Filling tetraspace with concentric tori

Discussion of shapes with curves and holes in various dimensions.

Filling tetraspace with concentric tori

Postby Teragon » Sun Jan 03, 2016 2:55 pm

In 4D there is an interesting possibility for a gapless filling of tetraspace using only concentric tori.

The image below shows 3D sections through different 4D objects. In the row at the bottom the objects are rotated by 90° in the yw-plane, so that the left-right direction is interchanged with the invisible w-direction.


The pictures in the center represent two objects I'm calling cyltori, tori with a cylindrical cross section where the top-/bottom surface of the cylinder is pointing to the center of the torus. The cyltori are aligned in a way that all vectors in the 2-plane of the loop of one torus are perpendicular to all vectors in the 2-plane of the loop of the other torus. The two loop-planes are intersecting at the common center point of both tori. The contact surface of both cyltori is a Clifford torus and the enclosed volume has the shape of a duocylinder as depicted on the left side. The Clifford torus is the edge of the duocylinder.

We can now wrap a tiger with a square as a cross section around both of them (right side). I'm calling this object a square-tiger. It turns out that the two cyltori together with the sqare-tiger form another duocylinder. So we can repeat the same game with bigger tori at the outside and smaller tori at the inside. Each layer is identical to the previous one, except for a uniform stetching factor of 3.

We could also build the filling around a central duocylinder or fuse two cyltori and square-tiger to hollow duocylinders. Here it becomes evident that the square-tiger and the tiger in general is nothing more that a bunch of concentric Clifford tori with different loop diameters wrapped around each other. In other words we can create a tiger by parameterizing the loop radii r1 and r2 of a Clifford torus, so that the 2D cross-section of the tiger is equal to the shape traced out by (r1, r2). In the case of a square-tiger, the inner and outer borders of 1 and r2 determine the inner and outer radii of the tiger loops. Similary we can create a cyltorus by varying r1 between 0 and the radius of the cylinder cross-section and r2 between the inner and outer radius of the desired cyltorus. A duocylinder is formed if we vary r1 and r2 between 0 and the radii of the duocylinder. All three classes of shapes - duocylinder/spherinder/glome, cyltorus/spheritorus and square-tiger/tiger - are parameterized by the equations for the Clifford torus.

Yet another possible filling for tetraspace would be using cyltori and square-tigers with identical cross sections around a central duocylinder by varying the radii of their loops.

I think this is a nice example to see how different 4D objects are interrelated.
What is deep in our world is superficial in higher dimensions.
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