How are these figures called?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

How are these figures called?

Postby joan » Sat Jan 30, 2016 4:13 pm

Hi,

I don't really know which words to search for to find the name of these basic figures.
Start with a triangle on plane XY, and second triangle on plane ZW. Connect vertices from one triangle to the other.
So this should be a polyhedron embedded in 4-space I guess. Somehow I feel it's not a simple triangular prism. Or is it?
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Re: How are these figures called?

Postby wendy » Sun Jan 31, 2016 1:33 am

It's a bitrianguar tegum.
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Re: How are these figures called?

Postby joan » Sun Jan 31, 2016 9:36 pm

Thank you!
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Re: How are these figures called?

Postby Prashantkrishnan » Sun Oct 02, 2016 5:52 pm

Do these triangles intersect at a point or are they completely outside each other?
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Re: How are these figures called?

Postby Klitzing » Sun Oct 02, 2016 7:19 pm

The bitriangular tegum has vertices (up to common radial scaling)
Code: Select all
(sin(k*120°), cos(k*120°), 0, 0) for k=1,2,3 plus
(0, 0, sin(n*120°), cos(n*120°)) for n=1,2,3.

Accordingly the centers of both triangles indeed do coincide.

These triangles are mutually perpendicular, i.e. no further point touches.
Further, as the tegum product asks for the subsequent hull operation onto the components, these spanning triangles themselves become completely internal.

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Re: How are these figures called?

Postby wendy » Mon Oct 03, 2016 1:14 am

If you are familiar with complex numbers and analytic geometry, then using complex numbers in N-dimesional geometry, doubles the value of N of each space, so lines are 2d, hedrices are 4d, &c. Reflection (r -> -r) is replaced by r = r cis(wt) where cis = cos + i sin, and wt is omega (speed) * time, means in every even dimension things go around a centre.

The regular polytopes of p(4)2 have edges with p vertives, gives a 4d reflex of the p-gns in a bi-p-gonal prism, The dual is 2(4)p has edges that have 2 vertices, that run from diameter to diameter. 2(3)2(4)p has triangles wedged in the x,y,z axis. Likewise, the 5{3}5 has 120 edges, by considering the pentagons around the edges that the vertices fall in, when the figure is rotated 6/120 of a circle. This is the basis of the Poincare pentagonal tegum i showed John Conway.

The figure Richard gives of the thatched (tegum-sum) inverted tri-triangular prisms, is but the 2(4)3(3)3 with three orthogonal diameters removed. A similar effect is see in 3,3,5 = f3,4,3 + GAP.
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