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)

Visual Perception of Symmetry - II

Let us now revisit the  flag experiment with the language we have introduced up to now. The experiment consisted of ranking the flags within each row in the figure below:

Each row, as we now recognize, is a symmetry orbit generated by

D_2={1,h,o,v}.

 The flags in columns B,C, and D are generated from the flag in column A by applying a horizontal reflection, a  double reflection, and a vertical reflection respectively.

The rankings were purely subjective, as there were no other directions given at the time of the   flag experiment.

The following table summarizes the frequency counts for the number of occasions in which the first choice is the flag with the row label and the second choice is the flag with the column label in that table. For example, in 14 occasions, flag B was the first choice and flag C was the second choice. I will refer to these frequency counts as transition counts, and write BC=14 to indicate the transition count just described.

With this is mind, a direct inspection of the flags in any one of the orbits shows that the transitions

• BC, CB, AD, DA are associated with vertical reflections,
• AB, BA, CD, DC are associated with  horizontal reflections,
• AC, CA, BD, DB are associated with  double reflections.

The posterior probability densities associated with these symmetry transitions of first-to-second preference, based on the joint frequency counts above, are shown in Figure 1. It is evident that the most likely pairing of first-to-second preferences is associated with those flags that are vertical reflection of each other.  

Figure 1.

Moreover, the vertical transitions form a symmetry orbit, as illustrated below:

For example, starting in (A,D) you move to (D,A) by a vertical reflection on each of the component flags; move to (B,C) by a horizontal reflection and to (C,B) by a double reflection.  The transitions were arranged in the diagram in a way that they are related by their relative positions in the square. The fellow diagram shows the corresponding transition counts, obtained from the summary data table at the top of the page.

These vertical transition data are, therefore, indexed by an orbit of  D_2. However, the assignment of data to the labels in D_2 is arbitrary, in the following way:

If we start the orbit in (A,D) then

• x_1 = 12;
• x_v = 16;
• x_o = 11;
• x_h = 14.

However, if we start the orbit in (B,C), then:

• x_1 = 14;
• x_v = 11;
• x_o = 16;
• x_h = 12.

Clearly, since each D_2 orbit has 4 points, there are 4 possible starting points and hence four relabelings of the data.

Similarly, the horizontal transition counts are labeled by a symmetry orbit:

And finally, the orbit for the the double reflection counts:

These transitions orbits, together, are illustrated here,

in relation to the joint frequency counts table shown at the top of this page.

The question to be investigated is this:

What are the summaries of the frequency counts that do NOT depend on a particular relabeling?

We have alluded to the answer, the orbit invariants for that specific orbit, here.  In all that is coming, the aim is exploring ways of systematically determining the invariants for most types of elementary orbits. Stay tuned.

Assignments:
[1] Following the discussion of the vertical transitions orbit, write the data assigned to the orbit with (A,D) as its starting point in the form of the symbolic sum

X=12*1 + 16*v + 11*o + 14*h. 

Then, referring to the multiplication table of D_2, evaluate the left multiplications

• h*X 
• v*X
• o*X

and verify that all regular relabellings of a D_2 orbit can be obtained that way.

Posted: 05/14/2011.
Last revised: 01/28/2015.
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetry orbits of the triangle

These six images

illustrate the symmetry orbit of the three-color triangle. Note that:

• Any two triangles in different rows are single-reflection images of each other;
• Any two triangles in the same row are double-reflection images of each other;
• Any two triangles in the same row are rotated images of each other.

As a consequence, the symmetry orbit of the triangle is generated by single and double reflections, or, equivalently, by reflections and rotations. 

The separation angle between adjacent reflection axes in the triangle is 2Pi/6 and is referred to as its dihedral angle. The resulting rotations are in angles that are multiple of twice the dihedral angle.

The rotations in one row are in the opposite direction relative to the rotations in the other row. This may hint us to connecting symmetry orbits with space orientation: For example, the arms of  a transparent clock hanging in a transparent wall rotate in opposite directions when viewed from opposing sides of its wall. Conversely, your position relative to the wall may be determined or label by the direction of rotating arms.
More to come on this.

Any one of the six triangles generates all the remaining ones after the reflections and iterated reflections are applied to that initial triangle. Therefore, the initial choice, or the generating element, is arbitrary. This property of arbitrariness is present in any symmetry orbit and will be explored later on in future postings.

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)