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)

The Sloan Fonts classification study

The 10 distinct letters appearing (in Roman font) in the chart below are used in the assessment of visual acuity. The actual charts are set in Sloan fonts and are referred to as  Snellen Charts.

The letters composing each line were selected [1] to balance the between-line overall difficulty in correctly identifying the five letters.

The table below lists the Sloan letters (in Roman font), their estimated probability (p) of being correctly identified by normal-acuity subjects, their corresponding entropy, and their -log contrast sensitivity:

We now refer to the subgroups introduced earlier on here and obtain the follow classification of the fonts according to the (largest) subgroup of symmetries afforded by each font:

• K, R        ------->    { 1 },
• Z, N, S    ------->    { 1, o },
• V             ------->    { 1, v },
• D, C        ------->    { 1, h },
• H, O        -------->  { 1, v, h, o }.

Each font in each class is the symmetry orbit of that subgroup. Shortly, we may say that the font has the symmetry of that subgroup.

Moreover, now we may associate to each subgroup a number, such as the maximum entropy of the fonts with the symmetry of that subgroup. From the above table where the entropies are listed we then obtain:

• K, R        ------->    { 1 }             -----> 0.669,
• Z, N, S    ------->    { 1, o }         -----> 0.693,
• V             ------->    { 1, v }         -----> 0.656,
• D, C        ------->    { 1, h }         -----> 0.687,
• H, O        -------->  { 1, v, h, o }  -----> 0.692.

This is then our first example of subgroups indexed by numerical evaluations. Later on we will remark that this experimental situation can be treated in analogy with the previous case, where the elements of a given group were indexed by numerical evaluations.

=====  Reference Cited =====

[1]  Ferris FL 3rd, Freidlin V, Kassoff A, Green SB, Milton RC. Relative letter and position difficulty on visual acuity  charts from the Early Treatment Diabetic Retinopathy Study. Am J Ophthalmol. 1993 Dec 15;116(6):735-40. 

Last revised on 03/01/2011

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