: A World of Symmetry
1709:. New York: Academic Press.
1703:Miller, Willard Jr. (1972).
1573:3-dimensional model geometry
738: = 3, this is the
603:(180°) rotation; see below.
340:
320:
294:
266:
249:
204:
180:
7:
2180:Sinden, Richard R. (1994).
2046:Kosmann-Schwarzbach, Yvette
2032:Geometry: Euclid and Beyond
1853:. Brazos Press. p. 70.
1589:
1556:/Γ for some model geometry
1525:A model geometry is called
1341:Scale symmetry and fractals
1289:The group of all bijective
783:with translational symmetry
10:
2356:
2182:DNA structure and function
2090:Martin, George E. (1982),
2030:Hartshorne, Robin (2000).
1642:undergroundmathematics.org
1456:projective transformations
1258:that is a scalar times an
1244:similarity transformations
1223:
1195:, with approximately 10.5
1033:
919:
869:
786:
659:
475:
359:
18:
2133:Bottema, O, and B. Roth,
2000:Geometry: Plane and Fancy
1996:Singer, David A. (1998).
1948:American Mathematical Soc
1765:"2.6 Reflection Symmetry"
1548:is a diffeomorphism from
1438:, and hierarchy of their
1430:associated an underlying
1280:This adds, e.g., oblique
1270:is considered a symmetry.
1115:Infinite helical symmetry
967:(not to be confused with
953:), the symmetry group is
854:Glide reflection symmetry
589:inversion through a point
344:
324:
298:
262:
256:
198:
188:
163:orthogonal transformation
21:Geometric symmetry (book)
2222:Introduction to Geometry
2146:George R. McGhee (2006)
2100:, Springer, p. 64,
1462:, which is preserved in
1315:reflection on the plane.
1234:Non-isometric symmetries
1204:Double rotation symmetry
880:symmetry (also called a
711:special orthogonal group
159:Cartan–Dieudonné theorem
77:symmetric under rotation
2267:2 (1892–1893): 215–249.
2135:Theoretical Kinematics,
1864:Bassarear, Tom (2011).
1623:. Springer. p. 28.
1466:, is not meaningful in
1138:n-fold helical symmetry
904:Rotoreflection symmetry
555:in a certain system of
413:with this symmetry are
1880:Johnson, N. W. Johnson
1688:Freitag, Mark (2013).
1398:, is another example.
1354:
1298:Möbius transformations
1291:affine transformations
1248:affine transformations
1221:
1167:
1134:-fold helical symmetry
1110:
1107:Boerdijk–Coxeter helix
1098:
1079:
917:
884:symmetry in 3D, and a
867:
844:
789:Translational symmetry
784:
771:Translational symmetry
675:
473:
35:
2265:Bull. N. Y. Math. Soc
2226:John Wiley & Sons
2015:Joshi, A. W. (2007).
1912:Dieck, Tammo (2008).
1426:With every geometry,
1348:
1211:
1168:
1104:
1088:
1077:
978:). A special case is
911:
861:
845:
778:
765:rotational invariance
669:
557:Cartesian coordinates
471:
362:Reflectional symmetry
356:Reflectional symmetry
349:Rotary transflection
89:reflectional symmetry
29:
1824:. Springer. p.
1669:mathbitsnotebook.com
1311:reflections such as
1152:
914:pentagonal antiprism
800:
740:rotation group SO(3)
100:geometric transforms
16:Geometrical property
1984:Natural Biodynamics
1619:Martin, G. (1996).
1584:Thurston geometries
1542:geometric structure
1468:projective geometry
1448:incidence structure
1432:group of symmetries
1385:, well seen in the
1335:projective geometry
1282:reflection symmetry
982: = 1, an
728:orthogonal matrices
662:Rotational symmetry
656:Rotational symmetry
337:Rotary translation
329:Rotary translation
178:
81:rotational symmetry
2237:William Thurston.
1916:Algebraic Topology
1601:Symmetric relation
1403:patterns in nature
1377:. As conceived by
1368:allometric scaling
1355:
1353:has scale symmetry
1331:Euclidean geometry
1222:
1163:
1162:
1111:
1099:
1080:
1078:A continuous helix
918:
868:
840:
839:
785:
676:
605:Antipodal symmetry
474:
450:fundamental domain
368:bilateral symmetry
176:
36:
32:bilateral symmetry
2303:978-0-387-94153-0
2063:978-0-387-87867-6
1564:acting freely on
1417:Abstract symmetry
1379:Benoît Mandelbrot
1307:This adds, e.g.,
1260:orthogonal matrix
1250:represented by a
1173:is a multiple of
936:improper rotation
928:rotary reflection
922:improper rotation
754:Noether's theorem
684:direct isometries
353:
352:
2347:
2308:
2307:
2289:
2283:
2276:
2270:
2255:
2249:
2235:
2229:
2215:
2209:
2202:
2196:
2195:
2177:
2171:
2165:
2157:
2151:
2144:
2138:
2131:
2125:
2118:
2112:
2110:
2087:
2081:
2078:Timeless Reality
2074:
2068:
2067:
2042:
2036:
2035:
2027:
2021:
2020:
2012:
2006:
2005:
2003:
1993:
1987:
1986:World Scientific
1980:
1974:
1973:
1955:
1949:
1942:
1936:
1935:
1919:
1909:
1903:
1902:
1894:
1888:
1887:
1876:
1870:
1869:
1861:
1855:
1854:
1846:
1840:
1839:
1823:
1820:Tissue Mechanics
1813:
1807:
1806:
1785:
1779:
1778:
1776:
1775:
1769:CK-12 Foundation
1761:
1755:
1754:
1752:
1751:
1742:. Archived from
1736:
1730:
1729:
1727:
1726:
1717:. Archived from
1700:
1694:
1693:
1685:
1679:
1678:
1676:
1675:
1661:
1652:
1651:
1649:
1648:
1634:
1625:
1624:
1616:
1499:simply connected
1491:William Thurston
1327:Erlangen program
1276:
1257:
1172:
1170:
1169:
1164:
1030:Helical symmetry
969:symmetric groups
878:glide reflection
872:Glide reflection
849:
847:
846:
841:
813:
812:
763:. For more, see
761:conservation law
758:angular momentum
748:
737:
726:
716:
705:
695:
681:
632:
626:
617:
598:
585:point reflection
582:
578:
570:
551:
489:
478:Point reflection
179:
175:
136:Euclidean spaces
42:, an object has
2355:
2354:
2350:
2349:
2348:
2346:
2345:
2344:
2330:
2329:
2316:
2311:
2304:
2290:
2286:
2277:
2273:
2256:
2252:
2236:
2232:
2216:
2212:
2203:
2199:
2192:
2178:
2174:
2158:
2154:
2145:
2141:
2132:
2128:
2119:
2115:
2108:
2088:
2084:
2075:
2071:
2064:
2054:Springer-Verlag
2043:
2039:
2028:
2024:
2013:
2009:
1994:
1990:
1981:
1977:
1970:
1956:
1952:
1943:
1939:
1932:
1910:
1906:
1895:
1891:
1877:
1873:
1862:
1858:
1847:
1843:
1836:
1814:
1810:
1803:
1786:
1782:
1773:
1771:
1763:
1762:
1758:
1749:
1747:
1738:
1737:
1733:
1724:
1722:
1701:
1697:
1686:
1682:
1673:
1671:
1663:
1662:
1655:
1646:
1644:
1636:
1635:
1628:
1617:
1613:
1609:
1592:
1544:on a manifold
1502:smooth manifold
1488:
1486:Thurston's view
1464:affine geometry
1458:. A concept of
1444:Euclidean group
1424:
1419:
1343:
1300:which preserve
1274:
1268:self-similarity
1255:
1236:
1228:
1206:
1153:
1150:
1149:
1089:A regular skew-
1038:
1032:
1013:
996:(angle of 360°/
994:
976:
962:
924:
906:
874:
856:
808:
804:
801:
798:
797:
791:
773:
746:
735:
718:
714:
703:
693:
679:
664:
658:
630:
624:
615:
614:if and only if
596:
580:
576:
573:affine subspace
560:
549:
540:
530:
521:
514:
505:
498:
487:
480:
466:
447:
435:
364:
358:
333:Double rotation
314:Rotoreflection
121:Euclidean group
117:
111:of the object.
46:if there is an
24:
17:
12:
11:
5:
2353:
2343:
2342:
2328:
2327:
2322:
2315:
2314:External links
2312:
2310:
2309:
2302:
2284:
2271:
2269:
2268:
2250:
2230:
2218:H.S.M. Coxeter
2210:
2197:
2190:
2172:
2152:
2139:
2126:
2113:
2106:
2082:
2069:
2062:
2037:
2022:
2007:
1988:
1975:
1968:
1950:
1937:
1930:
1904:
1889:
1871:
1856:
1841:
1834:
1808:
1801:
1780:
1756:
1731:
1695:
1680:
1653:
1626:
1610:
1608:
1605:
1604:
1603:
1598:
1591:
1588:
1495:model geometry
1487:
1484:
1423:
1420:
1418:
1415:
1411:virtual worlds
1387:Mandelbrot set
1342:
1339:
1319:
1318:
1317:
1316:
1294:
1287:
1286:
1285:
1271:
1235:
1232:
1214:clifford torus
1205:
1202:
1201:
1200:
1178:
1161:
1158:
1128:
1031:
1028:
1022:For more, see
1020:
1019:
1008:
992:
987:
974:
957:
943:
932:rotoreflection
920:Main article:
905:
902:
870:Main article:
864:frieze pattern
855:
852:
838:
834:
830:
826:
822:
819:
816:
811:
807:
787:Main article:
781:frieze pattern
772:
769:
660:Main article:
657:
654:
640:(P stands for
627: = 3
553:
552:
545:
535:
526:
519:
510:
503:
476:Main article:
465:
462:
445:
433:
426:symmetry group
417:and isosceles
411:quadrilaterals
360:Main article:
357:
354:
351:
350:
347:
345:
343:
339:
338:
335:
330:
327:
325:
323:
319:
318:
317:Transflection
315:
312:
311:Transflection
309:
307:Rotoreflection
304:
303:Transflection
301:
299:
297:
293:
292:
289:
286:
283:
280:
277:
274:
271:
269:
265:
264:
261:
258:
255:
252:
248:
247:
242:
237:
232:
227:
222:
217:
212:
207:
203:
202:
197:
192:
187:
182:
133:solid geometry
129:plane geometry
116:
113:
109:symmetry group
52:transformation
15:
9:
6:
4:
3:
2:
2352:
2341:
2338:
2337:
2335:
2326:
2323:
2321:
2318:
2317:
2305:
2299:
2295:
2288:
2281:
2275:
2266:
2262:
2261:
2259:
2254:
2248:
2247:0-691-08304-5
2244:
2240:
2234:
2227:
2223:
2219:
2214:
2207:
2201:
2193:
2191:9780126457506
2187:
2183:
2176:
2169:
2163:
2156:
2149:
2143:
2136:
2130:
2123:
2117:
2109:
2107:9780387906362
2103:
2099:
2095:
2094:
2086:
2079:
2073:
2065:
2059:
2055:
2051:
2047:
2041:
2033:
2026:
2018:
2011:
2002:
2001:
1992:
1985:
1979:
1971:
1969:0-521-29964-0
1965:
1961:
1954:
1947:
1941:
1933:
1931:9783037190487
1927:
1923:
1918:
1917:
1908:
1900:
1893:
1885:
1881:
1875:
1867:
1860:
1852:
1845:
1837:
1835:9780387368252
1831:
1827:
1822:
1821:
1812:
1804:
1802:0-691-02374-3
1798:
1794:
1790:
1789:Weyl, Hermann
1784:
1770:
1766:
1760:
1746:on 2012-07-23
1745:
1741:
1735:
1721:on 2010-02-17
1720:
1716:
1712:
1708:
1707:
1699:
1691:
1684:
1670:
1666:
1660:
1658:
1643:
1639:
1633:
1631:
1622:
1615:
1611:
1602:
1599:
1597:
1594:
1593:
1587:
1585:
1581:
1577:
1574:
1569:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1538:
1536:
1532:
1528:
1523:
1521:
1517:
1513:
1510:
1506:
1503:
1500:
1496:
1492:
1483:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1414:
1412:
1408:
1404:
1399:
1397:
1392:
1388:
1384:
1383:magnification
1380:
1376:
1371:
1369:
1365:
1361:
1352:
1347:
1338:
1336:
1332:
1328:
1324:
1314:
1310:
1306:
1305:
1303:
1299:
1296:The group of
1295:
1292:
1288:
1283:
1279:
1278:
1272:
1269:
1265:
1261:
1253:
1249:
1245:
1242:The group of
1241:
1240:
1239:
1231:
1227:
1219:
1215:
1210:
1198:
1194:
1190:
1186:
1182:
1179:
1176:
1159:
1156:
1147:
1143:
1139:
1135:
1133:
1129:
1125:
1124:cross section
1120:
1116:
1113:
1112:
1108:
1103:
1096:
1092:
1087:
1083:
1076:
1072:
1069:
1068:coiling angle
1065:
1064:angular speed
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1027:
1025:
1017:
1012:
1007:
1003:
999:
995:
988:
985:
981:
977:
970:
966:
961:
956:
952:
948:
944:
941:
940:
939:
937:
933:
929:
923:
915:
910:
901:
899:
895:
892:
887:
886:transflection
883:
879:
873:
865:
860:
851:
836:
832:
828:
824:
817:
809:
805:
796:
790:
782:
777:
768:
766:
762:
759:
755:
750:
743:
741:
734: 1. For
733:
729:
725:
721:
712:
707:
699:
697:
689:
685:
673:
668:
663:
653:
651:
647:
643:
639:
638:
628:
621:
613:
608:
606:
602:
594:
590:
586:
579: =
574:
571:-dimensional
568:
564:
558:
548:
544:
538:
534:
529:
525:
518:
513:
509:
502:
497:
496:
495:
493:
485:
479:
470:
461:
459:
455:
451:
443:
439:
431:
427:
422:
420:
416:
412:
408:
404:
399:
395:
393:
389:
384:
383:perpendicular
379:
377:
371:
369:
363:
348:
346:
341:
336:
334:
331:
321:
316:
313:
308:
302:
300:
295:
290:
287:
278:
275:
267:
250:
246:
243:
241:
238:
236:
231:
226:
223:
221:
218:
216:
211:
205:
201:
196:
191:
186:
181:
174:
173:reflections.
172:
168:
164:
160:
155:
153:
149:
145:
141:
137:
134:
130:
126:
122:
112:
110:
106:
103:mathematical
101:
96:
94:
93:line symmetry
90:
86:
82:
78:
73:
69:
65:
61:
57:
53:
49:
45:
41:
33:
28:
22:
2293:
2287:
2279:
2278:Tian Yu Cao
2274:
2264:
2258:Klein, Felix
2253:
2238:
2233:
2221:
2213:
2205:
2200:
2181:
2175:
2166:See section
2161:
2155:
2147:
2142:
2134:
2129:
2121:
2116:
2092:
2085:
2077:
2072:
2049:
2040:
2031:
2025:
2016:
2010:
1999:
1991:
1983:
1978:
1959:
1953:
1945:
1940:
1915:
1907:
1898:
1892:
1883:
1874:
1865:
1859:
1850:
1844:
1819:
1811:
1792:
1783:
1772:. Retrieved
1768:
1759:
1748:. Retrieved
1744:the original
1734:
1723:. Retrieved
1719:the original
1705:
1698:
1689:
1683:
1672:. Retrieved
1668:
1645:. Retrieved
1641:
1620:
1614:
1583:
1579:
1575:
1570:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1539:
1534:
1530:
1526:
1524:
1515:
1511:
1504:
1494:
1489:
1479:
1425:
1422:Klein's view
1400:
1372:
1356:
1320:
1302:cross-ratios
1237:
1229:
1180:
1146:double helix
1141:
1137:
1131:
1130:
1114:
1081:
1067:
1042:
1039:
1021:
1015:
1010:
1005:
1001:
997:
990:
979:
972:
964:
959:
954:
950:
946:
935:
931:
927:
925:
897:
893:
891:frieze group
885:
877:
875:
795:translations
792:
751:
744:
723:
719:
708:
700:
677:
634:
609:
604:
588:
566:
562:
554:
546:
542:
536:
532:
527:
523:
516:
511:
507:
500:
481:
438:point groups
423:
400:
396:
380:
376:mirror image
372:
365:
291:Translation
285:Translation
279:Translation
273:Translation
244:
239:
234:
229:
224:
219:
214:
209:
206:Reflections
199:
194:
189:
184:
170:
166:
156:
148:translations
118:
97:
85:plane figure
80:
76:
43:
37:
1460:parallelism
1452:cross-ratio
1428:Felix Klein
1366:(so-called
1323:Felix Klein
1095:perspective
1014:, for even
1000:); for odd
882:glide plane
732:determinant
688:orientation
612:orientation
492:involutions
442:involutions
263:Reflection
260:Reflection
257:Reflection
254:Reflection
140:reflections
79:or to have
56:translation
1774:2019-12-06
1750:2013-04-16
1725:2009-09-28
1674:2019-12-06
1647:2019-12-06
1607:References
1440:invariants
1266:is added,
1224:See also:
1197:base pairs
1185:irrational
1056:drill bits
1036:Screw axis
1034:See also:
989:The group
963:of order 2
672:triskelion
494:, such as
484:isometries
458:half-space
454:half-plane
430:isomorphic
419:trapezoids
125:isometries
72:invariance
68:reflection
2220:(1961,9)
1791:(1982) .
1520:Lie group
1509:Lie group
1394:trees in
1351:Julia set
1309:inversive
1264:homothety
1160:θ
1091:apeirogon
984:inversion
926:In 3D, a
876:In 2D, a
591:. On the
407:isosceles
403:triangles
144:rotations
54:(such as
48:operation
2340:Symmetry
2334:Category
2048:(2010).
1793:Symmetry
1590:See also
1480:a priori
1476:subgroup
1450:and the
1396:dioramas
1375:fractals
1360:elephant
1246:; i.e.,
1140:, where
637:symmetry
587:, or an
522:, ..., −
288:Rotation
282:Rotation
276:Rotation
152:subgroup
64:rotation
44:symmetry
40:geometry
1596:Fractal
1527:maximal
1262:. Thus
1048:springs
1044:Helical
648:into a
541:, ...,
506:, ...,
157:By the
60:scaling
2300:
2245:
2188:
2104:
2060:
1966:
1928:
1832:
1799:
1715:589081
1713:
1472:groups
1436:groups
1362:and a
1313:circle
1254:
1252:matrix
1060:augers
1058:, and
1054:toys,
1052:Slinky
642:parity
618:is an
515:) ↦ (−
448:. The
432:with C
409:, the
392:circle
388:square
245:Affine
235:Affine
225:Affine
215:Affine
107:, the
1497:is a
1391:coast
1364:mouse
1218:torus
1212:A 4D
1119:helix
730:with
593:plane
575:. If
452:is a
436:(see
415:kites
240:Point
230:Point
220:Point
210:Point
161:, an
105:group
2298:ISBN
2243:ISBN
2186:ISBN
2102:ISBN
2058:ISBN
1964:ISBN
1926:ISBN
1830:ISBN
1797:ISBN
1711:OCLC
1552:to
1389:. A
1175:360°
1105:The
894:p11g
670:The
620:even
601:turn
401:The
123:of "
1922:261
1826:152
1529:if
1514:on
1325:'s
1321:In
1193:DNA
934:or
713:SO(
486:of
456:or
428:is
378:).
165:in
131:or
91:or
66:or
50:or
38:In
2336::
2096:,
2056:.
1924:.
1828:.
1767:.
1667:.
1656:^
1640:.
1629:^
1571:A
1540:A
1413:.
1349:A
1337:.
1304:.
1191:.
1050:,
1026:.
993:nh
975:2n
930:,
912:A
900:.
862:A
779:A
767:.
742:.
722:×
702:E(
698:.
692:E(
635:P-
539:+1
531:,
460:.
421:.
342:5
322:4
296:3
268:2
251:1
200:4D
195:3D
190:2D
185:1D
146:,
142:,
62:,
58:,
2306:.
2228:.
2194:.
2170:.
2111:.
2066:.
1972:.
1934:.
1838:.
1805:.
1777:.
1753:.
1728:.
1677:.
1650:.
1580:X
1576:X
1566:X
1562:G
1558:X
1554:X
1550:M
1546:M
1535:X
1531:G
1516:X
1512:G
1505:X
1293:.
1284:.
1275:A
1256:A
1157:m
1142:n
1132:n
1097:.
1016:n
1011:n
1009:2
1006:C
1002:n
998:n
991:C
980:n
973:C
965:n
960:n
958:2
955:S
951:n
947:n
898:Z
837:a
833:+
829:p
825:=
821:)
818:p
815:(
810:a
806:T
747:m
736:m
724:m
720:m
715:m
704:m
696:)
694:m
680:m
631:m
625:m
616:k
597:m
595:(
581:m
577:k
569:)
567:k
565:−
563:m
561:(
550:)
547:m
543:x
537:k
533:x
528:k
524:x
520:1
517:x
512:m
508:x
504:1
501:x
499:(
488:m
446:2
434:s
171:n
167:n
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
