Re: Dekany in VRML (without sound clips)

I've been investigating Dekany in VRML some more, and found that all relative sizes of external octahedron and central
tetrahedron give valid perspective views.

When both have the same edge lengths, then it will serve as non perspective view (from infinity) towards centre, either
through octahedron, or through tetrahedron face.

When one is larger than another, will serve as perspective view through whichever has larger edges.

I've used this to make a perspective view on the dekany with five non intersecting tetrahedral faces:

http://members.nbci.com/_XMCM/tune_smithy/decany/decany_template_xx_0p3_filled_tetrahedra.wrl

At present, plan to use this for the first example sounding dekany. Usefully, the five external tetrahedra are
thin enough so that the spheres show through them from the other sied, so you can click on the triads for
the inward facing faces from outside it.

The red and magenta spheres in mid-air are mid points of faces shown as vacant so that one can see into the centre.

As before, spheres are coloured according to geometrical position.

All red ones are at centre of faces of external octahedron, all yellow ones are centres of the inward facing faces of
the tetrahedra (showing through on outside because of thinness of the tetrahedra), and all dark blue ones are at centres
of faces of the central tetrahedron. All cyan ones are mid-points of edges.

This is view through the octahedron face towards the central tetrahedron, which is the one furthest away in the fourth
spatial dimension. Thus, it is smaller because of 4D distance, and its edges are slightly smaller than those of the
octahedron.

The other tetrahedra are "squashed" because they are seen tilted in the 4th dimension rather than face on.

You can also easily see the five octahedral faces of the truncated simplex - the external one of course, and four
stretched out ones, each joining one of the vacant triangles of outer octahedron to one of the triangular faces of the
inner tetrahedron - each of these octahedra has four dark blue faces, one red one, and four vacant faces.

To make it easier for to generate these perspective views in VRML, made a file for anyone to use:

http://members.nbci.com/_XMCM/tune_smithy/decany/decany_template_av_filled_tetrahedra.txt

To use this to make a template VRML decany with any relative size of outer octahedron and central tetrahedron, first
save as a .wrl file.

Now you need to do some search and replace of text:

Select a value for xx, - (xx,xx,xx) will be one of vertices of centre tetrahedron, (where octahedron has coords
(1,0,0), (0,1,0),...)

Here xx = 0.5 is the non perspective view.

then replace

(1+2*xx)/3, (1+xx)/3, xx/3, (1+xx)/2, xx/2, and xx everywhere by appropriate values

So for instance with the ex. of xx = 0.3, these were replaced by
0.533333, 0.433333, 0.1, 0.65, 0.15, and 0.3.

(you can't just replace xx because use of formulae to specify coordinates is not part of VRML spec, so replace the
formulae first, then remaining instances of xx at the end.)

Result will be the VRML template for the perspective view.

This method will generate all 3D perspective views through either an octahedral or tetrahedral face of the decany
looking towards the centre.

Explanation of how to add MIDI clips is as before:

http://members.nbci.com/_XMCM/tune_smithy/decany/decany_template_expo.txt

So go ahead and use this if anyone else wants to have a go as well (or write program to do so).

Robert