by **wendy** » Wed Feb 12, 2014 9:52 am

Negative dimensions are necessarily quantum in nature, and can not be detected from the real world. It is only when one says that this bit is X and this bit is not X, that the bits that are X call down to a 'wessian' (being) of 'X'. This existance happens at a dimension one less than that of points.

Endoanalysis

This was a little experiment done, to determine whether solids really have an existance separate from the surrounding space, or whether they were a device of our creation. The results are quite interesting, because the gulf between 'real densities' (which measure how many times space is occupied), and "integer densities" (which tells us that there are things points and lines etc there), is never breached. Polytopes are devices of our mind, and negative dimensions are what gives them a thing to 'belong' to: a handle.

In the simplest case, one might consider that space has a varying density. We can not directly detect this density, but instead see the divergence of it: that is, we see when it goes from 1 to 0. So we get in maths-speak Out-vector = divergence of density, volume = moment of out-vector.

It is possible to consider for example, nebulus figures, like the gaussian distribution exp{-x^2-y^2-z^2}. Solids have discrete boundaries, where the density jumps by 'integers' from, eg 1 to 0. Things like the pentagram have density jumps from 2 to 1 (the core to the points of the star).

For polytopes, all such jumps occur in a small set of planes. Since the plane contains itself a polytope, we can equate the density on the plane with the transverse out-vector. That is, if the density drops from 2 to 1, the outvector has a strength of 1, and thus the surface-density is also 1. One can then for example, find the density of the pentagon (core = 2, points = 1), the surface is completely unit density.

Because the surface lies in a plane, which is a space of lesser dimension, we can continue the calculations. The density of the edges is 1 all the way to the vertices, and 0 outside the vertices. There is no change of density on the line where the two edges cross at the core, so the "surface" of the line is where on the line, it changes from 1 to 0 (ie at the ends of the points).

Endo-analisys can not tell us that the out-vector is due to one or two or three elements. All it tells us is that eg, the density drops from "2.0" to "1.0". or "1.0" to "0.0".

In order to "create" a pentagon, we need to take a marker and draw points and lines and things. In other words, the decision that "2.0" is the result of one element, (as in the pentagram), or two elements, (eg in the hexagram), is a concious decision which we need to make. Even in the hexagram, we could consider the thing as a single element, with two surfaces. And herein lies the paradox.

A hexagram could be seen as two overlapping triangles or a single thing with a disjoint surface. But depending on what we call it, means that it descends down to two wessians (or nulloids), or one. Where it is two overlapping triangles, there are two separate incidence diagrams, two volumes, and two anti-volumes (or wessians). When it is read as one, there is a single incidence diagram, and a d2 volume and a d2 anti-volume (wessian).

The Empty Set

Norman Johnson posited that it might be an 'empty set'. One can see that there can be several different things, each having its own wessian. This is different to the way the empty set happen. The same empty set exists for all families of sets. Moreover, it is possible for wessians to have different densities, because the wessian acts as an anti-bulk, or dual of the bulk.