Symmetry orbits and space orientation

Here is a simple illustration connecting the notions of symmetry orbit  and  space orientation.  Imagine the following image pasted with its center at the crossing of the equator and the prime meridian of a transparent glass globe, matching its standard cardinal directions. The image is one-sided except in its central area, where it is two-sided.

An observer is allowed to survey the image by walking full-circle around the equator and full-circle around the prime meridian, so that what is seen on the left, right, up, down, front, and back sides of the image relative to the observer's position can be annotated. The globe is stationary relative to the surveyor. What is annotated is the number of disks and circles as the observer approaches the image from the cardinal positions. Here 

is what is the view from the other side of the globe, or the back side of the image, and here

is what the observer reported.

The arrows indicate the front / back side viewing of the image. For example, the observer approaching the image from the East would have seen 16 circles in the front central area, 16 disks up, 25 down, 9 on the left, and 4 on the right side of the image. When at the other side of the globe the observer would have seen 4 circles in back central area, 25 disks up, 16 down, 9 on the left, and 4 on the right side of the image, when again approaching the image from the East.

It turns out that the observer's  path has enough information to map it to a path along a symmetry orbit of the square. That is, a dihedral D_4 orbit. Each up/down, left/right, front/back observed data vector (U,D,L,R,F,B) with the reported number of disks and circles is then indexed by a point in that orbit. 

This will be (somewhat) more evident if we now hold the observer stationary and do a two-step mechanism (the dihedral trick):

• Rotate (say clockwise) the globe along the central axis through the equator-prime meridian intersection (call it the x-axis) by 90, 180, 270, 360 degrees, thus rotating the image around its center;
• Rotate the globe (either direction) 180 deg along the N-S (z) axis, then repeat the four-fold rotations described above. The projections of the 8 transformations onto the z-y plane reproduces the planar (D_4) rotations and reversals.  

The standard question, then, is: what are the summaries of the reported data that remain invariant under the D_4 relabeling? In the present context:  What are the summaries of the data that do not depend on the up/down, left/right, in front/back relabeling? Shortly: What are are the orbit invariants? Or yet: What are the invariants that resolve the arbitrariness in the labels? The corresponding questions formulated for D_2 were discussed in the context of  visual field data. 

We do not have all the tools to develop the complete set of invariants for D_4 yet. However, as we shall see coming along these postings,  the D_4 orbits have exactly 5 invariant summaries. In the present posting we will just enunciate 2 of them. Here they are:

The orbit invariant on the top combines within rotation variation and within-reversals variation, whereas the other one compares rotations with reversals, briefly stating it. Both invariants define one-dimensional subspaces for the data. The first invariant thus resolve the arbitrariness in the left-right, up-down orientation. The relabeling of the planar orientation has the effect of at most changing the sign of the summary (+/- 56). The second invariant resolves the arbitrariness in the front-back orientation. Again, the summary (+/- 48) stays in a one-dimensional subspace. 

These two invariant (subspaces) account for two of the eight dimensions afforded by D_4. As we move along the remaining invariants will be introduced. When all invariants are available, the inverse problem of recovering the original data along the orbit can be effected.  

First revised 06/08/2011
Text with this color was revised in  06/09/2011

These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Field orbits

The following is a symmetry orbit in which the elements are planar vectors:

To generate the orbit, we may start with any one of its vectors in red, rotating it counterclockwise in steps of 90 degrees, producing the vectors in red; Then we may reflect the original starting vector with respect to the x-axis, and rotated it the same say, thus producing the remaining vectors (black color) in the orbit.  The resulting orbit thus has the symmetries of the square, introduced earlier on  here. Clearly, the orbit would have been the same regardless of its initial condition (generating vector) and direction of rotation.

The orbit above has the symmetry of the square. We will indicate those symmetries with the symbol D_4. We will refer to the symmetries of the regular polygons as dihedral symmetries,  We will write D_3 for the triangle, D_4 for the square and so on, for D_5, D_6, .... We extend the notation to D_2 as well, although in this case it  indicates the symmetries of the rectangle.

These symmetries are the planar line reflections and double reflections (or rotations), or equivalently, the n-fold rotations and rotations preceded or followed by a reflection. These always come in pairs, and referred to as rotations and reversals.

Rotations and reversals have, intrinsically, the same reflexivity found in the naming of left- and right-handed objects. It is perhaps better not to try to say it in words. It gets rather silly very fast.

Field orbits are simple designs to gather experimental results that are then indexed by the points in the orbit.

For example, dihedral symmetry aspects in a visual field can be studied by its response to a dihedral orbit of the square by embedding a set of rotations in the visual field, say:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

and the companion reversals:

0 0 0 26 27 24 21 0 0 0

0 0 23 23 26 28 25 22 0 0

0 27 29 27 27 28 29 29 28 0

28 29 29 29 31 20 20 28 28 29

26 26 29 30 33 34 28 27 28 27

28 30 30 31 32 33 33 4 27 28

27 29 31 31 32 32 30 29 28 27

0 30 31 31 28 29 32 30 30 0

0 0 28 32 29 28 29 28 0 0

0 0 0 26 27 28 29 0 0 0

Any summary of the visual field over each dihedral field vector, such as its gradient, average, or extreme values, is then indexed by the dihedral symmetries. The summaries may, of course, be in more than one dimension. Visual fields are usually obtained from fellow eyes, so each orbit may have a fellow orbit obtained from the fellow eye, thus producing a pair of numbers in each point of the dihedral field. 

Similarly, below are the dihedral field orbits of D_5, D_6, and D_10.

The question to be addressed in later postings is that of determining the orbit invariants for those experimental data, their broad interpretations, and plausible methods of statistical inference.

Last revised: May 15th, 2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)