The assorted planar patterns deserve to by divide by the revolution groups the leave them invariant, their symmetry groups. A mathematical analysis of these groups shows the there are specifically 17 different aircraft symmetry groups.Note that clicking a small image listed below will take you come a conversation of the linked symmetry team (as will picking the surname of the group in the headings below).
A short table of characteristics of the the contrary groups
There are sufficient characteristics noted in the table to distinguish the 17 different groups.SymmetrygroupIUCnotationLatticetypeRotationordersReflectionaxes

1  p1  parallelogrammatic  none  none 
2  p2  parallelogrammatic  2  none 
3  pm  rectangle  none  parallel 
4  pg  rectangle  none  none 
5  cm  rhombus  none  parallel 
6  pmm  rectangle  2  90° 
7  pmg  rectangle  2  parallel 
8  pgg  rectangle  2  none 
9  cmm  rhombus  2  90° 
10  p4  square  4  none 
11  p4m  square  4 +  45° 
12  p4g  square  4 *  90° 
13  p3  hexagon  3  none 
14  p31m  hexagon  3 *  60° 
15  p3m1  hexagon  3 +  30° 
16  p6  hexagon  6  none 
17  p6m  hexagon  6  30° 
 + = every rotation centers lie on have fun axes* = no all rotation centers on reflection axes 
The IUC notation is the notation because that the the contrary group adopted by the global Union of Crystallography in 1952.Symmetry group 1 (p1)This is the most basic symmetry group. It consists only that translations. There are neither reflections, glidereflections, no one rotations. The two translation axes may be lean at any kind of angle to every other. The lattice is parallelogrammatic, therefore a fundamental region for the symmetry team is the exact same as that for the translate in group, namely, a parallelogram.Symmetry team 2 (p2)This group differs just from the first group in the it includes 180° rotations, the is, rotations of bespeak 2. As in every symmetry teams there space translations, yet there no reflections nor glide reflections. The two translations axes might be inclined at any kind of angle to each other. The lattice is a parallelogrammatic. A an essential region because that the symmetry team is fifty percent of a parallel that is a basic region because that the translate in group.Symmetry team 3 (pm)This is the very first group that has reflections. The axes of reflection space parallel to one axis of translation and perpendicular to the other axis that translation. The lattice is rectangular. There space neither rotations nor glide reflections. A fundamental region for the translation team is a rectangle, and also one have the right to be favored that is separation by an axis the reflection so that among the fifty percent rectangles develops a an essential region because that the symmetry group.Symmetry team 4 (pg)This is the very first group that includes glide reflections. The direction the the glide reflection is parallel come one axis of translation and also perpendicular come the other axis the translation. There are neither rotations no one reflections. The lattice is rectanglular, and a rectangular fundamental region because that the translation group can be liked that is breakup by an axis of a glide reflection so that one of the fifty percent rectangles develops a an essential region because that the symmetry group.Symmetry group 5 (cm)This group includes reflections and also glide reflections v parallel axes. There room no rotations in this group. The translations might be skinny at any kind of angle to each other, however the axes of the reflect bisect the angle formed by the translations, therefore the basic region for the translation team is a rhombus. A an essential region because that the symmetry team is fifty percent the rhombus.Symmetry team 6 (pmm)This the contrary group includes perpendicular axes that reflection. There room no glidereflections or rotations. The lattice is rectanglular, and a rectangle deserve to be preferred for the an essential region the the translation group so that a quarterrectangle of it is a fundamental region for the the opposite group.Symmetry group 7 (pmg)This group consists of both a reflection and also a rotation of order 2. The centers of rotations do not lie on the axes that reflection. The lattice is rectangular, and also a quarterrectangle the a fundamental region because that the translation team is a an essential region for the the contrary group.Symmetry group 8 (pgg)This group contains no reflections, yet it has actually glidereflections and halfturns. There space perpendicular axes because that the glide reflections, and the centers of the rotations perform not lie on these axes. Again, the lattice is rectangular, and a quarterrectangle that a fundamental region because that the translation group is a fundamental region because that the symmetry group.Symmetry group 9 (cmm)This team has perpendicular have fun axes, together does team 6(pmm), yet it likewise has rotations of bespeak 2. The centers the the rotations execute not lied on the have fun axes. The lattice is rhomic, and a 4 minutes 1 of a an essential region because that the translation team is a an essential region for the the contrary group.Symmetry group 10 (p4)This is the an initial group through a 90° rotation, that is, a rotation of order 4. It likewise has rotations of stimulate 2. The centers the the order2 rotations room midway in between the centers the the order4 rotations. There room no reflections. The lattice is square, and again, a quarter of a basic region because that the translation team is a an essential region because that the the opposite group.Symmetry team 11 (p4m)This team differs from 10 (p4) in that it also has reflections. The axes the reflection room inclined come each other by 45° for this reason that 4 axes of reflection pass with the centers of the order4 rotations. In fact, every the rotation centers lie on the enjoy axes. The lattice is square, and an eighth, a triangle, of a fundamental region because that the translation group is a an essential region for the symmetry group.Symmetry group 12 (p4g)This group additionally contains reflections and also rotations of assignment 2 and also 4. However the axes the reflection are perpendicular, and also none the the rotation centers lied on the enjoy axes. Again, the lattice is square, and an eighth of a square an essential region that the translation team is a basic region because that the symmetry group.Symmetry team 13 (p3)This is the simplest team that has a 120°rotation, the is, a rotation of stimulate 3, and the first one whose lattice is hexagonal.Symmetry group 14 (p31m)This group contains reflections (whose axes are inclined at 60° to one another) and rotations of bespeak 3. Several of the centers of rotation lied on the have fun axes, and some carry out not. The lattice is hexagonal.Symmetry group 15 (p3m1)This group is comparable to the last in that it has reflections and also order3 rotations. The axes the the reflections are again inclined in ~ 60° to one another, yet for this group all of the centers of rotation lied on the enjoy axes. Again, the lattice is hexagonal.Symmetry group 16 (p6)This group includes 60° rotations, that is, rotations of stimulate 6. It also contains rotations of orders 2 and 3, but no reflections. That is lattice is hexagonal.Symmetry team 17 (p6m)This most facility group has actually rotations of stimulate 2, 3, and 6 as well as reflections. The axes that reflection accomplish at all the centers that rotation. At the centers of the order6 rotations, six reflection axes meet and also are inclined in ~ 30° to one another.
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'The Wonder Years': Who Sang The Wonder Years Theme Song That Is Played The lattice generator is hexagonal.Up come the table that contentsBack come groupsOn to history© 1994, 1997.David E. Joyce department of Mathematics and Computer ScienceClark UniversityWorcester, MA 01610
These papers are located at http://www.sdrta.net.edu/~djoyce/wallpaper/