Secret renders

Higher-dimensional geometry (previously "Polyshapes").

Secret renders

Postby Secret » Thu Apr 07, 2011 3:22 pm

Image

Guess what this is (hint: note the cross section)
(If the interpretation is correct, then this is really that shape)

4D tori
Image

From left to right:
3-torus (ditorus) (top: revolving xyz 2-torus, bottom: revolving xyw 2-torus (clifford torus/flat torus))
Toraspherinder (top: glue spherinderal hose into a loop, bottom: glue spherinderal hose in the inside (toraspherinder dual/inside out toraspherinder))
Torinder/toracublinder
Tiger/Toraduocylinder/Spherated bicircular comb

(note the 3-torus made from clifford torus is the same as the Tiger in the projections as the only difference between a clifford torus projection and a duocylinder projection is the presence of a cap at the hole of the torus (the cap is from part of the surface of another bounding 2-torus)

(More cooming soon)
Last edited by Secret on Tue Apr 12, 2011 2:19 pm, edited 3 times in total.
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Re: a sneak peak

Postby wendy » Fri Apr 08, 2011 7:16 am

spherated bi-circular comb. Yes, it really does exist.
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Re: a sneak peak

Postby Secret » Fri Apr 08, 2011 9:39 am

Oops
I don't understand what 'comb' means even after looking the description in the polygloss
Mind elaborate?
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Re: a sneak peak

Postby wendy » Sat Apr 09, 2011 7:49 am

comb = product by repetition of the surface. This reduces the dimension.

For example, for a polygon-polygon product, unwrap each polygon, to get a chain of lines. The product of repetition is the cartesian product here, so you get a grid of squares. You then roll these up to a solid that is covered by them: ie a torus.

For the particular figure in question here.

A cartesian product of two circles would give a 'duocylinder' or bi-circular prism. The margin between the faces is the product of the surfaces of the circle, becomes the 'comb' or product of two circle-surfaces.

Spheration is the process, where if x is within a distance of r from some structure, it is part of the solid. This is what happens when points become balls, and lines become cylinders. Regardless of the thing being spherated, the thing becomes a solid.
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Re: a sneak peak

Postby Secret » Sat Apr 09, 2011 10:10 am

do you mean like this?

Image

also if I understand your post correctly: Yes it is the spherated bi circular comb (aka Tiger)
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Re: Secret renders

Postby wendy » Sun Apr 10, 2011 10:46 am

Yes, that's how comb works.

In four dimensions, you can take a 2d * 3d surface, to get two different torii.

Suppose you have a hexagon ** dodecahedron.

You could take dodecahedral struts (long prisms in 4d, with dodecahedral sections), and make a hexagon from these, which join. The surface would then be topologically 6*12 pentagonal prisms, arranged in stacks of 6, and a dodecahedron shape.

You can also take a dodecahedron-prism stack, and roll the outer pentagon-hexagon prisms so that they link top to bottom (kind of like rolling off a sock off a 4d cylinder), and then link the top and bottom together.

The two shapes have the same surface topologically (since one inverts to the other), but one surrounds a hexagon (which can not disappear without going through the surface), and the other contains a hollow dodecahedron, ditto.
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