2468:
either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and
2432:). And since every isometry can be expressed as a sequence of reflections, its inverse can be expressed as that sequence reversed. Notice that the cancellation of a pair of identical reflections reduces the number of reflections by an even number, preserving the parity of the sequence; also notice that the identity has even parity. Therefore all isometries form a group, and even isometries a subgroup. (Odd isometries do not include the identity, so are not a subgroup). This subgroup is a normal subgroup, because sandwiching an even isometry between two odd ones yields an even isometry.
2489:
2655:
2047:
797:
3043:
1226:
2500:
If two rotations share a fixed point, then we can swivel the mirror pair of the second rotation to cancel the inner mirrors of the sequence of four (two and two), leaving just the outer pair. Thus the composition of two rotations with a common fixed point produces a rotation by the sum of the angles
2126:
Three mirrors. If they are all parallel, the effect is the same as a single mirror (slide a pair to cancel the third). Otherwise we can find an equivalent arrangement where two are parallel and the third is perpendicular to them. The effect is a reflection combined with a translation parallel to the
2111:
Two distinct mirrors that do not intersect must be parallel. Every point moves the same amount, twice the distance between the mirrors, and in the same direction. No points are left fixed. Any two mirrors with the same parallel direction and the same distance apart give the same translation, so long
2504:
If two translations are parallel, we can slide the mirror pair of the second translation to cancel the inner mirror of the sequence of four, much as in the rotation case. Thus the composition of two parallel translations produces a translation by the sum of the distances in the same direction. Now
2480:
We thus have two new kinds of isometry subgroups: all translations, and rotations sharing a fixed point. Both are subgroups of the even subgroup, within which translations are normal. Because translations are a normal subgroup, we can factor them out leaving the subgroup of isometries with a fixed
2467:
Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity (which is both, in a trivial way). Reflection composed with
2415:
The identity is an isometry; nothing changes, so distance cannot change. And if one isometry cannot change distance, neither can two (or three, or more) in succession; thus the composition of two isometries is again an isometry, and the set of isometries is closed under composition. The identity
1971:
Alternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the
2906:
2096:
Two distinct intersecting mirrors have a single point in common, which remains fixed. All other points rotate around it by twice the angle between the mirrors. Any two mirrors with the same fixed point and same angle give the same rotation, so long as they are used in the correct
2820:
649:
2949:
2585:. The inner mirrors now coincide and cancel, and the outer mirrors are left parallel. Thus the composition of two non-parallel translations also produces a translation. Also, the three pivot points form a triangle whose edges give the head-to-tail rule of
1080:
75:
Informally, a
Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include:
2645:
This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.
2829:
1074:
is a point in the plane (the centre of rotation), and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. First, a rotation around the origin is given by
2022:
and the length of the added vector has a continuous distribution. A pure translation and a pure reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom.
1998:
In all cases we multiply the position vector by an orthogonal matrix and add a vector; if the determinant is 1 we have a rotation, a translation, or the identity, and if it is −1 we have a glide reflection or a reflection.
1038:
2330:
We can recognize which of these isometries we have according to whether it preserves hands or swaps them, and whether it has at least one fixed point or not, as shown in the following table (omitting the identity).
2757:
565:
1608:
1414:
3216:
2391:
Isometries requiring an odd number of mirrors — reflection and glide reflection — always reverse left and right. The even isometries — identity, rotation, and translation — never do; they correspond to
2705:
Translations do not fold back on themselves, but we can take integer multiples of any finite translation, or sums of multiples of two such independent translations, as a subgroup. These generate the
186:
1503:
640:
The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices (i.e. with determinant 1 and −1) forming orthogonal group
1786:
1964:
1875:
2082:, a single mirror causes left and right hands to switch. (In formal terms, topological orientation is reversed.) Points on the mirror are left fixed. Each mirror has a unique effect.
3256:
1309:
275:
792:{\displaystyle R_{0,\theta }(p)={\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &\mathbf {-} \cos \theta \end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\end{bmatrix}}.}
636:
3343:
3297:
3369:
3038:{\displaystyle {\begin{array}{rccc}g\colon &\mathbb {C} &\longrightarrow &\mathbb {C} \\&z&\mapsto &{\frac {f(z)-a}{\omega }}{\mbox{,}}\end{array}}}
916:
1221:{\displaystyle R_{0,\theta }(p)={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\end{bmatrix}}.}
935:
2716:
We can also combine these two kinds of discrete groups — the discrete rotations and reflections around a fixed point and the discrete translations — to generate the
2654:
2674:
consist of rotations by integer multiples of 72° (360° / 5), along with reflections in the five mirrors which perpendicularly bisect the edges. This is a group, D
1651:
1995:
is a special case of a translation, and also a special case of a rotation. It is the only isometry which belongs to more than one of the types described above.
67:
in two dimensions. It is generated by reflections in lines, and every element of the
Euclidean group is the composite of at most three distinct reflections.
2064:
Two reflections in the same mirror restore each point to its original position. All points are left fixed. Any pair of identical mirrors has the same effect.
824:
2488:
86:
Turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet, then after turning the sheet over, we see the
439:
324:
117:
However, folding, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like bending, stretching, or twisting.
1051:
2007:
1512:
1335:
2901:{\displaystyle {\begin{array}{ccc}\mathbb {C} &\longrightarrow &\mathbb {C} \\z&\mapsto &a+\omega {\overline {z}}{\mbox{,}}\end{array}}}
3371:, that is, provided the direct isometry is not a pure translation. As stated by Cederberg, "A direct isometry is either a rotation or a translation."
2666:). Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite. For example, the
3144:
137:
3434:
1423:
3391:
2725:
1311:), with determinant 1 (the other possibility for orthogonal matrices is −1, which gives a mirror image, see below). They form the special
2046:
3474:
2019:
1717:
2408:; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce.
32:, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types:
2136:
Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation.
2815:{\displaystyle {\begin{array}{ccc}\mathbb {C} &\longrightarrow &\mathbb {C} \\z&\mapsto &a+\omega z\end{array}}}
1884:
1795:
3220:
where the last expression shows the mapping equivalent to rotation at 0 and a translation. Therefore, given direct isometry
2428:, we must also have an inverse for every element. To cancel a reflection, we merely compose it with itself (Reflections are
2451:, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a
3479:
203:
2736:, are incompatible with a discrete lattice of translations. (Each higher dimension also has only a finite number of such
3380:
578:
3071:
is either the identity or the conjugation, and the statement being proved follows from this and from the fact that
870:
2732:, we have only 7 distinct frieze groups and 17 distinct wallpaper groups. For example, the pentagon symmetries, D
2079:
2031:
Reflections, or mirror isometries, can be combined to produce any isometry. Thus isometries are an example of a
1650:
3469:
2452:
2429:
2015:
2011:
332:
103:
41:
2143:
An isometry is completely determined by its effect on three independent (not collinear) points. So suppose
3396:
3223:
2737:
1976:
1326:
to the origin, then performing the rotation around the origin, and finally translating the origin back to
1252:
1059:
832:
823:
95:
33:
3399:, the statement that the midpoints of corresponding pairs of points in an isometry of lines are collinear
2728:
with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to
3484:
3386:
323:
2472:
proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.
1968:
that is, we obtain the same result if we do the translation and the reflection in the opposite order.)
1050:
3302:
1621:
60:
3261:
3348:
2425:
1635:
99:
56:
37:
2456:
850:
292:
3454:
3426:
2954:
3430:
1231:
130:
129:
of the
Euclidean plane is a distance-preserving transformation of the plane. That is, it is a
2682:, of half the size, omitting the reflections. These two groups are members of two families, D
2002:
A "random" isometry, like taking a sheet of paper from a table and randomly laying it back, "
3418:
2482:
2440:
2032:
1659:
1312:
107:
45:
315:: the notations for the types of isometries listed below are not completely standardised.)
2749:
2721:
2706:
2586:
2401:
2397:
64:
29:
2834:
2762:
815:-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.
2448:
1246:
3383:, a characterization of isometries as the transformations that preserve unit distances
83:
Rotating the sheet by ten degrees around some marked point (which remains motionless).
3463:
3419:
3097:
This is obviously related to the previous classification of plane isometries, since:
2405:
2003:
1507:
Alternatively, a rotation around the origin is performed, followed by a translation:
1235:
2717:
2710:
2444:
87:
2740:, but the number grows rapidly; for example, 3D has 230 groups and 4D has 4783.)
2729:
2699:
2662:
The subgroups discussed so far are not only infinite, they are also continuous (
2417:
1033:{\displaystyle T_{v}(p)={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\end{bmatrix}}.}
415:
362:
2626:
The subgroup structure suggests another way to compose an arbitrary isometry:
2663:
2505:
suppose the translations are not parallel, and that the mirror sequence is A
2421:
1242:
2318:. Thus at most three reflections suffice to reproduce any plane isometry.
311:
It can be shown that there are four types of
Euclidean plane isometries. (
2671:
2667:
25:
17:
2185:; we can generate a sequence of mirrors to achieve this as follows. If
1612:
A rotation can be seen as a composite of two non-parallel reflections.
1042:
A translation can be seen as a composite of two parallel reflections.
2615:
2433:
2319:
560:{\displaystyle t=(p-c)\cdot v=(p_{x}-c_{x})v_{x}+(p_{y}-c_{y})v_{y},}
1603:{\displaystyle R_{c,\theta }(p)=c-R_{0,\theta }c+R_{0,\theta }(p).}
1409:{\displaystyle R_{c,\theta }=T_{c}\circ R_{0,\theta }\circ T_{-c},}
2724:. Curiously, only a few of the fixed-point groups are found to be
2533:; and, reassociating, we are free to pivot this inner pair around
2469:
1649:
1049:
822:
322:
2416:
isometry is also an identity for composition, and composition is
2199:
are distinct, choose their perpendicular bisector as mirror. Now
111:
2404:
of isometries. Neither the full group nor the even subgroup are
106:
respectively. There is one further type of isometry, called a
3211:{\displaystyle z\mapsto \omega (z-p)+p=\omega z+p(1-\omega )}
2043:
In the
Euclidean plane, we have the following possibilities.
807:, or equivalently, a reflection in a line making an angle of
2653:
2487:
2045:
1972:
combination is itself just a reflection in a parallel line.
2297:″; and if it is not in place, a final mirror through
1700:
are a combination of a reflection in the line described by
857:
have the effect of shifting the plane in the direction of
2635:
If the isometry is odd, use the mirror; otherwise do not.
2014:, as long as θ and the direction of the added vector are
3345:
as the center for an equivalent rotation, provided that
1642:, a subgroup of the full group of Euclidean isometries.
1626:
The set of translations and rotations together form the
373:
is for "flip".) have the effect of reflecting the point
181:{\displaystyle M:{\textbf {R}}^{2}\to {\textbf {R}}^{2}}
2006:" is a rotation or a glide reflection (they have three
3025:
2888:
2561:. Again reassociating, we pivot the first pair around
2537:. If we pivot 90°, an interesting thing happens: now A
1498:{\displaystyle R_{c,\theta }(p)=c+R_{0,\theta }(p-c).}
1180:
1117:
966:
751:
686:
3351:
3305:
3264:
3226:
3147:
2952:
2832:
2760:
2752:, the isometries of the plane are either of the form
1887:
1798:
1720:
1515:
1426:
1338:
1255:
1083:
938:
873:
652:
581:
442:
206:
140:
3363:
3337:
3291:
3250:
3210:
3037:
2918:with |ω| = 1. This is easy to prove: if
2900:
2814:
1958:
1869:
1780:
1602:
1497:
1408:
1303:
1220:
1032:
910:
791:
630:
559:
269:
180:
2010:). This applies regardless of the details of the
644:(2). In the case of a determinant of −1 we have:
2698:> 1. Together, these families constitute the
2420:; therefore isometries satisfy the axioms for a
2476:Translation, rotation, and orthogonal subgroups
2459:, of the even subgroup and the quotient group.
2213:; and we will pass all further mirrors through
55:The set of Euclidean plane isometries forms a
2658:Dihedral group of regular pentagon symmetries
2439:Since the even subgroup is normal, it is the
1781:{\displaystyle G_{c,v,w}=T_{w}\circ F_{c,v},}
8:
2638:If necessary, rotate around the fixed point.
2630:Pick a fixed point, and a mirror through it.
112:classification of Euclidean plane isometries
1959:{\displaystyle G_{c,v,w}(p)=F_{c,v}(p+w);}
1870:{\displaystyle G_{c,v,w}(p)=w+F_{c,v}(p).}
3350:
3315:
3304:
3263:
3225:
3146:
3135:Note that a rotation about complex point
3024:
2997:
2978:
2977:
2966:
2965:
2953:
2951:
2887:
2877:
2850:
2849:
2838:
2837:
2833:
2831:
2778:
2777:
2766:
2765:
2761:
2759:
2545:′ intersect at a 90° angle, say at
1926:
1892:
1886:
1843:
1803:
1797:
1763:
1750:
1725:
1719:
1576:
1554:
1520:
1514:
1465:
1431:
1425:
1394:
1375:
1362:
1343:
1337:
1322:can be accomplished by first translating
1292:
1276:
1263:
1254:
1201:
1187:
1175:
1112:
1088:
1082:
1013:
1000:
986:
973:
961:
943:
937:
878:
872:
803:-axis followed by a rotation by an angle
772:
758:
746:
724:
681:
657:
651:
586:
580:
548:
535:
522:
506:
493:
480:
441:
205:
172:
166:
165:
155:
149:
148:
139:
80:Shifting the sheet one inch to the right.
2678:, with 10 elements. It has a subgroup, C
2333:
3409:
3139:is obtained by complex arithmetic with
3128:are rotations (when |ω| = 1);
2220:, leaving it fixed. Call the images of
2112:as they are used in the correct order.
1696:is non-null a vector perpendicular to
2513:(the first translation) followed by B
2336:
569:and then we obtain the reflection of
7:
3392:Coordinate rotations and reflections
3251:{\displaystyle z\mapsto \omega z+a,}
1304:{\displaystyle GG^{T}=G^{T}G=I_{2}.}
270:{\displaystyle d(p,q)=d(M(p),M(q)),}
2573:, and pivot the second pair around
167:
150:
49:
1708:, followed by a translation along
14:
2492:Translation addition with mirrors
2127:mirror. No points are left fixed.
631:{\displaystyle F_{c,v}(p)=p-2tv.}
3131:the conjugation is a reflection.
725:
3338:{\displaystyle p=a/(1-\omega )}
2744:Isometries in the complex plane
3332:
3320:
3292:{\displaystyle p(1-\omega )=a}
3280:
3268:
3230:
3205:
3193:
3169:
3157:
3151:
3067:. It is then easy to see that
3009:
3003:
2992:
2972:
2863:
2844:
2791:
2772:
2027:Isometries as reflection group
1950:
1938:
1916:
1910:
1861:
1855:
1827:
1821:
1594:
1588:
1538:
1532:
1489:
1477:
1449:
1443:
1106:
1100:
955:
949:
890:
884:
675:
669:
604:
598:
541:
515:
499:
473:
461:
449:
261:
258:
252:
243:
237:
231:
222:
210:
161:
1:
3421:A Course in Modern Geometries
3417:Cederberg, Judith N. (2001).
3364:{\displaystyle \omega \neq 1}
2262:′, bisect the angle at
911:{\displaystyle T_{v}(p)=p+v,}
799:which is a reflection in the
2882:
2501:about the same fixed point.
2370:
2354:
2342:
357:is a point in the plane and
2371:
2358:
3501:
1619:
2910:for some complex numbers
2622:Nested group construction
2377: Translation
2355:
2338:
2080:through the looking-glass
1684:is a point in the plane,
861:. That is, for any point
381:that is perpendicular to
190:such that for any points
3475:Euclidean plane geometry
2641:If necessary, translate.
2269:with a new mirror. With
2012:probability distribution
401:. To find a formula for
385:and that passes through
22:Euclidean plane isometry
3381:Beckman–Quarles theorem
2738:crystallographic groups
1638:under composition, the
1230:These matrices are the
3365:
3339:
3293:
3252:
3212:
3118:functions of the type
3101:functions of the type
3039:
2902:
2816:
2659:
2493:
2234:under this reflection
2051:
1960:
1879:(It is also true that
1871:
1782:
1655:
1640:group of rigid motions
1604:
1499:
1410:
1305:
1222:
1055:
1034:
912:
828:
793:
632:
561:
418:to find the component
328:
271:
182:
94:These are examples of
3441:, quote from page 151
3425:. Springer. pp.
3387:Congruence (geometry)
3366:
3340:
3294:
3253:
3213:
3040:
2903:
2817:
2657:
2581:″ pass through
2569:″ pass through
2491:
2132:Three mirrors suffice
2050:Isometries as mirrors
2049:
2020:uniformly distributed
1979:isometry, defined by
1961:
1872:
1783:
1653:
1616:Rigid transformations
1605:
1500:
1411:
1306:
1223:
1053:
1035:
913:
826:
794:
633:
562:
326:
272:
183:
50:§ Classification
3480:Euclidean symmetries
3349:
3303:
3262:
3224:
3145:
2950:
2830:
2758:
2521:(the second). Then A
1885:
1796:
1718:
1688:is a unit vector in
1622:Rigid transformation
1513:
1424:
1336:
1253:
1081:
936:
871:
650:
579:
440:
204:
138:
3397:Hjelmslev's theorem
2944:and if one defines
2529:must cross, say at
2039:Mirror combinations
1790:or in other words,
1634:. This set forms a
1632:rigid displacements
1418:or in other words,
1232:orthogonal matrices
414:, we first use the
71:Informal discussion
3361:
3335:
3289:
3248:
3208:
3035:
3033:
3029:
2898:
2896:
2892:
2812:
2810:
2660:
2650:Discrete subgroups
2498:
2494:
2457:semidirect product
2413:
2141:
2052:
2008:degrees of freedom
1956:
1867:
1778:
1656:
1600:
1495:
1406:
1318:A rotation around
1301:
1218:
1209:
1169:
1056:
1030:
1021:
908:
829:
789:
780:
740:
628:
557:
397:or the associated
329:
293:Euclidean distance
267:
178:
3436:978-0-387-98972-3
3115:are translations;
3028:
3022:
2891:
2885:
2496:
2411:
2384:
2383:
2380:Glide reflection
2339:Preserves hands?
2255:is distinct from
2139:
1660:Glide reflections
1646:Glide reflections
338:mirror isometries
169:
152:
121:Formal definition
110:(see below under
46:glide reflections
3492:
3455:Plane Isometries
3442:
3440:
3424:
3414:
3370:
3368:
3367:
3362:
3344:
3342:
3341:
3336:
3319:
3298:
3296:
3295:
3290:
3257:
3255:
3254:
3249:
3217:
3215:
3214:
3209:
3127:
3114:
3093:
3066:
3059:
3053:is an isometry,
3052:
3044:
3042:
3041:
3036:
3034:
3030:
3026:
3023:
3018:
2998:
2985:
2981:
2969:
2943:
2928:
2917:
2913:
2907:
2905:
2904:
2899:
2897:
2893:
2889:
2886:
2878:
2853:
2841:
2821:
2819:
2818:
2813:
2811:
2781:
2769:
2722:wallpaper groups
2613:
2483:orthogonal group
2334:
2311:will flip it to
2119:Glide reflection
2033:reflection group
1965:
1963:
1962:
1957:
1937:
1936:
1909:
1908:
1876:
1874:
1873:
1868:
1854:
1853:
1820:
1819:
1787:
1785:
1784:
1779:
1774:
1773:
1755:
1754:
1742:
1741:
1654:Glide reflection
1609:
1607:
1606:
1601:
1587:
1586:
1565:
1564:
1531:
1530:
1504:
1502:
1501:
1496:
1476:
1475:
1442:
1441:
1415:
1413:
1412:
1407:
1402:
1401:
1386:
1385:
1367:
1366:
1354:
1353:
1313:orthogonal group
1310:
1308:
1307:
1302:
1297:
1296:
1281:
1280:
1268:
1267:
1234:(i.e. each is a
1227:
1225:
1224:
1219:
1214:
1213:
1206:
1205:
1192:
1191:
1174:
1173:
1099:
1098:
1039:
1037:
1036:
1031:
1026:
1025:
1018:
1017:
1005:
1004:
991:
990:
978:
977:
948:
947:
921:or in terms of (
917:
915:
914:
909:
883:
882:
798:
796:
795:
790:
785:
784:
777:
776:
763:
762:
745:
744:
728:
668:
667:
637:
635:
634:
629:
597:
596:
573:by subtraction,
566:
564:
563:
558:
553:
552:
540:
539:
527:
526:
511:
510:
498:
497:
485:
484:
276:
274:
273:
268:
187:
185:
184:
179:
177:
176:
171:
170:
160:
159:
154:
153:
108:glide reflection
3500:
3499:
3495:
3494:
3493:
3491:
3490:
3489:
3470:Crystallography
3460:
3459:
3451:
3446:
3445:
3437:
3416:
3415:
3411:
3406:
3377:
3347:
3346:
3301:
3300:
3260:
3259:
3222:
3221:
3143:
3142:
3119:
3102:
3072:
3061:
3054:
3048:
3032:
3031:
2999:
2995:
2990:
2983:
2982:
2975:
2970:
2963:
2948:
2947:
2930:
2919:
2915:
2911:
2895:
2894:
2866:
2861:
2855:
2854:
2847:
2842:
2828:
2827:
2824:or of the form
2809:
2808:
2794:
2789:
2783:
2782:
2775:
2770:
2756:
2755:
2750:complex numbers
2746:
2735:
2693:
2687:
2681:
2677:
2652:
2624:
2619:
2590:
2587:vector addition
2580:
2568:
2556:
2552:
2544:
2540:
2528:
2524:
2520:
2516:
2512:
2508:
2478:
2465:
2437:
2402:Euclidean group
2398:normal subgroup
2389:
2387:Group structure
2328:
2323:
2317:
2310:
2303:
2296:
2289:
2282:
2275:
2268:
2261:
2254:
2247:
2240:
2233:
2226:
2219:
2212:
2205:
2198:
2191:
2184:
2177:
2170:
2163:
2156:
2149:
2134:
2078:As Alice found
2041:
2029:
1991:for all points
1922:
1888:
1883:
1882:
1839:
1799:
1794:
1793:
1759:
1746:
1721:
1716:
1715:
1679:
1648:
1624:
1618:
1572:
1550:
1516:
1511:
1510:
1461:
1427:
1422:
1421:
1390:
1371:
1358:
1339:
1334:
1333:
1288:
1272:
1259:
1251:
1250:
1240:
1208:
1207:
1197:
1194:
1193:
1183:
1176:
1168:
1167:
1156:
1144:
1143:
1129:
1113:
1084:
1079:
1078:
1069:
1048:
1020:
1019:
1009:
996:
993:
992:
982:
969:
962:
939:
934:
933:
874:
869:
868:
844:
821:
779:
778:
768:
765:
764:
754:
747:
739:
738:
722:
710:
709:
698:
682:
653:
648:
647:
582:
577:
576:
544:
531:
518:
502:
489:
476:
438:
437:
413:
395:reflection axis
352:
321:
309:
291:) is the usual
202:
201:
164:
147:
136:
135:
123:
90:of the picture.
73:
65:Euclidean group
30:Euclidean plane
12:
11:
5:
3498:
3496:
3488:
3487:
3482:
3477:
3472:
3462:
3461:
3458:
3457:
3450:
3449:External links
3447:
3444:
3443:
3435:
3408:
3407:
3405:
3402:
3401:
3400:
3394:
3389:
3384:
3376:
3373:
3360:
3357:
3354:
3334:
3331:
3328:
3325:
3322:
3318:
3314:
3311:
3308:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3258:one can solve
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3133:
3132:
3129:
3116:
3021:
3017:
3014:
3011:
3008:
3005:
3002:
2996:
2994:
2991:
2989:
2986:
2984:
2980:
2976:
2974:
2971:
2968:
2964:
2962:
2959:
2956:
2955:
2884:
2881:
2876:
2873:
2870:
2867:
2865:
2862:
2860:
2857:
2856:
2852:
2848:
2846:
2843:
2840:
2836:
2835:
2807:
2804:
2801:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2780:
2776:
2774:
2771:
2768:
2764:
2763:
2745:
2742:
2733:
2713:of the plane.
2709:of a periodic
2700:rosette groups
2689:
2683:
2679:
2675:
2651:
2648:
2643:
2642:
2639:
2636:
2632:
2631:
2623:
2620:
2578:
2566:
2554:
2550:
2542:
2538:
2526:
2522:
2518:
2514:
2510:
2506:
2495:
2477:
2474:
2464:
2461:
2453:direct product
2449:quotient group
2410:
2388:
2385:
2382:
2381:
2378:
2375:
2369:
2368:
2365:
2362:
2357:
2353:
2352:
2347:
2341:
2340:
2337:
2327:
2324:
2315:
2308:
2301:
2294:
2287:
2283:now in place,
2280:
2273:
2266:
2259:
2252:
2245:
2238:
2231:
2224:
2217:
2210:
2203:
2196:
2189:
2182:
2175:
2168:
2161:
2154:
2147:
2138:
2133:
2130:
2129:
2128:
2123:
2122:
2121:
2120:
2114:
2113:
2108:
2107:
2106:
2105:
2099:
2098:
2093:
2092:
2091:
2090:
2084:
2083:
2075:
2074:
2073:
2072:
2066:
2065:
2061:
2060:
2059:
2058:
2040:
2037:
2028:
2025:
1955:
1952:
1949:
1946:
1943:
1940:
1935:
1932:
1929:
1925:
1921:
1918:
1915:
1912:
1907:
1904:
1901:
1898:
1895:
1891:
1866:
1863:
1860:
1857:
1852:
1849:
1846:
1842:
1838:
1835:
1832:
1829:
1826:
1823:
1818:
1815:
1812:
1809:
1806:
1802:
1777:
1772:
1769:
1766:
1762:
1758:
1753:
1749:
1745:
1740:
1737:
1734:
1731:
1728:
1724:
1667:
1647:
1644:
1620:Main article:
1617:
1614:
1599:
1596:
1593:
1590:
1585:
1582:
1579:
1575:
1571:
1568:
1563:
1560:
1557:
1553:
1549:
1546:
1543:
1540:
1537:
1534:
1529:
1526:
1523:
1519:
1494:
1491:
1488:
1485:
1482:
1479:
1474:
1471:
1468:
1464:
1460:
1457:
1454:
1451:
1448:
1445:
1440:
1437:
1434:
1430:
1405:
1400:
1397:
1393:
1389:
1384:
1381:
1378:
1374:
1370:
1365:
1361:
1357:
1352:
1349:
1346:
1342:
1300:
1295:
1291:
1287:
1284:
1279:
1275:
1271:
1266:
1262:
1258:
1238:
1217:
1212:
1204:
1200:
1196:
1195:
1190:
1186:
1182:
1181:
1179:
1172:
1166:
1163:
1160:
1157:
1155:
1152:
1149:
1146:
1145:
1142:
1139:
1136:
1133:
1130:
1128:
1125:
1122:
1119:
1118:
1116:
1111:
1108:
1105:
1102:
1097:
1094:
1091:
1087:
1067:
1047:
1044:
1029:
1024:
1016:
1012:
1008:
1003:
999:
995:
994:
989:
985:
981:
976:
972:
968:
967:
965:
960:
957:
954:
951:
946:
942:
931:
930:
929:) coordinates,
907:
904:
901:
898:
895:
892:
889:
886:
881:
877:
865:in the plane,
840:
820:
817:
788:
783:
775:
771:
767:
766:
761:
757:
753:
752:
750:
743:
737:
734:
731:
727:
723:
721:
718:
715:
712:
711:
708:
705:
702:
699:
697:
694:
691:
688:
687:
685:
680:
677:
674:
671:
666:
663:
660:
656:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
595:
592:
589:
585:
556:
551:
547:
543:
538:
534:
530:
525:
521:
517:
514:
509:
505:
501:
496:
492:
488:
483:
479:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
405:
393:is called the
344:
320:
317:
308:
307:Classification
305:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
209:
198:in the plane,
175:
163:
158:
146:
143:
122:
119:
92:
91:
84:
81:
72:
69:
13:
10:
9:
6:
4:
3:
2:
3497:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3456:
3453:
3452:
3448:
3438:
3432:
3428:
3423:
3422:
3413:
3410:
3403:
3398:
3395:
3393:
3390:
3388:
3385:
3382:
3379:
3378:
3374:
3372:
3358:
3355:
3352:
3329:
3326:
3323:
3316:
3312:
3309:
3306:
3286:
3283:
3277:
3274:
3271:
3265:
3245:
3242:
3239:
3236:
3233:
3227:
3218:
3202:
3199:
3196:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3166:
3163:
3160:
3154:
3148:
3140:
3138:
3130:
3126:
3122:
3117:
3113:
3109:
3105:
3100:
3099:
3098:
3095:
3091:
3087:
3083:
3079:
3075:
3070:
3064:
3057:
3051:
3045:
3019:
3015:
3012:
3006:
3000:
2987:
2960:
2957:
2945:
2941:
2937:
2933:
2926:
2922:
2908:
2879:
2874:
2871:
2868:
2858:
2825:
2822:
2805:
2802:
2799:
2796:
2786:
2753:
2751:
2743:
2741:
2739:
2731:
2727:
2723:
2719:
2718:frieze groups
2714:
2712:
2708:
2703:
2701:
2697:
2692:
2686:
2673:
2670:of a regular
2669:
2665:
2656:
2649:
2647:
2640:
2637:
2634:
2633:
2629:
2628:
2627:
2621:
2618:
2617:
2611:
2608:
2604:
2601:
2597:
2594:
2588:
2584:
2576:
2572:
2564:
2560:
2553:′ and B
2549:, and so do B
2548:
2536:
2532:
2502:
2490:
2486:
2484:
2475:
2473:
2471:
2462:
2460:
2458:
2455:, but only a
2454:
2450:
2446:
2442:
2436:
2435:
2431:
2427:
2423:
2419:
2409:
2407:
2403:
2399:
2396:, and form a
2395:
2394:rigid motions
2386:
2379:
2376:
2374:
2366:
2363:
2361:
2356:Fixed point?
2351:
2348:
2346:
2343:
2335:
2332:
2325:
2322:
2321:
2314:
2307:
2300:
2293:
2286:
2279:
2272:
2265:
2258:
2251:
2244:
2237:
2230:
2223:
2216:
2209:
2202:
2195:
2188:
2181:
2174:
2167:
2160:
2153:
2146:
2137:
2131:
2125:
2124:
2118:
2117:
2116:
2115:
2110:
2109:
2103:
2102:
2101:
2100:
2095:
2094:
2088:
2087:
2086:
2085:
2081:
2077:
2076:
2070:
2069:
2068:
2067:
2063:
2062:
2056:
2055:
2054:
2053:
2048:
2044:
2038:
2036:
2034:
2026:
2024:
2021:
2017:
2013:
2009:
2005:
2004:almost surely
2000:
1996:
1994:
1990:
1986:
1982:
1978:
1973:
1969:
1966:
1953:
1947:
1944:
1941:
1933:
1930:
1927:
1923:
1919:
1913:
1905:
1902:
1899:
1896:
1893:
1889:
1880:
1877:
1864:
1858:
1850:
1847:
1844:
1840:
1836:
1833:
1830:
1824:
1816:
1813:
1810:
1807:
1804:
1800:
1791:
1788:
1775:
1770:
1767:
1764:
1760:
1756:
1751:
1747:
1743:
1738:
1735:
1732:
1729:
1726:
1722:
1713:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1678:
1674:
1670:
1666:
1663:, denoted by
1662:
1661:
1652:
1645:
1643:
1641:
1637:
1633:
1629:
1628:rigid motions
1623:
1615:
1613:
1610:
1597:
1591:
1583:
1580:
1577:
1573:
1569:
1566:
1561:
1558:
1555:
1551:
1547:
1544:
1541:
1535:
1527:
1524:
1521:
1517:
1508:
1505:
1492:
1486:
1483:
1480:
1472:
1469:
1466:
1462:
1458:
1455:
1452:
1446:
1438:
1435:
1432:
1428:
1419:
1416:
1403:
1398:
1395:
1391:
1387:
1382:
1379:
1376:
1372:
1368:
1363:
1359:
1355:
1350:
1347:
1344:
1340:
1331:
1329:
1325:
1321:
1316:
1314:
1298:
1293:
1289:
1285:
1282:
1277:
1273:
1269:
1264:
1260:
1256:
1248:
1244:
1237:
1236:square matrix
1233:
1228:
1215:
1210:
1202:
1198:
1188:
1184:
1177:
1170:
1164:
1161:
1158:
1153:
1150:
1147:
1140:
1137:
1134:
1131:
1126:
1123:
1120:
1114:
1109:
1103:
1095:
1092:
1089:
1085:
1076:
1073:
1066:
1063:, denoted by
1062:
1061:
1052:
1045:
1043:
1040:
1027:
1022:
1014:
1010:
1006:
1001:
997:
987:
983:
979:
974:
970:
963:
958:
952:
944:
940:
928:
924:
920:
919:
918:
905:
902:
899:
896:
893:
887:
879:
875:
866:
864:
860:
856:
852:
848:
843:
839:
836:, denoted by
835:
834:
825:
818:
816:
814:
810:
806:
802:
786:
781:
773:
769:
759:
755:
748:
741:
735:
732:
729:
719:
716:
713:
706:
703:
700:
695:
692:
689:
683:
678:
672:
664:
661:
658:
654:
645:
643:
638:
625:
622:
619:
616:
613:
610:
607:
601:
593:
590:
587:
583:
574:
572:
567:
554:
549:
545:
536:
532:
528:
523:
519:
512:
507:
503:
494:
490:
486:
481:
477:
470:
467:
464:
458:
455:
452:
446:
443:
435:
433:
429:
425:
421:
417:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
351:
347:
343:
340:, denoted by
339:
335:
334:
325:
318:
316:
314:
306:
304:
302:
298:
294:
290:
286:
282:
277:
264:
255:
249:
246:
240:
234:
228:
225:
219:
216:
213:
207:
199:
197:
193:
188:
173:
156:
144:
141:
133:
132:
128:
120:
118:
115:
113:
109:
105:
101:
97:
89:
85:
82:
79:
78:
77:
70:
68:
66:
62:
58:
53:
51:
47:
43:
39:
35:
31:
27:
23:
19:
3485:Group theory
3420:
3412:
3219:
3141:
3136:
3134:
3124:
3120:
3111:
3107:
3103:
3096:
3089:
3085:
3081:
3077:
3073:
3068:
3062:
3055:
3049:
3046:
2946:
2939:
2935:
2931:
2924:
2920:
2909:
2826:
2823:
2754:
2748:In terms of
2747:
2715:
2704:
2695:
2690:
2684:
2661:
2644:
2625:
2609:
2606:
2602:
2599:
2595:
2592:
2582:
2574:
2570:
2562:
2558:
2546:
2534:
2530:
2503:
2499:
2479:
2466:
2445:homomorphism
2438:
2414:
2400:of the full
2393:
2390:
2372:
2359:
2349:
2344:
2329:
2312:
2305:
2298:
2291:
2284:
2277:
2270:
2263:
2256:
2249:
2248:′. If
2242:
2241:′ and
2235:
2228:
2221:
2214:
2207:
2200:
2193:
2186:
2179:
2172:
2165:
2158:
2151:
2144:
2142:
2135:
2042:
2030:
2001:
1997:
1992:
1988:
1984:
1980:
1974:
1970:
1967:
1881:
1878:
1792:
1789:
1714:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1676:
1672:
1668:
1664:
1658:
1657:
1639:
1631:
1627:
1625:
1611:
1509:
1506:
1420:
1417:
1332:
1327:
1323:
1319:
1317:
1229:
1077:
1071:
1064:
1058:
1057:
1041:
932:
926:
922:
867:
862:
858:
854:
846:
841:
837:
833:Translations
831:
830:
819:Translations
812:
811:/2 with the
808:
804:
800:
646:
641:
639:
575:
570:
568:
436:
431:
427:
423:
419:
410:
406:
402:
398:
394:
390:
386:
382:
378:
377:in the line
374:
370:
366:
358:
354:
349:
345:
341:
337:
331:
330:
312:
310:
300:
296:
288:
284:
280:
278:
200:
195:
191:
189:
134:
126:
124:
116:
96:translations
93:
88:mirror image
74:
54:
34:translations
21:
15:
2730:isomorphism
2481:point, the
2463:Composition
2430:involutions
2418:associative
2367:Reflection
2326:Recognition
2104:Translation
2016:independent
1712:. That is,
1330:. That is,
827:Translation
434:direction,
416:dot product
389:. The line
363:unit vector
333:Reflections
319:Reflections
104:reflections
61:composition
48:(see below
42:reflections
3464:Categories
3404:References
3299:to obtain
2726:compatible
2694:, for any
2668:symmetries
2664:Lie groups
2071:Reflection
327:Reflection
3356:≠
3353:ω
3330:ω
3327:−
3278:ω
3275:−
3234:ω
3231:↦
3203:ω
3200:−
3182:ω
3164:−
3155:ω
3152:↦
3020:ω
3013:−
2993:↦
2973:⟶
2961::
2883:¯
2875:ω
2864:↦
2845:⟶
2803:ω
2792:↦
2773:⟶
2577:to make A
2565:to make B
2557:, say at
2422:semigroup
1757:∘
1584:θ
1562:θ
1548:−
1528:θ
1484:−
1473:θ
1439:θ
1396:−
1388:∘
1383:θ
1369:∘
1351:θ
1243:transpose
1165:θ
1162:
1154:θ
1151:
1141:θ
1138:
1132:−
1127:θ
1124:
1096:θ
1060:Rotations
1046:Rotations
736:θ
733:
726:−
720:θ
717:
707:θ
704:
696:θ
693:
665:θ
614:−
529:−
487:−
465:⋅
456:−
162:→
100:rotations
38:rotations
3375:See also
2672:pentagon
2424:. For a
2364:Rotation
2206:maps to
2089:Rotation
2057:Identity
1977:identity
1680:, where
1070:, where
1054:Rotation
845:, where
426:−
353:, where
295:between
127:isometry
26:isometry
18:geometry
3065:(1) = 1
3058:(0) = 0
2707:lattice
2406:abelian
2164:map to
1315:SO(2).
1249:, i.e.
1247:inverse
1245:is its
430:in the
28:of the
3433:
3429:–164.
3060:, and
2938:(1) −
2711:tiling
2616:Q.E.D.
2605:) = 2(
2598:) + 2(
2441:kernel
2434:Q.E.D.
2320:Q.E.D.
2290:is at
2097:order.
1692:, and
1241:whose
851:vector
399:mirror
279:where
102:, and
63:: the
59:under
44:, and
24:is an
3047:then
2688:and C
2541:and A
2525:and B
2497:Proof
2470:Euler
2447:to a
2443:of a
2426:group
2412:Proof
2140:Proof
1636:group
849:is a
361:is a
336:, or
57:group
3431:ISBN
3080:) =
2929:and
2914:and
2720:and
2304:and
2276:and
2227:and
2192:and
2018:and
1987:) =
1975:The
1704:and
365:in
313:Note
299:and
194:and
20:, a
3427:136
3123:→ ω
2942:(0)
2927:(0)
2517:, B
2509:, A
2360:Yes
2345:Yes
1630:or
1159:cos
1148:sin
1135:sin
1121:cos
1068:c,θ
853:in
730:cos
714:sin
701:sin
690:cos
422:of
369:. (
131:map
125:An
114:).
52:).
16:In
3466::
3110:+
3106:→
3094:.
3086:ωg
3084:+
2934:=
2923:=
2702:.
2614:.
2591:2(
2589::
2485:.
2373:No
2350:No
2178:,
2171:,
2157:,
2150:,
2035:.
925:,
303:.
287:,
98:,
40:,
36:,
3439:.
3359:1
3333:)
3324:1
3321:(
3317:/
3313:a
3310:=
3307:p
3287:a
3284:=
3281:)
3272:1
3269:(
3266:p
3246:,
3243:a
3240:+
3237:z
3228:z
3206:)
3197:1
3194:(
3191:p
3188:+
3185:z
3179:=
3176:p
3173:+
3170:)
3167:p
3161:z
3158:(
3149:z
3137:p
3125:z
3121:z
3112:z
3108:a
3104:z
3092:)
3090:z
3088:(
3082:a
3078:z
3076:(
3074:f
3069:g
3063:g
3056:g
3050:g
3027:,
3016:a
3010:)
3007:z
3004:(
3001:f
2988:z
2979:C
2967:C
2958:g
2940:f
2936:f
2932:ω
2925:f
2921:a
2916:ω
2912:a
2890:,
2880:z
2872:+
2869:a
2859:z
2851:C
2839:C
2806:z
2800:+
2797:a
2787:z
2779:C
2767:C
2734:5
2696:n
2691:n
2685:n
2680:5
2676:5
2612:)
2610:q
2607:p
2603:q
2600:c
2596:c
2593:p
2583:p
2579:1
2575:q
2571:q
2567:2
2563:p
2559:q
2555:2
2551:1
2547:p
2543:2
2539:1
2535:c
2531:c
2527:1
2523:2
2519:2
2515:1
2511:2
2507:1
2316:3
2313:q
2309:2
2306:q
2302:1
2299:q
2295:3
2292:p
2288:3
2285:p
2281:2
2278:p
2274:1
2271:p
2267:1
2264:q
2260:2
2257:p
2253:2
2250:q
2246:3
2243:p
2239:2
2236:p
2232:3
2229:p
2225:2
2222:p
2218:1
2215:q
2211:1
2208:q
2204:1
2201:p
2197:1
2194:q
2190:1
2187:p
2183:3
2180:q
2176:2
2173:q
2169:1
2166:q
2162:3
2159:p
2155:2
2152:p
2148:1
2145:p
1993:p
1989:p
1985:p
1983:(
1981:I
1954:;
1951:)
1948:w
1945:+
1942:p
1939:(
1934:v
1931:,
1928:c
1924:F
1920:=
1917:)
1914:p
1911:(
1906:w
1903:,
1900:v
1897:,
1894:c
1890:G
1865:.
1862:)
1859:p
1856:(
1851:v
1848:,
1845:c
1841:F
1837:+
1834:w
1831:=
1828:)
1825:p
1822:(
1817:w
1814:,
1811:v
1808:,
1805:c
1801:G
1776:,
1771:v
1768:,
1765:c
1761:F
1752:w
1748:T
1744:=
1739:w
1736:,
1733:v
1730:,
1727:c
1723:G
1710:w
1706:v
1702:c
1698:v
1694:w
1690:R
1686:v
1682:c
1677:w
1675:,
1673:v
1671:,
1669:c
1665:G
1598:.
1595:)
1592:p
1589:(
1581:,
1578:0
1574:R
1570:+
1567:c
1559:,
1556:0
1552:R
1545:c
1542:=
1539:)
1536:p
1533:(
1525:,
1522:c
1518:R
1493:.
1490:)
1487:c
1481:p
1478:(
1470:,
1467:0
1463:R
1459:+
1456:c
1453:=
1450:)
1447:p
1444:(
1436:,
1433:c
1429:R
1404:,
1399:c
1392:T
1380:,
1377:0
1373:R
1364:c
1360:T
1356:=
1348:,
1345:c
1341:R
1328:c
1324:c
1320:c
1299:.
1294:2
1290:I
1286:=
1283:G
1278:T
1274:G
1270:=
1265:T
1261:G
1257:G
1239:G
1216:.
1211:]
1203:y
1199:p
1189:x
1185:p
1178:[
1171:]
1115:[
1110:=
1107:)
1104:p
1101:(
1093:,
1090:0
1086:R
1072:c
1065:R
1028:.
1023:]
1015:y
1011:v
1007:+
1002:y
998:p
988:x
984:v
980:+
975:x
971:p
964:[
959:=
956:)
953:p
950:(
945:v
941:T
927:y
923:x
906:,
903:v
900:+
897:p
894:=
891:)
888:p
885:(
880:v
876:T
863:p
859:v
855:R
847:v
842:v
838:T
813:x
809:θ
805:θ
801:x
787:.
782:]
774:y
770:p
760:x
756:p
749:[
742:]
684:[
679:=
676:)
673:p
670:(
662:,
659:0
655:R
642:O
626:.
623:v
620:t
617:2
611:p
608:=
605:)
602:p
599:(
594:v
591:,
588:c
584:F
571:p
555:,
550:y
546:v
542:)
537:y
533:c
524:y
520:p
516:(
513:+
508:x
504:v
500:)
495:x
491:c
482:x
478:p
474:(
471:=
468:v
462:)
459:c
453:p
450:(
447:=
444:t
432:v
428:c
424:p
420:t
411:v
409:,
407:c
403:F
391:L
387:c
383:v
379:L
375:p
371:F
367:R
359:v
355:c
350:v
348:,
346:c
342:F
301:q
297:p
289:q
285:p
283:(
281:d
265:,
262:)
259:)
256:q
253:(
250:M
247:,
244:)
241:p
238:(
235:M
232:(
229:d
226:=
223:)
220:q
217:,
214:p
211:(
208:d
196:q
192:p
174:2
168:R
157:2
151:R
145::
142:M
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