Circulation, the allspace vector j and clifford space

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Circulation, the allspace vector j and clifford space

Postby wendy » Wed Jun 17, 2020 10:50 am

The various outvectors bear a striking resemblence to the electromagnetic fields, that I decided to have a closer look.

If we take a line, we might suppose that the vertices represent opposing forces, as in <------>, This is the outvector of a line.

A chain of edges would simply move the end tensions further apart, as in <---><--->. The middle vertex now has no tension on it, since the two cancel out each other. In two dimensions, we suppose that the line has also a radial component, ie <---^---> and >---v---<. To make a polygon, we use a cycle of the first vector, which would put the radial component pointing outwards. Then we say that \( \vec O = \nabla d\), where O is the out-vector, and d the density of the space.

The surface consists of the <---><---> arrangement, in perfect tension. The art of endoanalysis is to analyse such density patterns, where a face is a step of density, and the density of the face is the difference of the spaces on either side of it. It is possible to construct a real-number density of polytopes in this way.

Circulation on a polygon supposes that the edges carry a current, that is, say <---^----. A polygon made of these edges in an anticlockwise arrangement, would have no excess current, and the circulation would point outwards in the hedrix. In the chorix, we suppose a secondary vector points upwards, by the right-hand rule. A polytope made of anticlockwise polygons would have no surface current, and would have a common surface density, representing the polygon outvectors in equal tension.

The process repeats evermore. We also note that every surtope carries a parity that is transferred to its arroundings. That is, for a polygon in 4d, there is an implied ortho-polygon (the opposite in the duo-tegum, for example), that is also directed.

Electric fields

The striking example here is that where the electric field is radial, the magnetic field is circuitous. People have been jiggling various forms of maxwell's equations into something different, and here we are to do the same. We write charge as \( c\rho + j \vec J\), c is the speed of light, and we have \( \epsilon=1/zc,\ \mu = z/c \).

We suppose \(c\vec D + j \vec H\) is a radiant flux from single and moving charges. The corresponding field is \( \vec E + jc\vec B\). The constuant equation is then \( (c \vec D + j \vec H) = z (\vec E + j\vec B)\). This is in essence electric + j magnetic.

What this means, is that the radial force is a surface, seen by motion as a simple vector, and it is somehow converted into a vector, where the bivector is converted also,

Clifford Arithmetic

Cliffrord vectors are tensors of various dimensions, written as a product of vectors. aa = bb = cc = .. 1, and ab=-ba etc. Ie anticommutive. The vector represented by all-space is simply a product of all the letters to the dimension (eg 3d -> j = abc, 4d j = abcd, etc). The effect of multiplying a vector by j is to create the orthogonal vector. abc,ac = = aab = b, for example, converts ac bivector into b vector.

j is supposed to be sqrt(-1), but \(j^n = (-1)^{n \mbox{ DIV } 2 \mbox { MOD }2} \). This is the second remainder. = + aba.cbc = - aab.bcc = -1. But abcd.abcd = +1, since 4 in binary is 100.
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