by **wendy** » Fri Dec 04, 2009 8:24 am

1. non-vanishing surface.

A non-vanishing sphere is a test for holes. Essentially, vanishing means going to zero or infinity, without crossing any part outside of the space it falls in.

For example, in a plane, you can have two points. A circle that includes both can go to infinity, ie vanish. A circle that includes one, can neither vanish to zero or to infinity, so indicates a hole exists. In order to prevent such a loop forming, one has to patch the hole with a fabric that is orthogonal to the hole and spans the space. We create a line between the two points, and so all circles that now form must include either zero or none.

The surface of a sphere of n dimensions, does include an n-dimensional sphere, which is strictly immobile. To prevent this from happening, we need to include a point, which means that the surface no longer can contain an n-sphere.

2. The calculation for the circle-sphere torus is as follows:

We start in five dimensions with a prism, being the circle-sphere prism, each with an extra point.

1h.1e.1v.1n *# 1c.1h.0e.1v.1n The prism product ignores n, but includes the first term, so

we find 1 001 001 * 1 001 000 001 = 1 002 002 002 001 001 = 1p 2t 2c 2h 1e 1v 1n

The next step is to fold the thing flat from 5d to 4d, while preserving all the connections. This removes 1p 1t, since this t becomes the outside. Think schegel diagrams.

We get then 1t 2c 2h 1e 1v 1n

This contains the torus 1t 1c 1n, subtracting this we get 1c 2h 1e 1v as the hole consist.

The missing patch theory suggests that a patch will prevent the forming of non-vanishing spheres in any space, so let's see

1c, 1h are required to prevent non-vanishing loops on the outside/inside. Circles are prevented by 1c, the spheres are by 1h.

The remaining three holes 1h, 2e. 1v form inside the 3d surface. This space is a circle-sphere comb (the product of surfaces).

You need a full sphere 1h to stop circles running around the height. Adding this allows one to open the figure into a sphere-prism, the curved side. All circles on this will vanish, including those that go around the figure but complete spheres do not. You can see that a circle on a unmarked sphere, or a sphere with a point on it, can still vanish in the space of the sphere. To prevent these spheres forming, we need a line from top to bottom, which is another line.

We have now created two new surfaces 1h 1e, both of which admit holes. None the same they cross, and this crossing provides a point for these figures to prevent the surface forming as a non-vanishing sphere.

We could also demonstrate the piecewise construction of this, by

v.n create a point.

h.e create the circle, with its interior. The circle contains v. The dimensions are wx

c.h create an entirely different sphere, so that it lies in wyz space, in opposite w.

t.c The surface created by rotating the sphere around the centre of the circle, and the interior ditto.

depending on closure the t could contain the leading c or the h from the previous spaces.

The bits that are extraneous are the first three elements, the beginning of the second and third are in some 4-space

The dream you dream alone is only a dream

the dream we dream together is reality.