Signed Difference Analysis

(a.k.a. Cordant Zone Analysis)

Ralph James* & John Dunn**

*Physics Department, University of Western Australia.
**Dept. of Psychiatry & Behavioural Science, University of Western Australia

Signed difference analysis is a new technique developed by John Dunn and Ralph James for the testing and evaluation of higher dimensional non-metric data sets. An introduction is being prepared.

For a preprint please see John Dunn

Zone labelling convention: eg. in 5D {+,+,+,+,+} = 0; {-,+,-,+,+} = 5

Z(6,3) structures

If the grid restoring force is too strong we get 'wobbling' - higher values do not settle but 'become unstable.

Z(7,3) structures

Starting structure w-7a.mov

Same pattern with stronger restoring force. watch8.mov

Related structures evolved from the previous (w-7a.mov) by successive branching chiral flips.

Z(8,3) structures

A sample of 16 (out of the possible 128) distinguishably different topologies for 3D hypersurfaces in an 8D measure space:

Z(11,3) structures

As an example of some of the higher dimensional structures, here are two - the first generated randomly z-11a3c.mov , the second derived by hand to show certain imperfect 10-fold symmetry z-11bc.mov, z-11bs.mov

Z(13,3) structures

Here is a highly symmetric 3D surface embedded in 13D, disentangled by Igor with centre free z-13c.mov, or rim free z-13r.mov initialisation.

And now for something completely different - a 2D surface in 4D (projected here into 3D) that contains a singularity in the Gauss map. The data is produced from a 40x40 parameter array .

Mobius-Klein example:

this is a simple but peculiar 2D surface in 4D satisying

x^2 + y^2 + z^2 + t^2 = 1 and x*y = z*t.

This surface is also known as a 'spherical torus' (since it lies on a 3-sphere in 4D) or 'double cylinder' since it can be thought of as a piece of square flat paper rolled into a cylinder top-to-bottom and side-to-side at the same time - something only possible in the wierd world of 4D.

Zonal coverage in 2D parameter space is shown in the following image.

Following are some cross-sections through the zone coverage map for the Mobius-Klein surface.

Moving 'diagonally' from {-90,-90} to {90,90}, diag.mov.

Moving 'horizontally' from {-90,-50} to {90,-50}, p5.mov.

Moving 'vertically' from {-50,-90} to {-50,90}, 5p.mov.

Krantz example

The above image shows the isozonal structure of the 1D curve in 3D descibed by the equations:

x = - sin(p)
y = - cos(p)
z = exp(2*sin(p)/(1+p)) - exp(-p)

Trinary zone structure (August 99)

The following images each show four examples of Z(6,3) surfaces. x and y axes are the 123 occupied tri-zones, while the value plotted (greyscale) is the contiguity between each pair of zones.

Try this,or this.

3D surface in 6D  - trinary zone analysis, Ocober 1999.
Generating matrix:

 1 0 0 0 1 0 0 0 1 2 5 7 -3 4 11 -5 13 -3

View the following movies (NOTE - each file is about 1Mb !!).
A 'flawed' variant where the first '2' is replaced by '0'.

Another randomly generated Z(6,3)

Some gif images (about 20kb each) from Z(4,3) and Z(5,3).

And why not a glimpse of a 7D structure.

Some miscellaneous images from Dec 1999.

A spiral of contention!

The following image is of a spiral ribbon surface (2D in 3D) that is everywhere locally "homomestic" but globally panzonal! The parametrisation used is:
• x = A p cos q + B p sin q + A c q

• y = - A p cos q + B p sin p - A c q

• z = - B p cos q + B c q

where A = 0.5, B = 1/sqrt(2). c is the "pitch" of the spiral.
p,q are the source parameters with ranges:
• a < p < b

• 0 < q < qmax

This particular image used a = 1, b = 2, c = 0.2.

If the above doesn't make any sense try the introduction.