What shape is this?

Higher-dimensional geometry (previously "Polyshapes").

What shape is this?

Postby The Shadow » Tue Jan 17, 2017 8:57 pm

I've been messing around a bit with a 4d algebra, and I keep coming across a shape described by this formula:

(w-x)^2 + (y-z)^2 = c

What is it, and what is it like? When c = 0, it's a 2d plane (unless I'm mistaken), but what happens for other c? The 3d equivalent would be an infinite cylinder, but when I try to picture this in 4d I keep getting two intersecting cylinders, which I know isn't right.
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Re: What shape is this?

Postby wendy » Wed Jan 18, 2017 12:31 am

The equation is for a duocyliner or bicircular priam. The surface ia two 3d toruses or cylnders joined end to end.
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Re: What shape is this?

Postby The Shadow » Wed Jan 18, 2017 1:04 am

Thanks, Wendy! It's going to take me a while to wrap my mind around that one - I think I have a handle on finite duocylinders, but my mind balks at an infinite one somehow.

Maybe you can help me with a closely-related shape too. The norm in this algebra turns out to be:

(w + x + y + z)(w - x + y - z)[(w - y)^2 + (x - z)^2] = c

What on earth is that thing? (The hypersurfaces of constant norm, I mean.)

EDIT: And believe it or not, that norm is multiplicative. I had to do it twice before I believed it myself.
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Re: What shape is this?

Postby The Shadow » Wed Jan 18, 2017 3:26 am

I just realized I could rewrite that norm as:

[(w + y)^2 - (x + z)^2][(w-y)^2 + (x - z)^2]

A simple coordinate transformation gives:

(w^2 - x^2)(y^2 + z^2)

Which is a little more clear; it's a hyperbola times a circle. But I still can't grasp exactly what that means...
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