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Geometric Models Of Physical Terms?


tetrahedron

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Hi again folks. I have a math problem some of you with advanced graphic skills might be able to solve.

 

The electronic charges of Standard Model fermions range across -3/3 for electrons to -2/3 for upper tier antiquarks, to -1/3 for lower tier matter quarks, 0 (+ or -?) for neutrinos, to +1/3 for lower tier antimatter quarks, +2/3 for upper tier quarks, to +3/3 for positrons. 

 

One can capture ALL these relationships redundantly on a cube embedded within a reference plane, with a body diagonal of the cube (one that runs through the cube center and two diametrically opposed vertices) lying in the plane as well. This diagonal serves as a rotational axis. Two vertices along the body diagonal already lie within the plane. When one of the other remaining vertices at an angle, using the rotational axis, of arctan(sqrt27) (plus or minus any multiple of 30 degrees) from the reference plane, then all normals from the 8 cubic vertices have relative lengths of 0, 1, 2, 3, three above the plane (for + charges) and thee below. This covers both matter and antimatter fermion charges. The squares of the sines of half of these angles (every other) are multiples of 1/28, related to the fact that we chose arctan(sqrt27) originally. The denominator is always one greater than the number under the square root. The intermediate angles have cosines with these relations.

 

Anyway, cubes are just one of the Platonic solids. What I want to know is, if a similar figure is created using a body diagonal embedded in a reference plane using other Platonic solids, will other simple sequences of relative lengths be created using sines or cosines of vertices of these solids not in the reference plane. I want to try to figure out if other properties of fermions can be similarly modeled in this fashion. Thanks.

 

Jess Tauber

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  • 2 weeks later...

Well, since I posted my query above I've tried several other Platonic solids, where a simplified perspective allows one to look down the axis created by the body diagonal and so you can represent the solid's vertices on a regular polygon. Equilateral triangles work, being the original format based on tetraheda withing the cube. But squares don't work. Pentagons don't work. Hexagons don't work. It may be that some other figures will, but I haven't tried them as yet. So perhaps the cubie-in-plane model really is a special case.

 

Jess Tauber

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