## The Roots of Unity Generalized to the Multi-torus in R^n

Discussion of shapes with curves and holes in various dimensions.

### The Roots of Unity Generalized to the Multi-torus in R^n

I have found a mathematical pattern, and formed an abstraction around it. We can now enter the world of multi-toruses in R^3, and it extends to multi-tigers. I don't know exactly what I've discovered. It may just be a clever approach, or a whole new algebraic theory of toratopes. I'm inclined to assume the former, to be on the safe side of such claims. Also, I'm not telling you this is how it works. My presentation here is what I think I understand about what I found. I am completely open to any and all corrections and suggestions.

What I've really actually found is the missing link that connects toratopes to complex numbers. I learned that ALL toratopes (esp. multi) are complex numbers generalized over the reals. The polygon symmetric multi-torus in R^3 is in fact the roots of unity generalized to R^2, then using a 3rd variable to form a smooth multi-torus. The 3D equation contains an exact solution on a 2D plane, of multiple quadratic equations of a circle in a perfect polygon array.

This new abstraction is able to derive a single un-factored equation for the solution as an array of circles, spheres, toruses, etc. I did work on this concept about 4 years ago, and I was on to something. But, not realizing how close I was, I stopped playing with equations, figuring that I hit a dead end. I was wrong. The way this abstraction derives equations is far better than any of my past crude attempts.

The second breakthrough was finally learning how to add in the 3rd variable to enclose the n-gon array of circles to form a multi-torus. It's a fairly simple process. First step is to form the n-gon array equation, second step is to implement the 3rd variable. Boom, instant multi-torus. The process also derives a torus equation properly, based on its solution ((I)I), a digon array of 2 circles.

But, first, let's talk about this un-factored equation for n-gon arrays of circles, and its connection to the roots of unity and complex polynomials.

Exhibit A:

The function Re[(x+iy)^n] : expand a complex polynomial (x+iy)^n , and remove the imaginary parts, leaving only the real parts behind:

x^2 -y^2
x^3 -3xy^2
x^4 -6x^2y^2 +y^4
x^5 -10x^3y^2 +5xy^4
x^6 -15x^4y^2 +15x^2y^4 -y^6
x^7 -21x^5y^2 +35x^3y^4 -7xy^6
x^8 -28x^6y^2 +70x^4y^4 -28x^2y^6 +y^8
x^9 -36x^7y^2 +126x^5y^4 -84x^3y^6 +9xy^8
x^10 -45x^8y^2 +210x^6y^4 -210x^4y^6 +45x^2y^8 -y^10

These equations are very important, and very related to the new abstraction. It's more to do with the signs, exponents, and groups of the x,y terms. Pay close attention to those. The coefficients, not so much.

Exhibit B:

The cartesian equation of a circle, of radius 'b' :

x^2 + y^2 = b^2

Converting to general circle in polar coordinates :

x = (r*cos(θ-t)-a)
y = (r*sin(θ-t))

(r*cos(θ-t)-a)^2 + (r*sin(θ-t))^2 = b^2

expands, simplifies to:

r^2+a^2-b^2 -2ar*cos(t-θ)

a = distance from origin

Reducing the equation further using substitution:

X(r) = r^2+a^2-b^2
Y(r) = ar

gives,

X - 2Y*cos(t-θ)

This provides us the most reduced form of the general circle in polar coordinates. It is independent of variables, radius parameter, and translate distance. The only important thing is angle. It takes the form of a trigonometric binomial, with a curious relation to our complex polynomial (x+iy)^n . In this new abstraction, we are to treat this reduced polar equation of a circle as a fundamental 'Unit of Solution' .

I also believe the sphere of multi-spheritorus, and torus of multi-ditorus may be expressed in binomial form. That's future research. The torus unit of solution for a polygonal multi-tiger is a trinomial, containing an angle parameter in two separate trig terms, at two different periodic rates. Polygon multi-tigers are double periodic! More on that later.

Exhibit C:

The Generating Function for Polygon Arrays of Circles:

W|A:

product_(i=1)^n (X -2Ycos(2iπ/n-θ))

Latex:

\prod_{i=1}^{n} \left(X -2Y\cdot \cos\left(\frac{2i\pi}{n} -\theta\right)\right)

n = n-gon array

This product function will generate a polynomial P(X,Y) for a polygon array of circles in reduced polar coordinates. The solutions of the polynomials are the trig binomials with exact 2iπ/n angle values, which further define rotated copies of a circle at a fixed distance around the origin.

Iterations of this function produce the following:

Degree-2

X^2 -2Y^2 -2Y^2cos(2θ)

2 circle solutions,
(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/2 - θ))

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

Degree-3

X^3 -3XY^2 -2Y^3cos(3θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/3 - θ))
(X - 2Y*cos(4π/3 - θ))

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

Degree-4

X^4 -4X^2Y^2 +2Y^4 -2Y^4cos(4θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/4 - θ))
(X - 2Y*cos(4π/4 - θ))
(X - 2Y*cos(6π/4 - θ))

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

Degree-5

X^5 -5X^3Y^2 +5XY^4 -2Y^5cos(5θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/5 - θ))
(X - 2Y*cos(4π/5 - θ))
(X - 2Y*cos(6π/5 - θ))
(X - 2Y*cos(8π/5 - θ))

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

Degree-6

X^6 -6X^4Y^2 +9X^2Y^4 -2Y^6 -2Y^6cos(6θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/6 - θ))
(X - 2Y*cos(4π/6 - θ))
(X - 2Y*cos(6π/6 - θ))
(X - 2Y*cos(8π/6 - θ))
(X - 2Y*cos(10π/6 - θ))

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

Degree-7

X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6 -2Y^7cos(7θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/7 - θ))
(X - 2Y*cos(4π/7 - θ))
(X - 2Y*cos(6π/7 - θ))
(X - 2Y*cos(8π/7 - θ))
(X - 2Y*cos(10π/7 - θ))
(X - 2Y*cos(12π/7 - θ))

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

Degree-8

X^8 -8X^6Y^2 +20X^4Y^4 -16X^2Y^6 +2Y^8 -2Y^8*cos(8θ)

(X - 2Y*cos(0 - θ))
(X - 2Y*cos(2π/8 - θ))
(X - 2Y*cos(4π/8 - θ))
(X - 2Y*cos(6π/8 - θ))
(X - 2Y*cos(8π/8 - θ))
(X - 2Y*cos(10π/8 - θ))
(X - 2Y*cos(12π/8 - θ))
(X - 2Y*cos(14π/8 - θ))

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

Collected together,

Code: Select all
X^2 -2Y^2                              -2Y^2cos(2θ)X^3 -3XY^2                             -2Y^3cos(3θ)X^4 -4X^2Y^2 +2Y^4                     -2Y^4cos(4θ)X^5 -5X^3Y^2 +5XY^4                    -2Y^5cos(5θ)X^6 -6X^4Y^2 +9X^2Y^4  -2Y^6           -2Y^6cos(6θ)X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6          -2Y^7cos(7θ)X^8 -8X^6Y^2 +20X^4Y^4 -16X^2Y^6 +2Y^8 -2Y^8cos(8θ)

General Form:

Even Degree
P1^n -n*P2 +A*P3 -B*P4 +C*P5 -D*P6 + ... ±2Y^n -2Y^n*cos(nθ)

Odd Degree
P1^n -n*P2 +A*P3 -B*P4 +C*P5 -D*P6 + ... ±n*XY^(n-1) -2Y^n*cos(nθ)

n - n-gon array
P1^n - 1st terms containing X
P2 - 2nd terms cont X^pY^q
A - coefficient of 3rd terms
P3 - 3rd terms cont X^pY^q
B - coeff of 4th terms
P4 - 4th terms cont X^pY^q
2Y^n*cos(nθ) - final term cont Y^n , θ

More Generally,

(Circle Terms) -2(Periodic Terms) = 0

The polynomials can be broken down into 2 distinct groups of terms: those containing all the X,Y terms, and the final Y^n*cos(nθ) term. The group of X,Y terms mostly define the product of circles (Circle Terms), while the last Y^n*cos(nθ) term mostly defines the periodic symmetry of their locations (centers).

This is more clear to us when considering Y = ar , which only contains the distance from origin 'a' , and a degree-1 sqrt(x^2+y^2) , which I believe only defines the circumradius of the polygon's vertices. Each n-gon array has a unique degree-n group of these terms, which are exactly the real parts of (x+iy)^n , or simplifying the expression (sqrt(x^2+y^2))^(n)*cos(n(arctan(y/x))) . I call these the 'Periodic Terms' .

Exhibit D:

Notice anything peculiar about these polynomials? If you don't see it yet, let's take the last -2Y^n*cos(nθ) periodic terms and convert them to cartesian, while leaving the rest of the X,Y circle terms alone:

Code: Select all
X^2 -2Y^2                              -2a^2(x^2 -y^2)X^3 -3XY^2                             -2a^3(x^3 -3xy^2)X^4 -4X^2Y^2 +2Y^4                     -2a^4(x^4 -6x^2y^2 +y^4)X^5 -5X^3Y^2 +5XY^4                    -2a^5(x^5 -10x^3y^2 +5xy^4)X^6 -6X^4Y^2 +9X^2Y^4  -2Y^6           -2a^6(x^6 -15x^4y^2 +15x^2y^4 -y^6)X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6          -2a^7(x^7 -21x^5y^2 +35x^3y^4 -7xy^6)X^8 -8X^6Y^2 +20X^4Y^4 -16X^2Y^6 +2Y^8 -2a^8(x^8 -28x^6y^2 +70x^4y^4 -28x^2y^6 +y^8)

Ignoring the coefficients (they come about for diff reasons), minding that X,Y are equations themselves (not x,y), and focusing on the signs, exponents, and groups of terms, we see that:

The overall structure of the equation is identical to a small sequence of its own terms

These polynomials posses a form of reflective symmetry in the algebraics that I've never seen before. It's all so obvious in retrospect:

- The trig binomial (X-2Y*cos(nθ))^n is the R^2 generalization of our complex polynomial (x+iy)^n .

- The multiplicative properties of i^n perform a rotation on the 2d complex plane.

- The trig term plays the role of an imaginary unit, as it rotates about the origin on the 2d real plane, the same exact way our roots of unity do, on the 2d complex plane.

What we've done, is broken down a circle equation into 2 parts, and recombined them in a way similar to complex exponentiation.

Not only are the real parts of (x+iy)^n within the equations, but they encode the overall equation structure itself.

Seriously, dude, what the hell type of voodoo magic is this. I would have never imagined such an abstraction could exist. This is what I accidentally discovered a few weeks ago, and developed a theory based on it. And, sure enough man, convert X,Y to their polar/cartesian forms, and the equations will produce perfect polygon arrays of circles up to as high an n-gon as you want. Just use that generating product function. This is why I call them the Fundamental Units of Solution, because they are a generalization of unique types of complex numbers.

Exhibit E:

The Roots of Unity Generalized to the Multi-Torus in R^3

Now that we've discussed some of the theory, let's dig into the maths and create some cool, new shapes. I've also been secretly using new names that now reflect the algebraics of multi-toruses. In place of n-prong, I've been using their exponential names, like the roots of unity. The 5-prong has degree-5 terms, quintic exponents, 5 exact solutions in a pentagonal array, 5-fold periodic symmetry, so it's all about the quintic. It is deservedly the Quintic Multi-torus!

3-prong - Cubic Multi-torus
4-prong - Quartic
5-prong - Quintic
6-prong - Hexic
7-prong - Heptic
8-prong - Octic
9-prong - Nonic
10-prong - Decic
etc...

Polygonal Multi-tiger
Polyhedral Multi-Tiger
Polychoral Multi-tiger
Polyteral
etc...

You don't have to adopt these, just listing them for less confusion.

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The gallery of pretty rotating multi-toruses:

Polygonal Multi-toruses in R^3

Also, here's a URL to copy and paste that will give you an interactive calcplot3d script, with all of these shapes in it:

Code: Select all
https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/?type=implicit;equation=(x^2+y^2+z^2)^3-3(x^2+y^2+z^2)^2-16(x^3-3xy^2+2z^2)+9(x^2+y^2+3);cubes=30;visible=true;fixdomain=true;xmin=-3;xmax=3;ymin=-3;ymax=3;zmin=-3;zmax=3;alpha=-1;format=normal;constcol=rgb(255,0,0)&type=implicit;equation=(x^2+y^2+z^2)^4-4(x^2+y^2+z^2)^3-30(x^2+y^2+z^2)^2-256(x^2+y^2+8z^2-16)-162(x^4+y^4-6x^2y^2);cubes=30;visible=false;fixdomain=true;xmin=-6;xmax=6;ymin=-6;ymax=6;zmin=-6;zmax=6;alpha=-1;format=normal;constcol=rgb(255,0,0)&type=implicit;equation=(x^2+y^2+z^2)^5-5(x^2+y^2+z^2)^4-70(x^2+y^2+z^2)^3-1050(x^2+y^2+z^2)^2-(3^3*5^4)(x^2+y^2+11z^2-3^2*5)-2^11(x^5-10x^3y^2+5xy^4);cubes=30;visible=false;fixdomain=true;xmin=-6;xmax=6;ymin=-6;ymax=6;zmin=-6;zmax=6;alpha=-1;format=normal;constcol=rgb(255,0,0)&type=implicit;equation=(x^2+y^2+z^2)^6-6(x^2+y^2+z^2)^5-81(x^2+y^2+z^2)^4-1172(x^2+y^2+z^2)^3-18225(x^2+y^2+z^2)^2-303750(x^2+y^2+9z^2)-2^13*(x^6-15x^4y^2+15x^2y^4-y^6)+15^6;cubes=30;visible=false;fixdomain=true;xmin=-6;xmax=6;ymin=-6;ymax=6;zmin=-6;zmax=6;alpha=-1;format=normal;constcol=rgb(255,0,0)&type=implicit;equation=(x^2+y^2+z^2)^7-7(x^2+y^2+z^2)^6-154(x^2+y^2+z^2)^5-3535(x^2+y^2+z^2)^4-84840(x^2+y^2+z^2)^3-2128896(x^2+y^2+z^2)^2-55738368(x^2+y^2+12z^2)-2*5^7(x^7-21x^5y^2+35x^3y^4-7xy^6)+4586471424;cubes=35;visible=false;fixdomain=true;xmin=-6;xmax=6;ymin=-6;ymax=6;zmin=-6;zmax=6;alpha=-1;format=normal;constcol=rgb(255,0,0)&type=window;hsrmode=3;nomidpts=true;anaglyph=-1;center=1.9302521750801496,8.676238297103318,-4.582250053480225,1;focus=0,0,0,1;up=-0.10372458602714334,-0.44634750245082516,-0.8888279458418308,1;transparent=false;alpha=140;twoviews=false;unlinkviews=false;axisextension=0.7;xaxislabel=x;yaxislabel=y;zaxislabel=z;edgeson=true;faceson=true;showbox=false;showaxes=false;showticks=true;perspective=true;centerxpercent=0.5159290856670771;centerypercent=0.5478214665249735;rotationsteps=30;autospin=true;xygrid=false;yzgrid=false;xzgrid=false;gridsonbox=true;gridplanes=false;gridcolor=rgb(128,128,128);xmin=-4;xmax=4;ymin=-4;ymax=4;zmin=-4;zmax=4;xscale=2;yscale=2;zscale=2;zcmin=-8;zcmax=8;zoom=0.539667;xscalefactor=1;yscalefactor=1;zscalefactor=1

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Deriving the Torus Polynomial

A reverse derivation of the 3D torus equation based on the polynomial of its solution of a digon array of circles, ((I)I) , and the spherical/polar conversion method, with adjustable z^2 term coeff.

Properties:
- Degree-4
- 3 variables
- 2 exact solutions on plane xy
- digonal symmetry
- genus-1 surface

Degree-2 Polynomial for a digon array of 2 circles,

P(X,Y) = X^2 -2Y^2 -2Y^2cos(2θ)

Convert to polar,

X = r^2+a^2-b^2 ; Y = ar

(r^2+a^2-b^2)^2 -2(ar)^2 -2(ar)^2cos(2θ)

^^^we can plot all of these in calcplot3d

Set a=2 , b=1 to get a reduced form of the polar equation,

r^4 - 8r^2*cos(2θ) -2r^2 +9

Exhibit F:

By this point, I have the n-gon array equation plotted with all z variables in place, and I'm ready to find (look for) the coefficient 'c' for the Lone z^2 Term. I'm using the parameter 'c' as an adjustable slider value in calcplot3d, my experimental geometry laboratory of choice.

Now, here's where we hack the system to make what we want. We are building shapes from the ground up. We are forming a single un-factored equation for a product of circles, and joining them in R^3. We cannot do a full spherical coordinate conversion to cartesian, because of the issues of arctan(z/sqrt(x^2+y^2+z^2)) being undefined when x=0 or y=0 , which is the polar regions on the z-axis. For the Circle Terms we use a spherical conversion for r, but the Periodic Terms get a polar conversion for r,θ. This is the first step in forming the polar regions on z.

The Partial Spherical/Polar Conversion

Circle Terms, containing only X, Y :

r = sqrt(x^2+y^2+z^2)

Periodic Terms, cont Y, θ :

r = sqrt(x^2+y^2)
θ = arctan(y/x)

r^4 - 8r^2*cos(2θ) -2r^2 +9

(sqrt(x^2+y^2+z^2))^4 - 8(sqrt(x^2+y^2))^2*cos(2(arctan(y/x))) -2(sqrt(x^2+y^2+z^2))^2 +9

(x^2+y^2+z^2)^2 - 8(x^2-y^2) -2(x^2+y^2+z^2) +9

The Coefficient for the Lone z^2 Term

The second, and also very important step in implementing the 3rd variable, is tracking down the lone z^2 term, and scaling its coeff higher than 1. It will always be located in the r^2 term of the 'super reduced form', that gets converted to cartesian. This is the most influential coeff for any of the z terms. What it does is fatten up the polar regions that smoothly join the infinite cylinders. Judging how the coeff of z^2 in the torus equation is exactly related to the -4a^2 coeff, I'm inclined to assume that's the best format for these equations.

But, the z^2 term can commute within the Periodic Terms, too. You could just as easily find a coeff for z^2 that makes a nice fit in there. I found that there is a nice integer value for each surface, within the range of 1 < c < 20 , up to the heptic (7-prong) multi-torus. It seems to be very related to the translate distance 'a' , but I haven't found a precise pattern. More future research on the list.

(x^2+y^2+z^2)^2 - 8(x^2-y^2) -2(x^2+y^2+c*z^2) +9

when c=5, we get a perfect torus!

(x^2+y^2+z^2)^2 - 8(x^2-y^2) -2(x^2+y^2+5z^2) +9

We could also simplify those two x^2, y^2 terms, but this is unique only to the torus equation, and strays from the concept I'm trying to sell. All other multi-toruses have higher degree periodic terms, with only one set of x^2, y^2, z^2 terms, so we can ignore this small particular.

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Deriving the Cubic Multi-torus Polynomial

Properties:
- Degree-6
- 3 variables
- 3 exact solutions on plane xy
- triangle symmetry
- genus-2 surface

Degree-3 Polynomial for a triangle array of 3 circles,

P(X,Y) = X^3 - 3XY^2 - 2Y^3cos(3θ)

Convert to polar,

(r^2+a^2-b^2)^3 -3(r^2+a^2-b^2)(ar)^2 -2(ar)^3cos(3θ)

Convert to spherical for Circle Terms, polar for Periodic Terms
r = (sqrt(x^2+y^2+z^2)) ; (sqrt(x^2+y^2))
θ = (arctan(y/x))

(x^2+y^2+z^2+a^2-b^2)^3 - 3a^2(x^2+y^2+z^2+a^2-b^2)(x^2+y^2+z^2) - 2a^3(x^3-3xy^2)

Add adj coeff +cz^2 in deg-3 Periodic Terms to form multi-torus in R^3,

(x^2+y^2+z^2+a^2-b^2)^3 - 3a^2(x^2+y^2+z^2+a^2-b^2)(x^2+y^2+z^2) - 2a^3(x^3-3xy^2+cz^2)

a=3 ; b=1 ; c=4 for best surface,

(x^2+y^2+z^2+a^2-1)^3 -3a^2(x^2+y^2+z^2+a^2-1)(x^2+y^2+z^2) -2a^3(x^3-3xy^2+4z^2) = 0

We can also use hard-set parameters in polar form, and get a super reduced version of the equation:

r^6 -3r^4 -16r^3*cos(3θ) -9r^2 +27 = 0

Then, convert and add a*z^2, find that a=2 for best surface. The actual best coeff for z^2 is -31, which won't tuck in nicely into any brackets, other than the Periodic Terms, and raising it to -32. Just trivial aesthetics.

(x^2+y^2+z^2)^3 -3(x^2+y^2+z^2)^2 -16(x^3-3xy^2+2z^2) +9(x^2+y^2+3) = 0

\begin{align*}
&(x^2+y^2+z^2)^3 -3(x^2+y^2+z^2)^2 -16(x^3-3xy^2+2z^2) +9(x^2+y^2+3) \\
\end{align*}

Expanded cubic multitorus polynomial

degree-6 terms
x^6 +y^6 +z^6

+3x^4y^2 +3x^2y^4 +3x^4z^2 +3x^2z^4 +3y^2z^4 +3y^4z^2 +6x^2y^2z^2

degree-4
-3x^4 -3y^4 -3z^4

-6x^2y^2 -6x^2z^2 -6y^2z^2

degree-3
-16x^3 +48xy^2

degree-2
+9x^2 +9y^2 -32z^2

degree-0
+27

---
It is a fairly nice and homogeneous equation, with the z variable intermixed with x and y. It is truly the extended cube-root analogue of the common 2-torus. Have a look at the main difference with the -32z^2 term, and the other coeffs on +9x^2, +9y^2. This is where we try to overcome the issue of a full spherical conversion.

General Form of The Cubic Multi-torus:

(x^2+y^2+z^2)^3 +(a^2-b^2)^3 -3b^2(x^2+y^2+z^2)^2 -2a^3(x^3-3xy^2+c*z^2) +3b^2(a^2-b^2)(x^2+y^2) = 0

c = most influential coefficient for forming the polar regions

For best surface, use: a=2 ; b=1 ; c=5/2 // a=3 ; b=1 ; c=4

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Deriving the Quartic Multi-torus Polynomial

Properties:
- Degree-8
- 3 variables
- 4 Solutions on xy plane
- rhombic symmetry
- genus-3 surface

Degree-4 Polynomial for a rhombic array of 4 circles,

P(X,Y) = X^4 -4X^2Y^2 +2Y^4 -2Y^4cos(4θ)

Convert to polar,

(r^2+a^2-b^2)^4 -4(r^2+a^2-b^2)^2(ar)^2 +2(ar)^4 -2(ar)^4cos(4θ) = 0

Use hard-set parameters a=3 , b=1, get super reduced equation:

r^8 -4r^6 -30r^4 -162r^4cos(4θ) -256r^2 +4096

Convert to spherical for Circle Terms, polar for Periodic Terms, use +az^2 for adj coeff,
r = (sqrt(x^2+y^2+z^2)) ; (sqrt(x^2+y^2))
θ = (arctan(y/x))

(x^2+y^2+z^2)^4 -4(x^2+y^2+z^2)^3 -30(x^2+y^2+z^2)^2 -256(x^2+y^2+az^2-16) -162(x^4-6x^2y^2+y^4)

Best value for smooth surface: a=8,

(x^2+y^2+z^2)^4 -4(x^2+y^2+z^2)^3 -30(x^2+y^2+z^2)^2 -256(x^2+y^2+8z^2-16) -162(x^4-6x^2y^2+y^4) = 0

\begin{align*}
&(x^2+y^2+z^2)^4 -4(x^2+y^2+z^2)^3 -30(x^2+y^2+z^2)^2 -256(x^2+y^2+8z^2-16) -162(x^4+y^4-6x^2y^2) \\
\end{align*}

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Deriving the Quintic Multi-torus Polynomial

Properties:
- Degree-10
- 3 variables
- 5 Solutions on xy plane
- pentagon symmetry
- genus-4 surface

Degree-5 Polynomial for a pentagon array of 5 circles,

P(X,Y) = X^5 -5X^3Y^2 +5XY^4 -2Y^5cos(5θ)

Convert to polar,

(r^2+a^2-b^2)^5 -5(r^2+a^2-b^2)^3(ar)^2 +5(r^2+a^2-b^2)(ar)^4 -2(ar)^5cos(5θ)

Use hard-set parameters a=4 , b=1 , reduce

r^10 -5r^8 -70r^6 -2^11r^5*cos(5θ) -1050r^4 -(3^3*5^4)r^2 +15^5

Convert to spherical for Circle Terms, polar for Periodic Terms. Add in special adj coeff 'a' for z^2 in (x^2+y^2-45) :

r = (sqrt(x^2+y^2+z^2)) ; (sqrt(x^2+y^2))
θ = (arctan(y/x))

(x^2+y^2+z^2)^5 -5(x^2+y^2+z^2)^4 -70(x^2+y^2+z^2)^3 -1050(x^2+y^2+z^2)^2 -(3^3*5^4)(x^2+y^2+a*z^2-45) -2^11(x^5 -10x^3y^2 +5xy^4)

Best value a=11,

(x^2+y^2+z^2)^5 -5(x^2+y^2+z^2)^4 -70(x^2+y^2+z^2)^3 -1050(x^2+y^2+z^2)^2 -(3^3*5^4)(x^2+y^2+11z^2-45) -2^11(x^5-10x^3y^2+5xy^4)

\begin{align*}
&\big(x^2+y^2+z^2\big)^5 -5\big(x^2+y^2+z^2\big)^4 -70\big(x^2+y^2+z^2\big)^3 -1050\big(x^2+y^2+z^2\big)^2 -3^35^4\big(x^2+y^2+11z^2-45\big)-2^{11}\big(x^5-10x^3y^2+5xy^4\big) \\
\end{align*}

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Deriving the Hexic Multi-torus Polynomial

Properties:
- Degree-12
- 3 variables
- 6 Solutions on xy plane
- hexagon symmetry
- genus-5 surface

Degree-6 Polynomial for a hexagon array of 6 circles,

P(X,Y) = X^6 -6X^4Y^2 +9X^2Y^4 -2Y^6 -2Y^6cos(6θ)

Convert to polar,

(r^2+a^2-b^2)^6 -6(r^2+a^2-b^2)^4(ar)^2 +9(r^2+a^2-b^2)^2(ar)^4 -2(ar)^6 -2(ar)^6cos(6θ) = 0

Use hard-set parameters a=4 , b=1 , reduce

r^12 -6r^10 -81r^8 -1172r^6 -18225r^4 -303750r^2 -2^13*r^6*cos(6θ) + 15^6

Convert to spherical for Circle Terms, polar for Periodic Terms, Add adj coeff +az^2 to form multi-torus in R^3,

r = (sqrt(x^2+y^2+z^2)) ; (sqrt(x^2+y^2))
θ = (arctan(y/x))

(x^2+y^2+z^2)^6 -6(x^2+y^2+z^2)^5 -81(x^2+y^2+z^2)^4 -1172(x^2+y^2+z^2)^3 -18225(x^2+y^2+z^2)^2 -303750(x^2+y^2+az^2)-2^13*(x^6-15x^4y^2+15x^2y^4-y^6) + 15^6

best surface, a=9

(x^2+y^2+z^2)^6 -6(x^2+y^2+z^2)^5 -81(x^2+y^2+z^2)^4 -1172(x^2+y^2+z^2)^3 -18225(x^2+y^2+z^2)^2 -303750(x^2+y^2+9z^2)-2^13*(x^6-15x^4y^2+15x^2y^4-y^6) + 15^6

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Deriving the Heptic Multi-torus Polynomial

Properties:
- Degree-14
- 3 variables
- 7 Solutions on xy plane
- heptagon symmetry
- genus-6 surface

Degree-7 Polynomial for a heptagon array of 7 circles,

P(X,Y) = X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6 -2Y^7cos(7θ)

Convert to polar,

(r^2+a^2-b^2)^7 -7(r^2+a^2-b^2)^5(ar)^2 +14(r^2+a^2-b^2)^3(ar)^4 -7(r^2+a^2-b^2)(ar)^6 -2(ar)^7*cos(7θ) = 0

Use hard-set parameters a=5 , b=1 , reduce

r^14 -7r^12 -154r^10 -3535r^8 -84840r^6 -2128896r^4 -55738368r^2 -2*5^7r^7*cos(7θ) +4586471424

Convert to spherical for Circle Terms, polar for Periodic Terms, Add adj coeff +az^2 to form multi-torus in R^3,

r = (sqrt(x^2+y^2+z^2)) ; (sqrt(x^2+y^2))
θ = (arctan(y/x))

(x^2+y^2+z^2)^7 -7(x^2+y^2+z^2)^6 -154(x^2+y^2+z^2)^5 -3535(x^2+y^2+z^2)^4 -84840(x^2+y^2+z^2)^3 -2128896(x^2+y^2+z^2)^2 -55738368(x^2+y^2+a*z^2) -2*5^7(x^7-21x^5y^2+35x^3y^4-7xy^6) +4586471424

a=12 for best surface,

(x^2+y^2+z^2)^7 -7(x^2+y^2+z^2)^6 -154(x^2+y^2+z^2)^5 -3535(x^2+y^2+z^2)^4 -84840(x^2+y^2+z^2)^3 -2128896(x^2+y^2+z^2)^2 -55738368(x^2+y^2+12z^2) -2*5^7(x^7-21x^5y^2+35x^3y^4-7xy^6) +4586471424

Creates a heptagon symmetric multi-torus! This is as far as I've gone for now. My head is still buried in theory and abstraction, and I want to keep hitting the iron while it's hot. I was trying to figure out the multi-tigers, which I detail at the end.

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Reference Tables

and other such organized data

Known Polygonal Multi-torus Equations

Cubic Multi-torus: a=2 ; b=1

(x^2+y^2+z^2)^3 -3(x^2+y^2+z^2)^2 +9(x^2+y^2+3) -2^4(x^3-3xy^2+2z^2)

Quartic Multi-torus: a=3 ; b=1

(x^2+y^2+z^2)^4 -4(x^2+y^2+z^2)^3 -30(x^2+y^2+z^2)^2 -2^8(x^2+y^2+8z^2-16) -162(x^4+y^4-6x^2y^2)

Quintic Multi-torus: a=4 ; b=1

(x^2+y^2+z^2)^5 -5(x^2+y^2+z^2)^4 -70(x^2+y^2+z^2)^3 -1050(x^2+y^2+z^2)^2 -16875(x^2+y^2+11z^2) -2^11(x^5-10x^3y^2+5xy^4) +15^5

Hexic Multi-torus: a=4 ; b=1

(x^2+y^2+z^2)^6 -6(x^2+y^2+z^2)^5 -81(x^2+y^2+z^2)^4 -1172(x^2+y^2+z^2)^3 -18225(x^2+y^2+z^2)^2 -303750(x^2+y^2+9z^2)-2^13*(x^6-15x^4y^2+15x^2y^4-y^6) + 15^6

Heptic Multi-torus: a=5 ; b=1

(x^2+y^2+z^2)^7 -7(x^2+y^2+z^2)^6 -154(x^2+y^2+z^2)^5 -3535(x^2+y^2+z^2)^4 -84840(x^2+y^2+z^2)^3 -2128896(x^2+y^2+z^2)^2 -55738368(x^2+y^2+12z^2) -2*5^7(x^7-21x^5y^2+35x^3y^4-7xy^6) +4586471424

What's neat about these equations are the symmetry and simplistic nature of them. These are fine examples of what's called a harmonic polynomial: a finite sum of radical polynomials. Which is exactly what they are in super reduced form. What really makes or breaks the shape are the very large, ultra-precise coefficients. If they deviate by ±1 , you won't get the exact solution of precise circles on plane xy. They will be microscopically elliptical or off center on the visual side, but algebraically a disaster, if you try to factor the solution. And we know this, since we reverse derived the shape from its exact solution, coefficients and all.

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Minimum Polynomials for the Topological N-Prong Multitorus in R^3

These are the absolute bare minimum terms required to define a topological n-prong multi-torus in R^3, for those who don't care about exact solutions of perfect circles on the xy plane. These will plot perfect n-gon arrays of ellipses instead. However, beyond 5/6 prong it forms thin cage bars joined by hemispherical blobs. on the other hand, it does give us a way to experiment with hypothetical 3D structures based on double periodic n,m-fold symmetries.

(x^2+y^2+z^2)^3 -20(x^3-3xy^2) +10(x^2+y^2+4z^2+2)

(x^2+y^2+z^2)^4 -160(x^4+y^4-6x^2y^2) -2^8(x^2+y^2+8z^2-16)

(x^2+y^2+z^2)^5 -2*10^3(x^5-10x^3y^2+5xy^4) -2*10^4(x^2+y^2+10z^2-20)

(x^2+y^2+z^2)^6 -2^13(x^6-15x^4y^2+15x^2y^4-y^6) -3^11(x^2+y^2+20z^2-64)

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Another curious equation that uses an embedded term function,

(x^3-3xy^2+c*z^2-5^3)^2 +(y^3-3yx^2)^2 +20t(x^2+y^2+d*z^2) + 5a*(x^2+y^2)z^2 =b^6

using a=1 , b=4.65 , c=6 , d=3 , t=7.5 and simplifying:

(x^3-3xy^2+6z^2-125)^2 +(y^3-3yx^2)^2 + 5(x^2+y^2)z^2 +140(x^2+y^2+3z^2) = 10^4

we get a pretty neat, not so bad 3-prong with a flatter polar region. This is using a method I was hoping would work universally on any shape. It might be extendable.

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A Closer Look At The Periodic Symmetry Terms

N-fold terms found in the multi-torus polynomials. These have the form r^n*cos(nθ) for an n-gon/n-prong.

Polar -> Cartesian Conversion for n-gon

(sqrt(x^2+y^2))^(n)*cos(n(arctan(y/x)))

Polar Form:

r^2*cos(2θ)
r^3*cos(3θ)
r^4*cos(4θ)
r^5*cos(5θ)
r^6*cos(6θ)
r^7*cos(7θ)
r^8*cos(8θ)
r^9*cos(9θ)
r^10*cos(10θ)
etc.

Cartesian Form:

x^2 -y^2
x^3 -3xy^2
x^4 -6x^2y^2 +y^4
x^5 -10x^3y^2 +5xy^4
x^6 -15x^4y^2 +15x^2y^4 -y^6
x^7 -21x^5y^2 +35x^3y^4 -7xy^6
x^8 -28x^6y^2 +70x^4y^4 -28x^2y^6 +y^8
x^9 -36x^7y^2 +126x^5y^4 -84x^3y^6 +9xy^8
x^10 -45x^8y^2 +210x^6y^4 -210x^4y^6 +45x^2y^8 -y^10
x^11 -55x^9y^2 +330x^7y^4 -462x^5y^6 +165x^3y^8 -11xy^10
x^12 -66x^10y^2 +495x^8y^4 -924x^6y^6 +495x^4y^8 -66x^2y^10 +y^12
x^13 -78x^11y^2 +715x^9y^4 -1716x^7y^6 +1287x^5y^8 -286x^3y^10 +13xy^12
x^14 -91x^12y^2 +1001x^10y^4 -3003x^8y^6 +3003x^6y^8 -1001x^4y^10 +91x^2y^12 -y^14
x^15 -105x^13y^2 +1365x^11y^4 -5005x^9y^6 +6435x^7y^8 -3003x^5y^10 +455x^3y^12 -15xy^14

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Known and Theoretical Polynomials for N-gon Array of Circles

X^2 -2Y^2 -2Y^2*cos(2θ)
X^3 -3XY^2 -2Y^3*cos(3θ)
X^4 -4X^2Y^2 +2Y^4 -2Y^4*cos(4θ)
X^5 -5X^3Y^2 +5XY^4 -2Y^5*cos(5θ)
X^6 -6X^4Y^2 +9X^2Y^4 -2Y^6 -2Y^6*cos(6θ)
X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6 -2Y^7*cos(7θ)
X^8 -8X^6Y^2 +20X^4Y^4 -16X^2Y^6 +2Y^8 -2Y^8*cos(8θ)
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Wolfram Alpha cannot go beyond this point. I just don't feel like dropping \$160 right now for the 1 year license for mathematica, but that's what I need at this point. I have some serious calculations to do, man.

Theoretical N-gon Equations : A,B,C = unknown coefficients

X^9 -9X^7Y^2 +27X^5Y^4 -A*X^3Y^6 +9XY^8 -2Y^9*cos(9θ) ; A=26?
X^10 -10X^8Y^2 +35X^6Y^4 -A*X^4Y^6 +B*X^2Y^8 -2Y^10 -2Y^10*cos(10θ) ; A=37? , B=23?
X^11 -11X^9Y^2 +44X^7Y^4 -A*X^5Y^6 +B*X^3Y^8 -11XY^10 -2Y^11*cos(11θ) ; A=49? , B=38?
X^12 -12X^10Y^2 +54X^8Y^4 -A*X^6Y^6 +B*X^4Y^8 -C*X^2Y^10 +2Y^12 -2Y^12*cos(12θ) ; A=??? , B=??? , C=???

There is order to the chaos of these coefficients. We need more data points to see the overall pattern, though.

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Polar Equations of Polygon Arrays of Circles

Digonal Array of 2 Circles:

X^2 -2Y^2 -2Y^2cos(2θ)

(r^2+a^2-b^2)^2 -2(ar)^2 -2(ar)^2cos(2θ)

Triangle Array of 3 Circles:

X^3 -3XY^2 -2Y^3cos(3θ)

(r^2+a^2-b^2)^3 -3(r^2+a^2-b^2)(ar)^2 -2(ar)^3cos(3θ)

Square Array of 4 Circles:

X^4 -4X^2Y^2 +2Y^4 -2Y^4cos(4θ)

(r^2+a^2-b^2)^4 -4(r^2+a^2-b^2)^2(ar)^2 +2(ar)^4 -2(ar)^4cos(4θ)

Pentagonal Array of 5 Circles:

X^5 -5X^3Y^2 +5XY^4 -2Y^5cos(5θ)

(r^2+a^2-b^2)^5 -5(r^2+a^2-b^2)^3(ar)^2 +5(r^2+a^2-b^2)(ar)^4 -2(ar)^5cos(5θ)

Hexagonal Array of 6 Circles:

X^6 -6X^4Y^2 +9X^2Y^4 -2Y^6 -2Y^6cos(6θ)

(r^2+a^2-b^2)^6 -6(r^2+a^2-b^2)^4(ar)^2 +9(r^2+a^2-b^2)^2(ar)^4 -2(ar)^6 -2(ar)^6cos(6θ)

Heptagonal Array of 7 Circles:

X^7 -7X^5Y^2 +14X^3Y^4 -7XY^6 -2Y^7cos(7θ)

(r^2+a^2-b^2)^7 -7(r^2+a^2-b^2)^5(ar)^2 +14(r^2+a^2-b^2)^3(ar)^4 -7(r^2+a^2-b^2)(ar)^6 -2(ar)^7*cos(7θ)

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Interesting Idea in Higher Abstraction Model:

The Nested Complex Polynomial : ((x+iy)^m +ix)^n

n-fold of the m-fold symmetry : [m,n]

plots of the equation :

Re[((x+iy)^2 +ix)^3] = a

x^6 -15x^4y^2 -12x^4y -3x^4 +15x^2y^4 +12x^2y^3 +3x^2y^2 -y^6 = a

^^^gives promising looking results in the arrangement of the 3 pairs of 2 hyperbolae when a<0. This abstraction does seem to derive equations for nested periodicity. Or, it could be a freshman's dream of the multi-torus geometer. One test is to form a minimum polynomial in 2D, and try to refine the 7 coefficients. Maybe it will give 6 circles with trigon of the digon symmetry. But, it fails to deliver clean plots of hyperbolae in anything higher than [2,3] .

Reducing the equation shows something interesting, though. It's a 4-period times a 2-period, that mimics a 3-period of a 2-period.

(x^4+y^4-x^2(14y^2+12y+3))(x^2-y^2)

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

Tiger is 2-fold of the 2-fold periodic, where if (x+iy)^2 means to translate on x first then revolve at 1/2 circle, then this one defines a tiger symmetry:

((x+iy)^2 +ix)^2

x^4 +i(4x^3y +2x^3 -4xy^3 -2xy^2) -6x^2y^2 -4x^2y -x^2 +y^4

Re parts,

x^4 -6x^2y^2 -4x^2y -x^2 +y^4 = a

plots curious square array of 4 hyperbolae

Even if these don't properly define a multi-tiger, they may still be useful to experiment with double periodic multi-toruses in R^3 .

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Now, here's what you really want to know about. What I discovered in trying to comprehend multi-tigers with this new approach.

The Equations of Multi-Tigers

In Reduced Cylindrical Coordinates of the form P(X,Y,Z) = 0

Torus equation, minor diameter 'c' , major diameter 'b' parallel to plane xz:

(x^2+y^2+z^2+b^2-c^2)^2 -4b^2(x^2+z^2) = 0

Converting to cylindrical coordinates, with translation on y-axis using 'a' , rotation on xy plane with 't'

x = (r*cos(θ-t))
y = (r*sin(θ-t)-a)
z = z

Cylindrical Equation for a torus that rotate/translates into positions of polygonal multi-tiger solutions in R^3:

((r*cos(θ-t))^2 +(r*sin(θ-t)-a)^2 +z^2 +b^2 -c^2)^2 -4b^2((r*cos(θ-t))^2 +z^2) = 0

Torus equation, expressed in consolidated form that separates all trig and non-trig terms

r^4+z^4+a^4+b^4+c^4 +2r^2(z^2+2a^2-c^2) +2z^2(a^2-b^2-c^2) +2(a^2b^2-a^2c^2-b^2c^2) -2(a^2+b^2)r^2*cos(2(t-θ)) +4ar(a^2+b^2-c^2+r^2+z^2)*sin(t-θ)

The Trigonometric Trinomial for Torus Solution of Polygonal Multi-Tigers

This form of the torus equation is independent of any variable, radius parameter, or translate distance. The one and only important thing here is angle. This is the fundamental Unit of Solution for all polygonal multi-tigers. It also introduces a new form of complex number, with some exotic type of double periodic imaginary part. I've been trying to generalize it to complex polynomials, exactly like ((x+iy)^m +ix)^n .

Unit of Solution for Polygonal Multi-tiger

X -2Y*cos(2(t-θ)) +4Z*sin(t-θ)

It's not hypercomplex since it's still on a 2D plane, or multi-complex since it's one angle value. It's one number, but in two different places of two different rates. It's the n-period of a fixed, perpendicular 2-period, combined. Instead of periodicity of a point around origin, this is a plane around an axis. Maybe it's a 2D imaginary number? Some kind of complex bivector, perhaps? Geometric algebra is the extension of complex numbers. Multitoruses are the roots of unity generalized over the reals. And, represented rather clearly in front of us is some extension of complex numbers.

So, multi-tigers may be the extension of complex numbers generalized over the reals. That's how mathematically insane they are. The unit of solution for a polyhedral multi-tiger likely has 4 parts to its analogous imaginary component, producing quadruple periodic equations. Unexplored territory, but not off limits with this new approach.

3D Cylindrical Conversion

These are the 3D versions of what will have a 4th variable properly implemented at some point. Although, using this form of abstraction, it makes no difference whether you use 1,2,3, or 4 variables in X, Y, or Z, because the overall polynomial P(X,Y,Z) will remain true and unchanged.

X(r,z) => (r^2+z^2-c^2)^2 +2a^2(2r^2+z^2+b^2-c^2) -2b^2(z^2+c^2) +a^4+b^4
Y(r,z) => (a^2+b^2)r^2
Z(r,z) => ar(r^2+z^2+a^2+b^2-c^2)

a = translate distance from origin , R1
b= major diameter of the torus solution, R2
c= minor diameter of the torus solution, R3

Simplified hard-set values of a = 4 , b = 2 , c = 1 :

Cylindrical Conversion
X(r,z) => r^4 +z^4 +2r^2z^2 +62r^2 +22z^2 +361
Y(r) => 20r^2
Z(r,z) => 4r^3 +4rz^2 +76r

Cartesian Conversion
X(x,y,z) = (x^2+y^2)^2 +z^2(2x^2+2y^2+z^2+22) +62(x^2+y^2) +361
Y(x,y) = 20(x^2+y^2)
Z(x,y,z) = 4*sqrt(x^2+y^2)(x^2+y^2+z^2+19)
θ = arctan(y/x)

The Polygonal Multi-Tiger Generating Function

n = n-gonal array of toruses

W|A
product_(i=1)^n (X -2Y*cos(2(2iπ/n-θ))) +4Z*sin(2iπ/n-θ)))

Latex
\prod_{i=1}^{n} \left(X -2Y\cdot \cos\left(2\left(\frac{2i\pi}{n} -\theta\right)\right) +4Z\cdot \sin\left(\frac{2i\pi}{n} -\theta\right)\right)

Iterations of this function produce the following:

Equation of Tiger, the Polygonal Quadratic Multi-Tiger:

2 Torus Solutions in digonal array,

X -2Y*cos(2(0-θ)) +4Z*sin(0-θ)
X -2Y*cos(2(π-θ)) +4Z*sin(π-θ)

Combine into:
P(X,Y,Z) = X^2 +2Y^2 +2Y^2*cos(4θ) -8Z^2 +4(2Z^2-XY)*cos(2θ)

another curious form:
(X -2Y*cos(2θ))^2 -(4Z*sin(θ))^2 = 0

3D slice in cartesian,
((x^2+y^2)^2 +2z^2(x^2+y^2) +2(11x^2+51y^2+11z^2) +z^4 +361)^2 -16(4y(x^2+y^2+z^2+19))^2 = 0

Equation of Mantis, the Polygonal Cubic Multi-Tiger:

3 Torus Solutions in triangular array,

X -2Y*cos(2(0-θ)) +4Z*sin(0-θ)
X -2Y*cos(2(2π/3-θ)) +4Z*sin(2π/3-θ)
X -2Y*cos(2(4π/3-θ)) +4Z*sin(4π/3-θ)

P(X,Y,Z) = X^3 -3XY^2 -2Y^3*cos(6θ) -12Z^2(X-2Y) +4Z(4Z^2-3XY+3Y^2)*sin(3θ)

3D slice in cylindrical,
(r^4 +2r^2z^2 +62r^2 +z^4 +22z^2 +361)^3 -3(r^4 +2r^2z^2 +62r^2 +z^4 +22z^2 +361)(20r^2)^2 -2(20r^2)^3*cos(6θ) -12(4r^3 +4rz^2 +76r)^2((r^4 +2r^2z^2 +62r^2 +z^4 +22z^2 +361)-2(20r^2)) +4(4r^3 +4rz^2 +76r)(4(4r^3 +4rz^2 +76r)^2-3(r^4 +2r^2z^2 +62r^2 +z^4 +22z^2 +361)(20r^2)+3(20r^2)^2)*sin(3θ) = 0

Very curious consolidated form,
(r^4+2r^2z^2+62r^2+z^4+22z^2+361)^3 -1200r^4(r^4+2r^2z^2+62r^2+z^4+22z^2+361) -16000r^6*cos(6θ) -192r^2(r^2+z^2+19)^2(r^4+2r^2z^2+22r^2+z^4+22z^2+361) +64r^3(r^2+z^2+19)(r^4+2r^2z^2-22r^2+z^4+278z^2+361)*sin(3θ) = 0

Might be able to form a cage structure very easily from this.

Equation of Spider, the Polygonal Quartic Multi-Tiger:

4 Torus Solutions in square (rhombic) array,

X -2Y*cos(2(0-θ)) +4Z*sin(0-θ)
X -2Y*cos(2(2π/4-θ)) +4Z*sin(2π/4-θ)
X -2Y*cos(2(4π/4-θ)) +4Z*sin(4π/4-θ)
X -2Y*cos(2(6π/4-θ)) +4Z*sin(6π/4-θ)

P(X,Y,Z) = X^4 -4X^2Y^2 +6Y^4 +2Y^4*cos(8θ) +32Z^4 -16Z^2((X-Y)^2+Y^2) -4((XY-4Z^2)^2-2(Y^2-2Z^2)^2)*cos(4θ)

Using a=6 , b=2 , c=1:
X = ((r^2+2(z^2+71))r^2+z^2(z^2+62)+1521)
Y = (40r^2)
Z = (6r(r^2+z^2+39))

3D slice in cylindrical,
((r^2+2(z^2+71))r^2+z^2(z^2+62)+1521)^4 -4((r^2+2(z^2+71))r^2+z^2(z^2+62)+1521)^2(40r^2)^2 +6(40r^2)^4 +2(40r^2)^4*cos(8θ) +32(6r(r^2+z^2+39))^4 -16(6r(r^2+z^2+39))^2((((r^2+2(z^2+71))r^2+z^2(z^2+62)+1521)-(40r^2))^2+(40r^2)^2) -4((((r^2+2(z^2+71))r^2+z^2(z^2+62)+1521)(40r^2)-4(6r(r^2+z^2+39))^2)^2-2((40r^2)^2-2(6r(r^2+z^2+39))^2)^2)*cos(4θ) = 0

Organized List of Known Multi-Tiger Equations as P(X,Y,Z)

X^2 +2Y^2 +2Y^2*cos(4θ) -8Z^2 +4(2Z^2-XY)*cos(2θ)

X^3 -3XY^2 -2Y^3*cos(6θ) -12Z^2(X-2Y) +4Z(4Z^2-3XY+3Y^2)*sin(3θ)

X^4 -4X^2Y^2 +6Y^4 +2Y^4*cos(8θ) +16Z^2(2Z^2-Y^2-(X-Y)^2) -4((XY-4Z^2)^2-2(Y^2-2Z^2)^2)*cos(4θ)

These equations come together nicely. Look at those 2 different periodic rates! For the Mantis, we can clearly see the 6-period of the hexagon array of circles as well as the 3-period of the whole torus in one expression. Wow, neat! This is why it could very well be a whole new theory, because the expressions are confirming things that we already know about, in an obvious algebraic way (to us now, at least).

Someway, somehow, these X, Y, Z terms combine together through some kind of exotic complex exponentiation that is off limits to my mind right now. I see a little order on the left with the X,Y terms, which resembles the circle arrays. But, the way the X,Y,Z terms combine on the right is pretty crazy, and I can barely make any sense of it.

Try them out, these equations plot polygonal arrays of perfect toruses, for now. And uh, yeah, the polynomial for Crab starts off with a quartic polynomial getting raised to the fifth power. Yeah, you guys are trying to comprehend multi-tigers, huh? You guys are totally nuts, man. These things are stupid crazy and we have barely tasted the 4th dimension.

- Philip Pugeau , 7/15/2019
It is by will alone, I set my donuts in motion
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### Re: The Roots of Unity Generalized to the Multi-torus in R^n

Have a look at this, where i derive the equations in equation C from first principles. It's one of the key things I use to walk higher dimensional space.

The second part of this series of papers is to be on the Fibonacci Hamster.
The dream you dream alone is only a dream
the dream we dream together is reality.

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### Re: The Roots of Unity Generalized to the Multi-torus in R^n

Thanks, Wendy! I'll have a closer look at it later. Most of what you're talking about is over my head, though. I do like the abstraction of defining x^3 - 2=0 as (1,0,0,0) = (0,0,2) . Does this have any real use with what I'm trying to do?
It is by will alone, I set my donuts in motion
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### Re: The Roots of Unity Generalized to the Multi-torus in R^n

All right, finally had a look at the sphere of multi-spheritorus, and the torus of multi-ditorus solutions, and how they consolidate in cylindrical coordinates.

For the sphere, we have:

Cylindrical equation for a sphere, rotating on plane xy:

(r*cos(θ-t)-a)^2 + (r*sin(θ-t))^2 + z^2 = b^2

expanded, consolidated:

r^2 +z^2 +a^2 -b^2 -2ar*cos(t-θ) = 0

Reduced into P(X,Y)

X = r^2 +z^2 +a^2 -b^2
Y = ar

X -2Y*cos(t-θ) = 0

This trig binomial is identical to the one for a circle, which means all known P(X,Y) for n-gon arrays will also work for higher spheres.

##################################

For the torus of multi-ditorus,

cylindrical coords that rotate on plane XY.

(sqrt((r*cos(θ-t)-a)^2 + (r*sin(θ-t))^2) -b)^2 +z^2 -c^2 = 0

Reduced form Trig Trinomial P(X,Y,Z),

X = r^4+z^4+a^4+b^4+c^4 +2r^2(z^2+2a^2-b^2-c^2) +2z^2(a^2+b^2-c^2) -2(a^2b^2+b^2c^2+a^2c^2)

Y = a^2r^2

Z = ar(r^2+z^2+a^2-b^2-c^2)

P(X,Y,Z) = X +2Y*cos(2(t-θ)) -4Z*cos(t-θ)

This is identical to the multi-tiger solutions, except for the -4Z*cos(t-θ). Not what I was hoping for, but the system will still work. It may be even easier to add the 4th variable into the 3D solutions, compared to the tigers. It was simple enough forming the first multi-spheritorus, so this should be the next one to try out.

########################################################

On the subject of forming a multi-spheritorus, I made the 3-prong a few days ago. It's a simple enough process, without much difficulty adding in the 4th variable. Exploring the shape was more rewarding than I was expecting, too. The plain quadratic (2-prong) spheritorus does not have much to offer in 3D slices, but a 3-prong version opens the door to a whole new variety of oblique translations and rotations. I usually don't make 14 animations for a single shape, but when I do, it was deserving.

Deriving the Cubic Multi-Spheritorus Polynomial

Degree-3 polynomial for trigonal array of 3 solutions (X -2Y*cos(t-θ)),

P(X,Y) = X^3 -3XY^2 -2Y^3*cos(3θ)

Cylindrical conversion of P(X,Y) for arrays of spheres,

X = r^2 +z^2 +a^2 -b^2
Y = ar

(r^2 +z^2 +a^2 -b^2)^3 -3(r^2 +z^2 +a^2 -b^2)(ar)^2 -2(ar)^3*cos(3θ) = 0

Hardset radius values a = 3 ; b = 1,

(r^2 +z^2 +3^2 -1^2)^3 -3(r^2 +z^2 +3^2 -1^2)(3r)^2 -2(3r)^3*cos(3θ) = 0

(r^2+z^2+8)^3 -27r^2(r^2+z^2+8) -54r^3*cos(3θ) = 0

Setting z=a to make 2d slice , and re-using z to find working 4th variable to relabel as w,

(r^2+a^2+8)^3 -27r^2(r^2+a^2+8) -54r^3*cos(3θ) = 0

Adding z^2 terms with every lone r^2, with adj coeff on -54b*z^2,

(r^2+a^2+z^2+8)^3 -27(r^2+z^2)(r^2+a^2+z^2+8) -54r^3*cos(3θ) -54b*z^2 = 0

b=4 for best surface,

(r^2+a^2+z^2+8)^3 -27(r^2+z^2)(r^2+a^2+z^2+8) -54r^3*cos(3θ) -216z^2 = 0

Relabel a -> z ; z -> w ; as a function f(r,θ,z,w),

(r^2+z^2+w^2+8)^3 -27(r^2+w^2)(r^2+z^2+w^2+8) -54r^3*cos(3θ) -216w^2 = 0

Convert to cartesian using Polar only: r^2 = x^2+y^2 ; r^3*cos(3θ) = x^3-3xy^2,

(x^2+y^2+z^2+w^2+8)^3 -27(x^2+y^2+w^2)(x^2+y^2+z^2+w^2+8) -54(x^3-3xy^2) -216w^2 = 0

Further consolidated,

(x^2+y^2+z^2+w^2+8)^3 -27(x^2+y^2+w^2)(x^2+y^2+z^2+w^2+8) -54(x^3-3xy^2+4w^2) = 0

^- Cartesian equation of a 3-prong multi-spheritorus in R^4

Rotate/Translate on XW, YW, ZW:

((x*sin(b)+a*cos(b))^2+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2+(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2+8)^3 -27((x*sin(b)+a*cos(b))^2+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2+(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2)((x*sin(b)+a*cos(b))^2+(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2+(z*sin(d)+(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*cos(d))^2+(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2+8) -54((x*sin(b)+a*cos(b))^3-3(x*sin(b)+a*cos(b))(y*sin(c)+(x*cos(b)-a*sin(b))*cos(c))^2+4(z*cos(d)-(y*cos(c)-(x*cos(b)-a*sin(b))*sin(c))*sin(d))^2) = 0

Exploring the Triple Rotate Equation:
-------------------------------------
a : translate
b : rotate on XW
c : rotate on YW
d : rotate on ZW

Single 360 Rotations:
---------------------

1. [0 < b < 2π] ; c=0 ; d=0 : hemi-torus <-> 2 blobs

2. [0 < b < 2π] ; c=π/2 ; d=0 : 2 blobs -> 3-prong MT -> 2 blobs -> alt 3-prong MT

3. [0 < b < 2π] ; c=π/2 ; d=π/2 : trigon of S^2 -> 2 blobs -> alt trigon of S^2

4. b=π/2 ; [0 < c < 2π] ; d=0 : hemi-torus <-> 3-prong MT

5. b=π/2 ; [0 < c < 2π] ; d=π/2 : trigon of S^2 <-> hemi-torus

6. b=π/2 ; c=π/2 ; [0 < d < 2π] : trigon of S^2 <-> 3-prong MT

Cool 360 rotation with addt'l tilt:
-----------------------------------

1. [0 < b < 2π] ; c=π/2 ; d=2.8 : 2 blobs <-> 3-prong MT oblq

2. [0 < b < 2π] ; c=3π/4 ; d=0 : 3 alternating prongs between 2 blobs

3. b=π/2 ; [0 < c < 2π] ; d=0.334 : hemi-torus -> 3-prong MT -> hemi-torus oblq

4. b= 1.27 ; [0 < c < 2π] ; d=0.468 : even cooler swirling, chasing 3 blobs

5. b=1.6 ; c=1.27 ; [0 < d < 2π] : trigon of S^2 <-> 3-prong MT oblq #2

Very cool oblique scans:
------------------------

1. [-2.5 < a < 2.5] ; b=1.27 ; c=1.8 ; d=0.334

2. [-2 < a < 2] ; b=π/2 ; c=1.2 ; d=0

3. [-2.5 < a < 2.5] ; b=1.8 ; c=π/2 ; d=0.267
It is by will alone, I set my donuts in motion
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