*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

w-6a.mov , w-6b.mov , w-6d.mov

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

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.

w-7b.mov , w-7c.mov , w-7d.mov , w-7e.mov , w-7f.mov , w-7g.mov , w-7h.mov , w-7j.mov , w-7k.mov

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

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 .

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.

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)

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 !!).

- Outer shell only.

- Inner network .

Another randomly generated Z(6,3)

- Outer shell only.

- Inner network .

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

- Z(4,3) pivot 0 .

- Z(4,3) pivot 2 .

- Z(4,3) pivot 8 .

- Z(5,3) pivot 1 .

- Z(5,3) pivot 61 .

- Z(5,3) tumble animation rotated by 15 degrees between successive images.

And why not a glimpse of a 7D structure.

Some miscellaneous images from Dec 1999.

- 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

p,q are the source parameters with ranges:

- a < p < b
- 0 < q < qmax

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