Some details for the hypertoroid.  A simple torus surface has a major and a minor diameter.  This toroid has four diameters of interest.  This looks like four normal toroids, grouped together.  The toroid  with points A, C and E is one of these four.  The two diameters for a normal toroid is A-C, and A-E.  These are the only points that look connected together.  The other points are actually connected also, but requires the cross section to be made along another axis.  For a the Hypertoroid view  of this figure, the four diameters are in descending diameter sizes are A-B, A-C, A-D and A-E.

A-B, in this HyperTorus manifold, is the diameter of the universe.  The diameter is related to the gravitational coupling constant. 

A-C, is the classical diameter of the electron.  It's size determines the mass and coupling constant for an electron.

A-D, is related to the fine structure constant.  It determines weak force coupling.

A-E, is smaller and relates to the mass and coupling for particles with a strong force component.

The family of particles that seem to have 3 levels can be envisioned with this model.  A-C is the lightest mode, A-D is heavier, while A-E is the heaviest.  Particle wavelengths tend to want to expand to fill available space.  But  it requires an orthogonal turn to make the large dimension available.

Cross section at the major diameter with highlighted points

To help step through the logic of the above 3D cross section of a hypertorus, a tour of a plane cutting though a 3D torus may help.

Plane cutting through a normal torus

A cutting plane's view results in two disconnected circles.  While in the original 3D view,  it can be seen that the two circles are part of the same surface.

A cross section view in the plane of the torus would look like a circle in a circle.

Plane cutting through the inner tangent point, is seen as a stretched figure eight on the plane.

Plane cutting through mid tube

A view of a 4d hypertoroid.  This looks like a tube with a tube.  Compare this with the circle within circle 2D cross section of a torus above.

There are many ways to make a 3D cross section view of a hypertoroid.  The one at the top of this page illustrates the 4 radii sizes of the torus.  But it fails to show the connections between all the points.  They all are connected.  Another view could possibly show all the points are connected, but the nice constant curves would not be seen.

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Last Updated on March 12, 2002

©2002 Robert D Rutkiewicz