## Bicircular tegum surface volume?

Higher-dimensional geometry (previously "Polyshapes").

### Bicircular tegum surface volume?

Hi everyone, I've lurked here occasionally but this is my first post.

I've been playing around with split-quaternions, and I'm particularly interested in a certain subset of them, restricted to a region bounded by a curious 4d shape I now know (from searching the forum) is called a "bicircular tegum". (If anyone's interested, I got here from the 2d split-complex numbers, in which the corresponding region is a square, which I guess would be a "bilinear tegum"?)

I've managed to work out that its hypervolume is (pi^2/6) r^4, one third that of a glome. But I need to know its surface volume too, and I thought I'd ask here before breaking out the integrals.
Mononian

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Joined: Wed Nov 02, 2016 3:23 pm

### Re: Bicircular tegum surface volume?

If you have its volume, the surface is no major hassle. You would find it by n.dV = r.ds where n is the dimension, dV is an incrementt of volume, r is a radial vector, and ds. is an element of surface.

The volume of <()()> is then, for a radius of 1, pi^2/6. The inradius r is 1/sqrt(2) gives v= 4/3 pi^2 sqrt(2), which is the volume of a tetrahedron wrapped on the surface.
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wendy
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### Re: Bicircular tegum surface volume?

Thanks! Though unfortunately I realized yesterday that I slipped up, and the tegum includes points I don't want it to. I'll have to figure out how to pare it down. Perhaps to a 16-cell, though I hope not.
Mononian

Posts: 10
Joined: Wed Nov 02, 2016 3:23 pm

### Re: Bicircular tegum surface volume?

You no chance are looking at a quarterion group that corresponds to the bi-polygonal tegum.

I generally hold there is this (infinite) groups, and Q2 (the 16-choron), Q3 (the 24-ch), Q4 (dual 24ch) and Q5 (3,3,5). One of the subgroups of Q3 is the bi-triangular group, where one limb is at sqrt(2) of the other. This corresponds to the bi-alternating group AA3, where Q4 and Q5 are the bialternating groups AA4 and AA5.
The dream you dream alone is only a dream
the dream we dream together is reality.

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wendy
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### Re: Bicircular tegum surface volume?

Oho! That is very useful information, thank you!

I'm a very amateur mathematician plinking around, so no surprise I'm trying to reinvent the wheel!

I'll have to check those groups still work in the split-quaternions, though.
Mononian

Posts: 10
Joined: Wed Nov 02, 2016 3:23 pm

### Re: Bicircular tegum surface volume?

I'm pretty much an amateur too. Just really good at it.
The dream you dream alone is only a dream
the dream we dream together is reality.

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wendy
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Posts: 1920
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia