Experimental Notation for Multi-Torus Geometry

[ICN5D]

Experimental Notation for Multi-Torus Geometry

After researching and playing around with various means of creating these shapes, I think I came up with an approach to define these wild possibilities. My idea is based on how the polygonal arrays of points, circles, spheres, toruses, etc. have relation to the roots of unity. I decided to encode the real and imaginary parts of these solutions into a syntax we can use to represent these arrays. The notation even extends into defining polyhedral arrays, as well. But, a simpler, reduced form can also be used, like the one you guys have come up with a while ago, i.e. ; (([II]-3)(II)) .

What I propose is just another way of looking at these cuts, and trying to emulate an abstraction (roots of unity) in a symbolic form, like what we have with toratope notation, and its relation to the equation.




Some Basic Rules:

[x] - the Real part, single point on x-axis

[(x)] - the Real part, a pair of 2 points reflected on the x-axis

[±x,(y)] - the 2D complex part, a conjugate pair of 2 points translated on x-axis by some +,- value, reflected on y-axis

[-x,-y,(z)] - the 3D complex part, i.e. a conj pair of 2 points translated on x and y, reflected on z-axis

[x][x+(y)]...[n] - n-array of points, even n-arrays have 2 real + (n-2) conjugate pairs ; odd arrays have 1 real + (n-1) conjugate pairs

([x][x+(y)]...[n]z) - smoothly connecting n-array of points to n-edge frame, with poles on the z-axis




Polygon Symmetric Arrays
-------------------------
[(x)] - digon array of 2 points
[x][-x,(y)] - triangle array of 3 points
[(x)][(y)] - square (rhombus) array of 4 points
[x][x,(y)][-x,(y)] - pentagonal array of 5 points
[(x)][x,(y)][-x,(y)] - hexagonal array of 6 points
[x][x,(y)][-x,(y)][-x,(y)] - heptagonal array of 7 points
[(x)][(y)][x,(y)][-x,(y)] - octagonal array of 8 points
etc...

-----------------------------------

EDIT:

Okay, after some more thought, I realized we need more information in the notation. We need to be able to distinguish between:

1. An array of 3 points
2. An enclosed tri-edge frame in 3D
3. A circle (or whatever) embedded within the tri-edge frame, to make 3-prong multi-torus

Here's my revised list of polyhedral arrays and 3-prong shapes, with new notation:

Polyhedron Symmetric Arrays
----------------------------
[x][y][-x,-y,(z)] - tetrahedral array of 4 points
([x][y][-x,-y,(z)]w) - 4-edge frame in R^4
((xz)([x][y][-x,-y,(z)]w)) - tet symmetric 4-p multi-tiger
[x,(y)][-x,(z)] - alternate tet array
(xz)([x,(y)][-x,(z)]w)) - alternate tet symmetric 4-p multi-tiger
[(x)][(y)][(z)] - octahedral array of 6 points
([(x)][(y)][(z)]w) - 6-edge frame in R^4
((xz)([(x)][(y)][(z)]w)) - - octahedral symmetric 6-p multi-tiger
[±x,±y,(z)] - cubic array of 8 points
([±x,±y,(z)]w) - 8-edge frame in R^4
((xy)([±x,±y,(z)]w)) - cube symmetric 8-p multi-tiger
etc...

3-Prong Symmetric Shapes
-------------------------
(xy) - circle
[x][-x,(y)] : triangle array of 3 points
[x][-x,(y)]z : triangle array of 3 vertical lines
([x][-x,(y)]z) : enclosed tri-edge frame, 1D surface in R^3 , () acts as 'bipyramid spheration' here
([x][-x,(y)]i) : slice of tri-edge, also defines trigon array of 3 points
(([x][-x,(y)]i)y) : triangle array of 3 circles , circle embedded within array of 3 points
(([x][-x,(y)]z)y) : 3-prong multi-torus , circle xy embedded within full tri-edge frame
(([x][-x,(y)]z)yw) : 3-prong multi-spheritorus
((([x][-x,(y)]z)y)w) : 3-prong multi-ditorus
(([x][-x,(y)](zw))y) : torus of the 3-prong multi-torus, by rot on plane zw, not mantis
((xy)([z][-z,(x)]w)) : 3-prong multi-tiger : Mantis , torus embedded in tri-edge frame

I derived mantis by taking a torus ((xy)z) and embedding it within a tri-edge frame ([z][-z,(x)]w) , by replacing z . The usage of the x,y,z terms is a necessary part of the notation, since it can define the arrays with some accuracy.

((xy)([z][-z,(x)]w)) - full mantis
((xy)([z][-z,(x)]i)) - triangular array of 3 toruses, one centered on z-axis in +z , plus a reflected pair in the xz 3rd and 4th quadrants

[ICN5D]

Of course, since this is still in the experimental phase, there's bound to be changes. I think it might be too cluttered, too many symbols that don't need to be used. So, here's a slightly more streamlined version, that reduces the symbols for the polytope arrays. For the most part, the expression that used to be [x][-x,(y)] is now [x,-x+(y)] .

[x] - the Real part, single point on x-axis

[(x)] - the Real part, a pair of 2 points reflected on the x-axis

[±x+(y)] - the 2D complex part, a conjugate pair of 2 points translated on x-axis by some +,- value, reflected on y-axis

[-x-y+(z)] - the 3D complex part, i.e. a conj pair of 2 points translated on x and y, reflected on z-axis

[x,x+(y),...,n] - n-array of points, even n-arrays have 2 real + (n-2) conjugate pairs ; odd arrays have 1 real + (n-1) conjugate pairs

([x,x+(y),...,n]z) - smoothly connecting n-array of points to n-edge frame, with poles on the z-axis



Polygon Symmetric Arrays
-------------------------
[(x)] - digon array of 2 points
[x,-x+(y)] - triangle array of 3 points
[(x),(y)] - square (rhombus) array of 4 points
[x,x+(y),-x+(y)] - pentagonal array of 5 points
[(x),x+(y),-x+(y)] - hexagonal array of 6 points
[x,x+(y),-x+(y),-x+(y)] - heptagonal array of 7 points **
[(x),(y),x+(y),-x+(y)] - octagonal array of 8 points
etc...

** May simplify this to [x,x+(y),2-x+(y)] , where 2-x+(y) represents the two conjugate pairs (4 total) in quadrants 2 and 3. Why repeat the expression when we can just use a number, right?


Polyhedron Symmetric Arrays
----------------------------
[x,y,-x-y+(z)] - tetrahedral array of 4 points
([x,y,-x-y+(z)]w) - 4-edge frame in R^4
((xz)([x,y,-x-y+(z)]w)) - tet symmetric 4-p multi-tiger
[x+(y),-x+(z)] - alternate tet array
(xz)([x+(y),-x+(z)]w)) - alternate tet symmetric 4-p multi-tiger
[(x),(y),(z)] - octahedral array of 6 points
([(x),(y),(z)]w) - 6-edge frame in R^4
((xz)([(x),(y),(z)]w)) - - octahedral symmetric 6-p multi-tiger
[±x±y+(z)] - cubic array of 8 points **
([±x±y+(z)]w) - 8-edge frame in R^4
((xy)([±x±y+(z)]w)) - cube symmetric 8-p multi-tiger
etc...

** using the ±x±y+(z) expression to define a square array of pairs, without using tons of symbols.

3-Prong Symmetric Shapes
-------------------------
(xy) - circle
[x,-x+(y)] : triangle array of 3 points
[x,-x+(y)]z : triangle array of 3 vertical lines
([x,-x+(y)]z) : enclosed 3-edge frame, 1D surface in R^3 , () acts as 'spheration' here
([x,-x+(y)]i) : slice of 3-edge, also defines trigon array of 3 points, most common form we will see
(([x,-x+(y)]i)y) : triangle array of 3 circles , circle embedded within array of 3 points
(([x,-x+(y)]z)y) : 3-prong multi-torus , circle xy embedded within full 3-edge frame
(([x,-x+(y)]z)yw) : 3-prong multi-spheritorus
((([x,-x+(y)]z)y)w) : 3-prong multi-ditorus
(([x,-x+(y)](zw))y) : torus of the 3-prong multi-torus, by rot on plane zw, not mantis
((xy)([z,-z+(x)]w)) : 3-prong multi-tiger : Mantis , torus embedded in 3-edge frame


Next, I want to illustrate how the slice notation works along with how the x,-x+(y) symbols work to define locations of points, circles, spheres, toruses, etc.