1767:
common bonds. Polyhedra containing inversion centers are known as centrosymmetric, while those without are non-centrosymmetric. Six-coordinate octahedra are an example of centrosymmetric polyhedra, as the central atom acts as an inversion center through which the six bonded atoms retain symmetry. Tetrahedra, on the other hand, are non-centrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. Polyhedra with an odd (versus even) coordination number are not centrosymmtric.
169:
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530:
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50:
1784:
oriented in space in a manner which contains an inversion center between the two. Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair. The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group.
1775:
fluorine. Distortions will not change the inherent geometry of the polyhedra—a distorted octahedron is still classified as an octahedron, but strong enough distortions can have an effect on the centrosymmetry of a compound. Disorder involves a split occupancy over two or more sites, in which an atom
1770:
Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry. Common irregularities found in crystallography include distortions and disorder. Distortion involves the warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic attraction
1766:
depending on the bonding angles. All crystalline compounds come from a repetition of an atomic building block known as a unit cell, and these unit cells define which polyhedra form and in what order. These polyhedra link together via corner-, edge- or face sharing, depending on which atoms share
1783:
which describe how the different polyhedra arrange themselves in space in the bulk structure. Of these thirty-two point groups, eleven are centrosymmetric. The presence of noncentrosymmetric polyhedra does not guarantee that the point group will be the same—two non-centrosymmetric shapes can be
1795:. The lack of symmetry via inversion centers can allow for areas of the crystal to interact differently with incoming light. The wavelength, frequency and intensity of light is subject to change as the electromagnetic radiation interacts with different energy states throughout the structure.
1776:
will occupy one crystallographic position in a certain percentage of polyhedra and the other in the remaining positions. Disorder can influence the centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center.
870:
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673:
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and non-centrosymmetric compounds. Crystal structures are composed of various polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as
2323:
2024:
1771:
between heteroatoms. For instance, a titanium center will likely bond evenly to six oxygens in an octahedra, but distortion would occur if one of the oxygens were replaced with a more
2604:
2524:
1067:
1157:
2449:+1) (it is in the non-identity component), and there is no natural sense in which it is a "farther point" than any other point in the non-identity component, but it does provide a
2243:
1611:
452:
1116:
2572:
1851:. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion; in physics 3-dimensional reflection through the origin is also called a
2389:
2180:
1017:
1405:
2095:
493:
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2650:
2627:
2132:
2059:
1929:
1906:
1949:
1165:
1819:. The applications for nonlinear materials are still being researched, but these properties stem from the presence of (or lack thereof) an inversion center.
608:
2417:
Analogously, it is a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have
2182:(from the representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the
1763:
1520:. These rotations are mutually commutative. Therefore, inversion in a point in even-dimensional space is an orientation-preserving isometry or
2794:
1759:
1279:
2756:"Orthogonal planes" meaning all elements are orthogonal and the planes intersect at 0 only, not that they intersect in a line and have
2402:
133:
67:
1779:
Centrosymmetry applies to the crystal structure as a whole, not just individual polyhedra. Crystals are classified into thirty-two
1649:
2853:
2670:
114:
2712:
2429:
though it is not unique in this: other maximal combinations of rotations (and possibly reflections) also have maximal length.
1755:
86:
71:
1417:
1745:
Molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In
2848:
1780:
2441:), reflection through the origin is the farthest point from the identity element with respect to the usual metric. In O(2
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2769:
This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the
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808:
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237:
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1586:(in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are
82:
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241:
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with an isometric reflection across each point. Symmetric spaces play an important role in the study of
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with reflection across the plane of rotation, perpendicular to the axis of rotation. In dimension
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of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation
2533:
913:
729:
690:
507:
412:
is applied to any involution of
Euclidean space, and the fixed set (an affine space of dimension
297:
2425:
with respect to the generating set of reflections, and reflection through the origin has length
2347:
1022:
2702:
2682:
2149:
1638:
869:
854:
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either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity
353:
352:, meaning that they have order 2 – they are their own inverse: applying them twice yields the
264:
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212:
107:
2770:
2722:
2717:
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1828:
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1734:
987:
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38:
31:
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on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity
2707:
2418:
2071:
1952:
1863:
1812:
1772:
1750:
1746:
1564:
1528:
1521:
1493:
1263:{\displaystyle {\begin{cases}x_{c}={\frac {x+x'}{2}}\\y_{c}={\frac {y+y'}{2}}\end{cases}}}
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909:
835:
401:
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Example of a 2-dimensional figure with central symmetry, invariant under point reflection
2632:
2609:
2114:
2041:
1911:
1888:
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1934:
1630:
849:
does not coincide with the origin, point reflection is equivalent to a special case of
668:{\displaystyle \mathrm {Ref} _{\mathbf {p} }(\mathbf {a} )=2\mathbf {p} -\mathbf {a} .}
529:
462:
281:
461:
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the
2842:
2406:
2035:
953:
1804:
921:
905:
717:
541:
535:
204:
928:
by negation. It is precisely the subgroup of the
Euclidean group that fixes the
873:
The composition of two offset point reflections in 2-dimensions is a translation.
2190:), and it is orientation-reversing in odd dimension, thus not an element of SO(2
1808:
276:
146:
49:
17:
834:
coincides with the origin, point reflection is equivalent to a special case of
2737:
2527:
2468:
2450:
466:
367:
968:
305:
2138:
orthogonal planes; note again that rotations in orthogonal planes commute.
857:
coinciding with P, and scale factor −1. (This is an example of non-linear
2111:
In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension
1811:, and is a useful non-linear crystal. KTP is used for frequency-doubling
1571:
1119:
805:
555:
458:. In dimension 1 these coincide, as a point is a hyperplane in the line.
408:, a plane in 3-space), with the hyperplane being fixed, but more broadly
319:
308:
293:
184:
1273:
Hence, the equations to find the coordinates of the reflected point are
259:
An object that is invariant under a point reflection is said to possess
838:: uniform scaling with scale factor equal to −1. This is an example of
546:
1582:
which is perpendicular to the axis; the result does not depend on the
1555:-dimensional subspace spanned by these rotation planes. Therefore, it
682:
is the origin, point reflection is simply the negation of the vector
315:
1578:
by an angle of 180°, combined with reflection in the plane through
1365:{\displaystyle {\begin{cases}x'=2x_{c}-x\\y'=2y_{c}-y\end{cases}}}
868:
826:
Point reflection as a special case of uniform scaling or homothety
145:
2655:
Reflection through the identity extends to an automorphism of a
904:
The set consisting of all point reflections and translations is
2398:
It equals the identity if and only if the characteristic is 2.
43:
1375:
Particular is the case in which the point C has coordinates
340:
is loose, and considered by some an abuse of language, with
279:
including a point reflection among its symmetries is called
30:"Central inversion" redirects here. Not to be confused with
2104:
orthogonal reflections (reflection through the axes of any
1470:
1358:
1256:
2530:. This is particularly confusing for even spin groups, as
1749:, the presence of inversion centers distinguishes between
952:
Point reflection across the center of a sphere yields the
177:
Dual tetrahedra that are centrally symmetric to each other
2061:
on the diagonal, and, together with the identity, is the
1408:
220:
263:; if it is invariant under point reflection through its
1737:, which can be thought of as a "inversion in a plane".
1535: + 1)-dimensional space, it is equivalent to
554:
In two dimensions, a point reflection is the same as a
1477:{\displaystyle {\begin{cases}x'=-x\\y'=-y\end{cases}}}
521:
2635:
2612:
2580:
2536:
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2350:
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2204:
2152:
2117:
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by −1. The operation commutes with every other
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1420:
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1282:
1168:
1128:
1075:
1025:
990:
611:
475:
422:
376:
2038:, it is represented in every basis by a matrix with
1827:
Inversion with respect to the origin corresponds to
207:
in which every point is reflected across a specific
74:. Unsourced material may be challenged and removed.
2644:
2621:
2598:
2566:
2518:
2445:+ 1), reflection through the origin is not in SO(2
2383:
2317:
2237:
2174:
2126:
2089:
2053:
2018:
1943:
1923:
1900:
1815:, utilizing a nonlinear optical property known as
1629:, and 1×. The group type is one of the three
1605:
1476:
1399:
1364:
1262:
1151:
1110:
1061:
1011:
667:
487:
446:
388:
2318:{\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}}
1803:(KTP). crystalizes in the non-centrosymmetric,
2108:); note that orthogonal reflections commute.
1787:Non-centrosymmetric insulating compounds are
1500:-dimensional space, the inversion in a point
8:
2312:
2303:
1758:, while five-coordinate environments can be
939:= 1, the point reflection group is the full
2669:Reflection through the identity lifts to a
2344:It preserves every quadratic form, meaning
1547:mutually orthogonal planes intersecting at
1516:mutually orthogonal planes intersecting at
318:); a point reflection through the object's
356:– which is also true of other maps called
2634:
2611:
2579:
2535:
2493:
2482:
2349:
2333:Together with the identity, it forms the
2250:
2203:
2166:
2151:
2116:
2073:
2043:
2019:{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}
1960:
1936:
1913:
1890:
1729:Closely related to inverse in a point is
1593:
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1343:
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375:
134:Learn how and when to remove this message
2599:{\displaystyle \operatorname {Spin} (n)}
2519:{\displaystyle -1\in \mathrm {Spin} (n)}
2194: + 1) and instead providing a
252:. The point of inversion is also called
2786:
2749:
1877:. Reflection through the origin is an
1551:, combined with the reflection in the 2
574:is even, and orientation-reversing if
304:, a point reflection is the same as a
1791:and can be useful for application in
1543:in each plane of an arbitrary set of
1512:in each plane of an arbitrary set of
980:Point reflection in analytic geometry
885:. Specifically, point reflection at
514:is defined with respect to a circle.
7:
1831:of the position vector, and also to
1823:Inversion with respect to the origin
1741:Inversion centers in crystallography
594:, the formula for the reflection of
72:adding citations to reliable sources
2134:, it is rotation by 180 degrees in
1955:. In three dimensions, this sends
2503:
2500:
2497:
2494:
2391:, and thus is an element of every
1862:refers to the point reflection of
1570:Geometrically in 3D it amounts to
620:
617:
614:
25:
2457:Clifford algebras and spin groups
1152:{\displaystyle {\overline {PP'}}}
924:of order 2, the latter acting on
751:The formula for the inversion in
2238:{\displaystyle O(2n+1)\to \pm 1}
1650:point groups in three dimensions
947:Point reflections in mathematics
889:followed by point reflection at
658:
650:
636:
626:
540:
528:
167:
158:
48:
1606:{\displaystyle {\overline {1}}}
893:is translation by the vector 2(
740:is the same as the vector from
447:{\displaystyle 1\leq k\leq n-1}
59:needs additional citations for
2593:
2587:
2561:
2552:
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2507:
2378:
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2363:
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2297:
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2208:
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2153:
2084:
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2013:
1986:
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1962:
1394:
1382:
1111:{\displaystyle C(x_{c},y_{c})}
1105:
1079:
1056:
1034:
1006:
994:
881:of two point reflections is a
785:* are the position vectors of
640:
632:
348:is widely used. Such maps are
1:
2477:be confused with the element
1908:, and can also be written as
1860:reflection through the origin
1781:crystallographic point groups
1633:types in 3D without any pure
2567:{\displaystyle -I\in SO(2n)}
1598:
1144:
526:
506:should not be confused with
366:refers to a reflection in a
27:Geometric symmetry operation
2393:indefinite orthogonal group
1875:Cartesian coordinate system
1797:Potassium titanyl phosphate
322:is the same as a half-turn
292:, a point reflection is an
240:: applying it twice is the
2880:
2733:Riemannian symmetric space
2713:Kovner–Besicovitch measure
2466:
2460:
2384:{\displaystyle Q(-v)=Q(v)}
1817:second-harmonic generation
1069:with respect to the point
36:
29:
2823:sites.math.washington.edu
1879:orthogonal transformation
1062:{\displaystyle P'(x',y')}
851:homothetic transformation
830:When the inversion point
246:homothetic transformation
236:A point reflection is an
2819:"Lab 9 Point Reflection"
2728:Reflection (mathematics)
2453:in the other component.
2184:special orthogonal group
2175:{\displaystyle (-1)^{n}}
704:with respect to a point
566:, point reflections are
267:, it is said to possess
244:. It is equivalent to a
233:are more commonly used.
37:Not to be confused with
2327:internal direct product
865:Point reflection group
728:*. In other words, the
716:is the midpoint of the
590:in the Euclidean space
242:identity transformation
2854:Functions and mappings
2795:"Reflections in Lines"
2646:
2623:
2600:
2568:
2520:
2385:
2319:
2239:
2176:
2128:
2091:
2055:
2020:
1945:
1925:
1902:
1813:neodymium-doped lasers
1607:
1574:about an axis through
1559:rather than preserves
1508:rotations over angles
1478:
1401:
1366:
1264:
1153:
1112:
1063:
1013:
1012:{\displaystyle P(x,y)}
874:
814:which has exactly one
669:
489:
448:
390:
151:
2698:Congruence (geometry)
2647:
2624:
2601:
2569:
2521:
2467:Further information:
2461:Further information:
2386:
2320:
2240:
2177:
2129:
2092:
2056:
2021:
1946:
1926:
1903:
1883:scalar multiplication
1853:parity transformation
1837:linear transformation
1833:scalar multiplication
1608:
1479:
1402:
1400:{\displaystyle (0,0)}
1367:
1265:
1154:
1113:
1064:
1014:
872:
859:affine transformation
840:linear transformation
812:affine transformation
670:
490:
449:
391:
149:
2849:Euclidean symmetries
2633:
2610:
2578:
2534:
2481:
2348:
2249:
2202:
2150:
2115:
2090:{\displaystyle O(n)}
2072:
2042:
1959:
1935:
1912:
1889:
1849:general linear group
1764:trigonal bipyramidal
1590:
1492:In even-dimensional
1418:
1379:
1280:
1166:
1126:
1118:, the latter is the
1073:
1023:
988:
609:
473:
420:
374:
344:preferred; however,
211:. When dealing with
68:improve this article
2411:signed permutations
2146:It has determinant
2100:It is a product of
1652:contain inversion:
1635:rotational symmetry
1527:In odd-dimensional
1019:and its reflection
973:Riemannian geometry
965:Riemannian manifold
524:
488:{\displaystyle n-1}
389:{\displaystyle n-1}
360:. More narrowly, a
273:centrally symmetric
2645:{\displaystyle -I}
2642:
2622:{\displaystyle -1}
2619:
2596:
2564:
2516:
2381:
2315:
2235:
2172:
2127:{\displaystyle 2n}
2124:
2087:
2054:{\displaystyle -1}
2051:
2016:
1941:
1924:{\displaystyle -I}
1921:
1901:{\displaystyle -1}
1898:
1829:additive inversion
1603:
1474:
1469:
1397:
1362:
1357:
1260:
1255:
1149:
1108:
1059:
1009:
914:semidirect product
875:
691:Euclidean geometry
678:In the case where
665:
522:
508:inversive geometry
485:
444:
386:
248:with scale factor
222:inversion symmetry
213:crystal structures
152:
83:"Point reflection"
2859:Clifford algebras
2799:new.math.uiuc.edu
2703:Estermann measure
2683:Affine involution
2665:grade involution.
1944:{\displaystyle I}
1881:corresponding to
1639:cyclic symmetries
1601:
1565:indirect isometry
1504:is equivalent to
1251:
1211:
1147:
855:homothetic center
853:: homothety with
598:across the point
552:
551:
400:– a point on the
254:homothetic center
217:physical sciences
197:central inversion
144:
143:
136:
118:
16:(Redirected from
2871:
2833:
2832:
2830:
2829:
2815:
2809:
2808:
2806:
2805:
2791:
2774:
2771:spectral theorem
2767:
2761:
2754:
2723:Parity (physics)
2718:Orthogonal group
2693:Clifford algebra
2688:Circle inversion
2657:Clifford algebra
2651:
2649:
2648:
2643:
2628:
2626:
2625:
2620:
2605:
2603:
2602:
2597:
2573:
2571:
2570:
2565:
2525:
2523:
2522:
2517:
2506:
2463:Clifford algebra
2390:
2388:
2387:
2382:
2339:orthogonal group
2324:
2322:
2321:
2316:
2244:
2242:
2241:
2236:
2181:
2179:
2178:
2173:
2171:
2170:
2133:
2131:
2130:
2125:
2106:orthogonal basis
2096:
2094:
2093:
2088:
2067:orthogonal group
2060:
2058:
2057:
2052:
2026:, and so forth.
2025:
2023:
2022:
2017:
1950:
1948:
1947:
1942:
1930:
1928:
1927:
1922:
1907:
1905:
1904:
1899:
1858:In mathematics,
1793:nonlinear optics
1760:square pyramidal
1733:in respect to a
1645: = 1.
1612:
1610:
1609:
1604:
1602:
1594:
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1511:
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1142:
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1117:
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1114:
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1103:
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1090:
1068:
1066:
1065:
1060:
1055:
1044:
1033:
1018:
1016:
1015:
1010:
984:Given the point
930:line at infinity
797:* respectively.
674:
672:
671:
666:
661:
653:
639:
631:
630:
629:
623:
544:
532:
525:
494:
492:
491:
486:
454:) is called the
453:
451:
450:
445:
404:, a line in the
395:
393:
392:
387:
346:point reflection
314:
269:central symmetry
251:
227:inversion center
189:point reflection
171:
162:
139:
132:
128:
125:
119:
117:
76:
52:
44:
39:Reflection point
32:Circle inversion
21:
18:Central symmetry
2879:
2878:
2874:
2873:
2872:
2870:
2869:
2868:
2864:Quadratic forms
2839:
2838:
2837:
2836:
2827:
2825:
2817:
2816:
2812:
2803:
2801:
2793:
2792:
2788:
2783:
2778:
2777:
2773:, for instance.
2768:
2764:
2755:
2751:
2746:
2708:Euclidean group
2679:
2661:main involution
2631:
2630:
2629:and 2 lifts of
2608:
2607:
2576:
2575:
2532:
2531:
2479:
2478:
2471:
2465:
2459:
2435:
2403:longest element
2346:
2345:
2247:
2246:
2245:, showing that
2200:
2199:
2162:
2148:
2147:
2144:
2113:
2112:
2070:
2069:
2040:
2039:
2032:
2030:Representations
1957:
1956:
1953:identity matrix
1933:
1932:
1910:
1909:
1887:
1886:
1864:Euclidean space
1843:: it is in the
1839:, but not with
1825:
1802:
1773:electronegative
1751:centrosymmetric
1747:crystallography
1743:
1725:
1718:
1711:
1699:
1689:
1674:
1664:
1628:
1621:
1588:
1587:
1540:
1539:rotations over
1529:Euclidean space
1522:direct isometry
1509:
1494:Euclidean space
1490:
1468:
1467:
1450:
1447:
1446:
1429:
1422:
1416:
1415:
1409:paragraph below
1377:
1376:
1356:
1355:
1339:
1325:
1322:
1321:
1305:
1291:
1284:
1278:
1277:
1254:
1253:
1239:
1232:
1217:
1214:
1213:
1199:
1192:
1177:
1170:
1164:
1163:
1135:
1131:
1124:
1123:
1122:of the segment
1095:
1082:
1071:
1070:
1048:
1037:
1026:
1021:
1020:
986:
985:
982:
961:symmetric space
949:
910:Euclidean group
867:
836:uniform scaling
828:
720:with endpoints
612:
607:
606:
586:Given a vector
584:
570:-preserving if
545:
533:
520:
471:
470:
418:
417:
398:affine subspace
372:
371:
334:
312:
302:Euclidean plane
290:Euclidean space
282:centrosymmetric
249:
231:centrosymmetric
193:point inversion
191:(also called a
181:
180:
179:
178:
174:
173:
172:
164:
163:
140:
129:
123:
120:
77:
75:
65:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
2877:
2875:
2867:
2866:
2861:
2856:
2851:
2841:
2840:
2835:
2834:
2810:
2785:
2784:
2782:
2779:
2776:
2775:
2762:
2758:dihedral angle
2748:
2747:
2745:
2742:
2741:
2740:
2735:
2730:
2725:
2720:
2715:
2710:
2705:
2700:
2695:
2690:
2685:
2678:
2675:
2641:
2638:
2618:
2615:
2606:there is both
2595:
2592:
2589:
2586:
2583:
2574:, and thus in
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2515:
2512:
2509:
2505:
2502:
2499:
2496:
2492:
2489:
2486:
2458:
2455:
2434:
2431:
2415:
2414:
2399:
2396:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2342:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2169:
2165:
2161:
2158:
2155:
2143:
2140:
2123:
2120:
2086:
2083:
2080:
2077:
2050:
2047:
2031:
2028:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1940:
1920:
1917:
1897:
1894:
1824:
1821:
1800:
1742:
1739:
1727:
1726:
1723:
1716:
1709:
1704:
1694:
1684:
1679:
1669:
1659:
1648:The following
1631:symmetry group
1626:
1617:
1600:
1597:
1489:
1486:
1485:
1484:
1471:
1466:
1463:
1460:
1456:
1453:
1449:
1448:
1445:
1442:
1439:
1435:
1432:
1428:
1427:
1425:
1396:
1393:
1390:
1387:
1384:
1373:
1372:
1359:
1354:
1351:
1346:
1342:
1338:
1335:
1331:
1328:
1324:
1323:
1320:
1317:
1312:
1308:
1304:
1301:
1297:
1294:
1290:
1289:
1287:
1271:
1270:
1257:
1250:
1245:
1242:
1238:
1235:
1229:
1224:
1220:
1216:
1215:
1210:
1205:
1202:
1198:
1195:
1189:
1184:
1180:
1176:
1175:
1173:
1146:
1141:
1138:
1134:
1107:
1102:
1098:
1094:
1089:
1085:
1081:
1078:
1058:
1054:
1051:
1047:
1043:
1040:
1036:
1032:
1029:
1008:
1005:
1002:
999:
996:
993:
981:
978:
977:
976:
957:
948:
945:
941:isometry group
866:
863:
827:
824:
771:
770:
676:
675:
664:
660:
656:
652:
648:
645:
642:
638:
634:
628:
622:
619:
616:
583:
580:
550:
549:
538:
519:
516:
484:
481:
478:
465:maps with all
463:diagonalizable
443:
440:
437:
434:
431:
428:
425:
385:
382:
379:
333:
330:
261:point symmetry
201:transformation
176:
175:
166:
165:
157:
156:
155:
154:
153:
142:
141:
56:
54:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2876:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2846:
2844:
2824:
2820:
2814:
2811:
2800:
2796:
2790:
2787:
2780:
2772:
2766:
2763:
2759:
2753:
2750:
2743:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2680:
2676:
2674:
2672:
2667:
2666:
2662:
2659:, called the
2658:
2653:
2639:
2636:
2616:
2613:
2590:
2584:
2581:
2558:
2555:
2549:
2546:
2543:
2540:
2537:
2529:
2510:
2490:
2487:
2484:
2476:
2470:
2464:
2456:
2454:
2452:
2448:
2444:
2440:
2432:
2430:
2428:
2424:
2420:
2412:
2408:
2407:Coxeter group
2404:
2400:
2397:
2394:
2375:
2369:
2366:
2360:
2357:
2351:
2343:
2340:
2336:
2332:
2331:
2330:
2328:
2309:
2306:
2300:
2294:
2291:
2288:
2285:
2279:
2276:
2273:
2267:
2264:
2261:
2258:
2252:
2232:
2229:
2220:
2217:
2214:
2211:
2205:
2197:
2193:
2189:
2185:
2167:
2159:
2156:
2141:
2139:
2137:
2121:
2118:
2109:
2107:
2103:
2098:
2081:
2075:
2068:
2064:
2048:
2045:
2037:
2036:scalar matrix
2029:
2027:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1977:
1974:
1971:
1968:
1965:
1954:
1938:
1918:
1915:
1895:
1892:
1884:
1880:
1876:
1872:
1868:
1865:
1861:
1856:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1822:
1820:
1818:
1814:
1810:
1806:
1798:
1794:
1790:
1789:piezoelectric
1785:
1782:
1777:
1774:
1768:
1765:
1761:
1757:
1752:
1748:
1740:
1738:
1736:
1732:
1722:
1715:
1708:
1705:
1703:
1697:
1693:
1688:
1683:
1680:
1678:
1672:
1668:
1662:
1658:
1655:
1654:
1653:
1651:
1646:
1644:
1640:
1636:
1632:
1625:
1620:
1616:
1595:
1585:
1581:
1577:
1573:
1568:
1566:
1562:
1558:
1554:
1550:
1546:
1538:
1534:
1530:
1525:
1523:
1519:
1515:
1507:
1503:
1499:
1495:
1487:
1464:
1461:
1458:
1454:
1451:
1443:
1440:
1437:
1433:
1430:
1423:
1414:
1413:
1412:
1410:
1391:
1388:
1385:
1352:
1349:
1344:
1340:
1336:
1333:
1329:
1326:
1318:
1315:
1310:
1306:
1302:
1299:
1295:
1292:
1285:
1276:
1275:
1274:
1248:
1243:
1240:
1236:
1233:
1227:
1222:
1218:
1208:
1203:
1200:
1196:
1193:
1187:
1182:
1178:
1171:
1162:
1161:
1160:
1139:
1136:
1132:
1121:
1100:
1096:
1092:
1087:
1083:
1076:
1052:
1049:
1045:
1041:
1038:
1030:
1027:
1003:
1000:
997:
991:
979:
974:
970:
966:
962:
958:
955:
954:antipodal map
951:
950:
946:
944:
943:of the line.
942:
938:
933:
931:
927:
923:
919:
915:
911:
907:
902:
900:
896:
892:
888:
884:
880:
871:
864:
862:
860:
856:
852:
848:
843:
841:
837:
833:
825:
823:
821:
817:
813:
810:
807:
803:
798:
796:
792:
788:
784:
780:
776:
769:
765:
761:
758:
757:
756:
754:
749:
747:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
700:
696:
692:
687:
685:
681:
662:
654:
646:
643:
605:
604:
603:
601:
597:
593:
589:
581:
579:
577:
573:
569:
565:
561:
557:
548:
543:
539:
537:
531:
527:
517:
515:
513:
509:
505:
500:
498:
482:
479:
476:
468:
464:
459:
457:
441:
438:
435:
432:
429:
426:
423:
415:
411:
407:
403:
399:
383:
380:
377:
369:
365:
364:
359:
355:
351:
347:
343:
339:
331:
329:
327:
326:
321:
317:
310:
307:
303:
299:
295:
291:
286:
284:
283:
278:
274:
270:
266:
262:
257:
255:
247:
243:
239:
234:
232:
228:
224:
223:
218:
214:
210:
206:
202:
198:
194:
190:
186:
170:
161:
148:
138:
135:
127:
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
2826:. Retrieved
2822:
2813:
2802:. Retrieved
2798:
2789:
2765:
2752:
2671:pseudoscalar
2668:
2664:
2660:
2654:
2474:
2472:
2446:
2442:
2438:
2436:
2426:
2422:
2416:
2191:
2187:
2145:
2135:
2110:
2101:
2099:
2033:
1866:
1859:
1857:
1826:
1805:orthorhombic
1786:
1778:
1769:
1744:
1728:
1720:
1713:
1706:
1701:
1695:
1691:
1686:
1681:
1676:
1670:
1666:
1660:
1656:
1647:
1642:
1623:
1618:
1614:
1579:
1575:
1569:
1556:
1552:
1548:
1544:
1536:
1532:
1526:
1517:
1513:
1505:
1501:
1497:
1491:
1374:
1272:
983:
936:
935:In the case
934:
925:
922:cyclic group
917:
906:Lie subgroup
903:
898:
894:
890:
886:
876:
846:
844:
831:
829:
819:
799:
794:
790:
786:
782:
778:
774:
772:
767:
763:
759:
752:
750:
745:
741:
737:
733:
725:
721:
718:line segment
713:
712:* such that
709:
705:
701:
694:
688:
683:
679:
677:
599:
595:
591:
587:
585:
575:
571:
563:
553:
536:parallelogon
523:2D examples
511:
503:
501:
496:
460:
455:
413:
409:
396:dimensional
361:
357:
354:identity map
345:
341:
337:
335:
324:
287:
280:
272:
268:
260:
258:
235:
230:
226:
221:
205:affine space
196:
192:
188:
182:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
2198:of the map
1869:across the
1841:translation
1809:space group
1584:orientation
1563:, it is an
1561:orientation
932:pointwise.
912:. It is a
883:translation
879:composition
818:, which is
816:fixed point
708:is a point
568:orientation
467:eigenvalues
358:reflections
350:involutions
332:Terminology
296:(preserves
277:point group
215:and in the
209:fixed point
2843:Categories
2828:2024-04-27
2804:2024-04-27
2781:References
2738:Spin group
2528:spin group
2473:It should
2469:Spin group
2451:base point
2401:It is the
2142:Properties
1756:tetrahedra
1731:reflection
1488:Properties
969:Lie groups
809:involutive
534:Hexagonal
410:reflection
368:hyperplane
363:reflection
338:reflection
300:). In the
238:involution
219:the terms
94:newspapers
2637:−
2614:−
2585:
2544:∈
2538:−
2491:∈
2485:−
2358:−
2307:±
2301:×
2230:±
2227:→
2196:splitting
2157:−
2046:−
2008:−
1999:−
1990:−
1984:↦
1916:−
1893:−
1675:for even
1599:¯
1462:−
1441:−
1407:(see the
1350:−
1316:−
1145:¯
806:isometric
695:inversion
655:−
512:inversion
504:inversion
502:The term
480:−
439:−
433:≤
427:≤
381:−
342:inversion
336:The term
311:(180° or
306:half-turn
271:or to be
2677:See also
2433:Geometry
2421:at most
2395:as well.
1931:, where
1799:, KTiOPO
1700:for odd
1572:rotation
1557:reverses
1531:, say (2
1455:′
1434:′
1330:′
1296:′
1244:′
1204:′
1140:′
1120:midpoint
1053:′
1042:′
1031:′
897: −
578:is odd.
560:composed
556:rotation
518:Examples
510:, where
416:, where
320:centroid
309:rotation
298:distance
294:isometry
185:geometry
124:May 2024
2526:in the
2437:In SO(2
2405:of the
2337:of the
2065:of the
1951:is the
1873:of the
1847:of the
1496:, say 2
920:with a
908:of the
802:mapping
582:Formula
547:Octagon
316:radians
199:) is a
108:scholar
2419:length
2335:center
2325:as an
2063:center
1871:origin
1845:center
1807:Pna21
1719:, and
1637:, see
804:is an
773:where
730:vector
693:, the
456:mirror
265:center
110:
103:
96:
89:
81:
2744:Notes
2034:As a
1735:plane
1641:with
963:is a
845:When
800:This
762:* = 2
732:from
699:point
697:of a
406:plane
115:JSTOR
101:books
2760:90°.
2582:Spin
2186:SO(2
1690:and
1665:and
971:and
877:The
793:and
781:and
724:and
402:line
325:spin
275:. A
187:, a
87:news
2663:or
2475:not
2409:of
1885:by
1762:or
916:of
901:).
861:.)
755:is
748:*.
744:to
736:to
689:In
602:is
499:).
288:In
229:or
203:of
195:or
183:In
70:by
2845::
2821:.
2797:.
2673:.
2652:.
2427:n,
2329:.
2097:.
1855:.
1712:,
1622:,
1613:,
1567:.
1524:.
1411:)
1159:;
959:A
842:.
822:.
789:,
777:,
766:−
686:.
328:.
285:.
256:.
250:−1
225:,
2831:.
2807:.
2640:I
2617:1
2594:)
2591:n
2588:(
2562:)
2559:n
2556:2
2553:(
2550:O
2547:S
2541:I
2514:)
2511:n
2508:(
2504:n
2501:i
2498:p
2495:S
2488:1
2447:r
2443:r
2439:r
2423:n
2413:.
2379:)
2376:v
2373:(
2370:Q
2367:=
2364:)
2361:v
2355:(
2352:Q
2341:.
2313:}
2310:I
2304:{
2298:)
2295:1
2292:+
2289:n
2286:2
2283:(
2280:O
2277:S
2274:=
2271:)
2268:1
2265:+
2262:n
2259:2
2256:(
2253:O
2233:1
2224:)
2221:1
2218:+
2215:n
2212:2
2209:(
2206:O
2192:n
2188:n
2168:n
2164:)
2160:1
2154:(
2136:n
2122:n
2119:2
2102:n
2085:)
2082:n
2079:(
2076:O
2049:1
2014:)
2011:z
2005:,
2002:y
1996:,
1993:x
1987:(
1981:)
1978:z
1975:,
1972:y
1969:,
1966:x
1963:(
1939:I
1919:I
1896:1
1867:R
1801:4
1724:h
1721:I
1717:h
1714:O
1710:h
1707:T
1702:n
1698:d
1696:n
1692:D
1687:n
1685:2
1682:S
1677:n
1673:h
1671:n
1667:D
1663:h
1661:n
1657:C
1643:n
1627:2
1624:S
1619:i
1615:C
1596:1
1580:P
1576:P
1553:N
1549:P
1545:N
1541:π
1537:N
1533:N
1518:P
1514:N
1510:π
1506:N
1502:P
1498:N
1465:y
1459:=
1452:y
1444:x
1438:=
1431:x
1424:{
1395:)
1392:0
1389:,
1386:0
1383:(
1353:y
1345:c
1341:y
1337:2
1334:=
1327:y
1319:x
1311:c
1307:x
1303:2
1300:=
1293:x
1286:{
1249:2
1241:y
1237:+
1234:y
1228:=
1223:c
1219:y
1209:2
1201:x
1197:+
1194:x
1188:=
1183:c
1179:x
1172:{
1137:P
1133:P
1106:)
1101:c
1097:y
1093:,
1088:c
1084:x
1080:(
1077:C
1057:)
1050:y
1046:,
1039:x
1035:(
1028:P
1007:)
1004:y
1001:,
998:x
995:(
992:P
975:.
956:.
937:n
926:R
918:R
899:p
895:q
891:q
887:p
847:P
832:P
820:P
795:X
791:X
787:P
783:x
779:x
775:p
768:x
764:p
760:x
753:P
746:X
742:P
738:P
734:X
726:X
722:X
714:P
710:X
706:P
702:X
684:a
680:p
663:.
659:a
651:p
647:2
644:=
641:)
637:a
633:(
627:p
621:f
618:e
615:R
600:p
596:a
592:R
588:a
576:n
572:n
564:n
497:n
483:1
477:n
442:1
436:n
430:k
424:1
414:k
384:1
378:n
370:(
313:π
137:)
131:(
126:)
122:(
112:·
105:·
98:·
91:·
64:.
41:.
34:.
20:)
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