Spheration defined?

[Keiji]

Okay, I have been searching the tetraspace forums for the last hour or so, but nothing seems to have turned up. Can anyone give me a clear definition for group notation ((A₁)(A₂)...(A_n)) where A_i is at least two-dimensional?

These are what I call the tigroids, based on the name "tiger". They're the only shapes which I cannot produce a CSG notation string for, and since they're all in 4D or higher, there's absolutely no way to visualize them without it.

[wendy]

Spheration is a process of replacing points, lines, etc by solids based on the prism-product of these and spheres.

For example see this, which is a sphereated cube. It is also one of the prototypes in the definition.

http://www.skypoint.com/members/jkm/ato ... omium.html

You will see that the points have been considerably spherated (ie their radius is huge), while the edges have been rendered as cylinders, literally the trace of a sphere moving along the line. Of course, in higher dimensions one can do different.

In the case of the tiger, or spherated bi-circular comb, one takes in 4d, the prism-product of the surfaces of orthogonal surfaces, and then consider every point that is a distance r from this figure, as the surface of the tiger.

[Keiji]

Firstly this is a ridiculously old post o.o

Secondly, did you even read my original post fully?

Can anyone give me a clear definition for group notation ((A₁)(A₂)...(A_n)) where A_i is at least two-dimensional?

[wendy]

The response was picked up in 'search for un-answered questions'. I did not look at the dates :s

Spheration is an effect applied to the a partial construction. This is what the name of the article says.

The crind product is a product of form rss(), applied to a radiant form.

The radiant form of a surface and a centre, supposes that the centre is 0, and the surface is 1. Any line drawn from the centre that strikes the surface, is measured in a linear scale using these units. Should the line strike twice, then it is counted as two separate lines, with different measures. One can then draw smaller and larger copies, eg at 0.7 or 1.3, by linking all of the 0.7 &c points together.

The normal practice is to centre the 0-point. For example for a circle of radius 1, the radiant function is simply the true radius. For a square of edge 2, the radiant function is max(abs(x), abs(y)).

An example of spheration applied to a figure of 2d is the crossing of two equal cylinders, being {[x,y],z}. For any plane containing the z axis, it strikes the square at some angle, giving a line somewhat longer than 2 units. The section in this axis and z, gives an ellipse of height z, and length longer than 2. For example, the one through the long diagonal gives an ellipse of axies 2 (z), and 2.828 (x+y). This is the crind-product of lines 2, 2.828.