Polynomials for Toratopes

Discussion of shapes with curves and holes in various dimensions.

Re: Polynomials for Toratopes

Postby PWrong » Thu Oct 09, 2014 2:21 pm

In my original calculations I had x y + z w = 0, so I must have made the mistake copying it in.

We can turn this figure 45 degrees in xy plane and 45 degrees in zw plane:
(x*sqrt(2)/2 + y*sqrt(2)/2) (x*sqrt(2)/2 - y*sqrt(2)/2) + (z*sqrt(2)/2 + w*sqrt(2)/2) (z*sqrt(2)/2 - w*sqrt(2)/2) = 0
1/2(x^2 + z^2 - y^2 - w^2) = 0
x^2 + z^2 - y^2 - w^2 = 0

I had a similar idea. If you apply that same coordinate transform to the other equation, you get something like x y + z w = 1/2.

The constant term can be shoved into the variables, so we're actually talking about the intersection of a 4D cone and a 4D hyperboloid, with one of them rotated by tau/8 in xy and zw.
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Re: Polynomials for Toratopes

Postby Marek14 » Thu Oct 09, 2014 4:02 pm

PWrong wrote:In my original calculations I had x y + z w = 0, so I must have made the mistake copying it in.

We can turn this figure 45 degrees in xy plane and 45 degrees in zw plane:
(x*sqrt(2)/2 + y*sqrt(2)/2) (x*sqrt(2)/2 - y*sqrt(2)/2) + (z*sqrt(2)/2 + w*sqrt(2)/2) (z*sqrt(2)/2 - w*sqrt(2)/2) = 0
1/2(x^2 + z^2 - y^2 - w^2) = 0
x^2 + z^2 - y^2 - w^2 = 0

I had a similar idea. If you apply that same coordinate transform to the other equation, you get something like x y + z w = 1/2.

The constant term can be shoved into the variables, so we're actually talking about the intersection of a 4D cone and a 4D hyperboloid, with one of them rotated by tau/8 in xy and zw.


Yes, exactly.
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Re: Polynomials for Toratopes

Postby PWrong » Fri Oct 10, 2014 2:59 am

So back to the main topic, I'm still not sure what we're trying to do here. The expanded polynomials don't seem very useful for doing anything. Especially since they get outrageously complicated very quickly.
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Re: Polynomials for Toratopes

Postby Marek14 » Fri Oct 10, 2014 5:50 am

PWrong wrote:So back to the main topic, I'm still not sure what we're trying to do here. The expanded polynomials don't seem very useful for doing anything. Especially since they get outrageously complicated very quickly.


I thought it was you who said that the functions shouldn't contain the square roots :)
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Re: Polynomials for Toratopes

Postby PWrong » Fri Oct 10, 2014 10:06 am

You corrected me on that :P
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Re: Polynomials for Toratopes

Postby Marek14 » Fri Oct 10, 2014 10:44 am

PWrong wrote:You corrected me on that :P


Of course, the full combination makes it easier (relatively) to create equations of cuts. It should be even possible to extrapolate equation of a toratope from the equations of its cuts as every term that contains less than all coordinates will appear in at least one cut equation. We actually probably could find out the tiger equation in this way...
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Re: Polynomials for Toratopes

Postby ICN5D » Fri Oct 10, 2014 9:11 pm

This is what Ive been talking about. These polynomials, all of them in fact, can be decomposed into products when setting any number of dimensions to zero. You will end up with several copies of a toratope poly, spaced apart by a diameter value. The intercept polys will contain all terms in the original, minus the dimensions. Using the cut algorithm and a little intuition, we can pre-derive the intercept polys, as a goal when rewriting and simplifying the parent equation. Thats what Ive been writing in the above posts, the intercepts in product form. It looks like another interesting problem to delve into. Toratopes seem to represent a very high degree of symmetry in their polys, since their intercepts can be written this way. I also feel the full equation can be derived through their intercepts.
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Re: Polynomials for Toratopes

Postby Marek14 » Fri Oct 10, 2014 9:33 pm

ICN5D wrote:This is what Ive been talking about. These polynomials, all of them in fact, can be decomposed into products when setting any number of dimensions to zero. You will end up with several copies of a toratope poly, spaced apart by a diameter value. The intercept polys will contain all terms in the original, minus the dimensions. Using the cut algorithm and a little intuition, we can pre-derive the intercept polys, as a goal when rewriting and simplifying the parent equation. Thats what Ive been writing in the above posts, the intercepts in product form. It looks like another interesting problem to delve into. Toratopes seem to represent a very high degree of symmetry in their polys, since their intercepts can be written this way. I also feel the full equation can be derived through their intercepts.


Well, they can't ALWAYS be decomposed into products with any number of dimension setting, since sometimes the cut is just a single toratope -- ((III)(III)) would, for example, need to be reduced by at least 2 dimensions before it can be decomposed.
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Re: Polynomials for Toratopes

Postby ICN5D » Sat Oct 11, 2014 2:39 am

Well, yes, this is true :)
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Re: Polynomials for Toratopes

Postby PWrong » Mon Oct 13, 2014 4:14 am

Maybe you can go the other way? Derive the total polynomial by figuring out the roots and multiplying the factors? I feel like there's more to it than that though. I'm on my phone and can't look at it closely yet.
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Re: Polynomials for Toratopes

Postby PWrong » Mon Oct 13, 2014 5:08 am

Never mind, that doesn't make sense
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Re: Polynomials for Toratopes

Postby Marek14 » Mon Oct 13, 2014 7:06 am

Well, let's have a look at deriving equations from cuts.

First of all, this is obviously non-unique -- the resulting polynomial can have any number of terms that contain ALL variables added without changing its cuts (since such terms will be set to 0 by every cut). But I would hypothesize that toratopes don't contain such terms -- they are the simplest shapes with given cuts. I don't have a proof, though...

So, if we look at a torus ((xy)z) with major radius a and minor radius b, the cuts are:

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

Expanding these gives:
y^4 + 2 y^2 z^2 + z^4 - 2 a^2 y^2 - 2 b^2 y^2 + 2 a^2 z^2 - 2 b^2 z^2 + a^4 - 2 a^2 b^2 + b^4
x^4 + 2 x^2 z^2 + z^4 - 2 a^2 x^2 - 2 b^2 x^2 + 2 a^2 z^2 - 2 b^2 z^2 + a^4 - 2 a^2 b^2 + b^4
x^4 + 2 x^2 y^2 + y^4 - 2 a^2 x^2 - 2 b^2 x^2 - 2 a^2 y^2 - 2 b^2 y^2 + a^4 - 2 a^2 b^2 + b^4

Which can then be consolidated into complete polynomial:
x^4 + 2 x^2 y^2 + 2 x^2 z^2 + y^4 + 2 y^2 z^2 + z^4 - 2 a^2 x^2 - 2 b^2 x^2 - 2 a^2 y^2 - 2 b^2 y^2 + 2 a^2 z^2 - 2 b^2 z^2 + a^4 - 2 a^2 b^2 + b^4

Which can be simplified to (x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 - z^2) - 2 b^2 (x^2 + y^2 + z^2) + (a^2 - b^2)^2

Would this work as an equation of a torus?
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Re: Polynomials for Toratopes

Postby Marek14 » Mon Oct 13, 2014 6:06 pm

Tried it in Calc3D -- the equation works, which means we can extend.

Torisphere ((xyz)w) with major radius a and minor radius b:
(y^2 + z^2 + w^2)^2 - 2 a^2 (y^2 + z^2 - w^2) - 2 b^2 (y^2 + z^2 + w^2) + (a^2 - b^2)^2
(x^2 + z^2 + w^2)^2 - 2 a^2 (x^2 + z^2 - w^2) - 2 b^2 (x^2 + z^2 + w^2) + (a^2 - b^2)^2
(x^2 + y^2 + w^2)^2 - 2 a^2 (x^2 + y^2 - w^2) - 2 b^2 (x^2 + y^2 + w^2) + (a^2 - b^2)^2
(x^2 + y^2 + z^2 - (a + b)^2)(x^2 + y^2 + z^2 - (a - b)^2)

Consolidated, we get

x^4 + 2 x^2 y^2 + 2 x^2 z^2 + 2 x^2 w^2 + y^4 + 2 y^2 z^2 + 2 y^2 w^2 + z^4 + 2 z^2 w^2 + w^4 - 2 a^2 x^2 - 2 b^2 x^2 - 2 a^2 y^2 - 2 b^2 y^2 - 2 a^2 z^2 - 2 b^2 z^2 + 2 a^2 w^2 - 2 b^2 w^2 + a^4 - 2 a^2 b^2 + b^4

or

(x^2 + y^2 + z^2 + w^2)^2 - 2 a^2 (x^2 + y^2 + z^2 - w^2) - 2 b^2 (x^2 + y^2 + z^2 + w^2) + (a^2 - b^2)^2

Which is, as expected, a simple extension of the original torus equation.

If we try for spheritorus ((xy)zw) with major radius a and minor radius b, we get:

((x + a)^2 + z^2 + w^2 - b^2)((x - a)^2 + z^2 + w^2 - b^2)
((y + a)^2 + z^2 + w^2 - b^2)((y - a)^2 + z^2 + w^2 - b^2)
(x^2 + y^2 + w^2)^2 - 2 a^2 (x^2 + y^2 - w^2) - 2 b^2 (x^2 + y^2 + w^2) + (a^2 - b^2)^2
(x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 - z^2) - 2 b^2 (x^2 + y^2 + z^2) + (a^2 - b^2)^2

By the way, the first two equations can be rewritten like this:
(x^2 + z^2 + w^2)^2 - 2 a^2 (x^2 - z^2 - w^2) + 2 b^2 (x^2 + z^2 + w^2) + (a^2 - b^2)^2
(y^2 + z^2 + w^2)^2 - 2 a^2 (y^2 - z^2 - w^2) + 2 b^2 (y^2 + z^2 + w^2) + (a^2 - b^2)^2

See how close the equations of two spheres are to the equation of torus? And the two concentric spheres, (x^2 + y^2 + z^2 - (a + b)^2)(x^2 + y^2 + z^2 - (a - b)^2) from torisphere? It can be also rewritten into more interesting form:
(x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 + z^2) - 2 b^2 (x^2 + y^2 + z^2) + (a^2 - b^2)^2

So the general equation for an arbitrary torus should be
(x1^2 + x2^2 + ... + xn^2 + y1^2 + y2^2 + ... + ym^2)^2 - 2 a^2 (x1^2 + x2^2 + ... + xn^2 - y1^2 - y2^2 - ... - ym^2) - 2 b^2 (x1^2 + x2^2 + ... + xn^2 + y1^2 + y2^2 + ... + ym^2) + (a^2 - b^2)^2

Different toruses in the same dimension differ in the number of plus and minus signs in the a^2 section, and this is robust enough to include even degenerate toruses formed by concentric or separate spheres, and probably even the empty sets.

So the full equation for the spheritorus is
(x^2 + y^2 + z^2 + w^2)^2 - 2 a^2 (x^2 + y^2 - z^2 - w^2) - 2 b^2 (x^2 + y^2 + z^2 + w^2) + (a^2 - b^2)^2
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Re: Polynomials for Toratopes

Postby Marek14 » Mon Oct 13, 2014 7:03 pm

I suspect that higher toratopes can be also expressed in some similar way, but so far they defy my calculations :)
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Re: Polynomials for Toratopes

Postby PWrong » Tue Oct 14, 2014 12:45 am

What do you mean by consolidate?
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Re: Polynomials for Toratopes

Postby Marek14 » Tue Oct 14, 2014 2:46 am

PWrong wrote:What do you mean by consolidate?


Well, simply said it means that I compare the polynomials for the cuts and put together all terms that appear in them. A term containing certain variables will appear in all cuts except those that have those variables cut (as its value is 0 there). It's like one of those puzzles where you have to find a 3D shape based on shadows it projects in various directions.

So, the torus equation (x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 - z^2) - 2 b^2 (x^2 + y^2 + z^2) + (a^2 - b^2)^2 has cuts

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

which can be all rewritten as products of two circles. The consolidation is the reverse process where you guess on the whole polynome from the cuts. In this case, it lead me to an equation that looks more like a traditional polynome (since the various degrees of variables are well-separated). It also suggests other possible ways of development by switching the signs around. Possibly could lead to some "hyperbolic toroid"...

So far I was not successful in finding the tiger equation using this, some terms appeared with different coefficients in different cuts, I probably made an error somewhere. But it should definitely include the terms (x^2 + y^2 + z^2 + w^2)^4 and (a^2 + b^2 - c^2)^4.
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Re: Polynomials for Toratopes

Postby ICN5D » Tue Oct 14, 2014 3:10 am

Wow, I see what you did there. You took all possible cuts, multiplied them out, and combined them into one equation. Some terms will be duplicates, others unique. By consolidating, you're adding everything together and combining the duplicates into one. This is some extremely awesome stuff right here. And, look at the full equation. It's perfectly balanced, and probably the best form to study. This method might make it easier to write a degree-16 for (((II)I)(II)) , (((II)(II))I) , and ((((II)I)I)I) .


Quite recently, I rewrote the torus equation another simplified form:

x^4 + y^4 + z^4 + a^4 + b^4 + 2(x^2y^2 + x^2z^2 + y^2z^2) - 2a^2(x^2 + y^2 - z^2 + b^2) - 2b^2(x^2 + y^2 + z^2)

It's interesting how many different ones there are for just 3D.
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Re: Polynomials for Toratopes

Postby Marek14 » Tue Oct 14, 2014 3:21 am

ICN5D wrote:Wow, I see what you did there. You took all possible cuts, multiplied them out, and combined them into one equation. Some terms will be duplicates, others unique. By consolidating, you're adding everything together and combining the duplicates into one. This is some extremely awesome stuff right here. And, look at the full equation. It's perfectly balanced, and probably the best form to study. This method might make it easier to write a degree-16 for (((II)I)(II)) , (((II)(II))I) , and ((((II)I)I)I) .


Quite recently, I rewrote the torus equation another simplified form:

x^4 + y^4 + z^4 + a^4 + b^4 + 2(x^2y^2 + x^2z^2 + y^2z^2) - 2a^2(x^2 + y^2 - z^2 + b^2) - 2b^2(x^2 + y^2 + z^2)

It's interesting how many different ones there are for just 3D.


Easier? Yes. Easy? No :)

Your form, just from looking, is equivalent to mine. But I think mine does a good job by treating the two kinds of variables (coordinates and radii) as fundamentally different and not mixing them up inside the terms.
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Re: Polynomials for Toratopes

Postby ICN5D » Tue Oct 14, 2014 3:41 am

Yeah, I would agree with that. All diameter terms are separated out, leaving behind dimensions. It reflects some very good symmetry between torisphere and spheritorus. Tiger not possible? Huh, even with four cuts of a column of tori? How about using circles, in six different coordinate planes?
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Re: Polynomials for Toratopes

Postby Marek14 » Tue Oct 14, 2014 6:09 am

ICN5D wrote:Yeah, I would agree with that. All diameter terms are separated out, leaving behind dimensions. It reflects some very good symmetry between torisphere and spheritorus. Tiger not possible? Huh, even with four cuts of a column of tori? How about using circles, in six different coordinate planes?


Tiger should be possible, but it's more complicated and I haven't managed to do it yet. You can try if you want :) The four cuts should look like this:

((z^2 + w^2 + (y + a)^2)^2 - 2 b^2 (z^2 + w^2 - (y + a)^2) - 2 c^2 (z^2 + w^2 + (y + a)^2) + (b^2 - c^2)^2)((z^2 + w^2 + (y - a)^2)^2 - 2 b^2 (z^2 + w^2 - (y - a)^2) - 2 c^2 (z^2 + w^2 + (y - a)^2) + (b^2 - c^2)^2)
((z^2 + w^2 + (x + a)^2)^2 - 2 b^2 (z^2 + w^2 - (x + a)^2) - 2 c^2 (z^2 + w^2 + (x + a)^2) + (b^2 - c^2)^2)((z^2 + w^2 + (x - a)^2)^2 - 2 b^2 (z^2 + w^2 - (x - a)^2) - 2 c^2 (z^2 + w^2 + (x - a)^2) + (b^2 - c^2)^2)
((x^2 + y^2 + (w + b)^2)^2 - 2 a^2 (x^2 + y^2 - (w + b)^2) - 2 c^2 (x^2 + y^2 + (w + b)^2) + (a^2 - c^2)^2)((x^2 + y^2 + (w - b)^2)^2 - 2 a^2 (x^2 + y^2 - (w - b)^2) - 2 c^2 (x^2 + y^2 + (w - b)^2) + (a^2 - c^2)^2)
((x^2 + y^2 + (z + b)^2)^2 - 2 a^2 (x^2 + y^2 - (z + b)^2) - 2 c^2 (x^2 + y^2 + (z + b)^2) + (a^2 - c^2)^2)((x^2 + y^2 + (z - b)^2)^2 - 2 a^2 (x^2 + y^2 - (z - b)^2) - 2 c^2 (x^2 + y^2 + (z - b)^2) + (a^2 - c^2)^2)

Starting from circles in six coordinate planes isn't any better, though. For starters, you can only use four planes (two have empty cuts). Also, you'd miss some terms this way since tiger does include terms with three coordinates which are zero in any coordinate plane cut.

And cuts for ditorus should be:
(((y + a)^2 + z^2 + w^2)^2 - 2 b^2 ((y + a)^2 + z^2 - w^2) - 2 c^2 ((y + a)^2 + z^2 + w^2) + (b^2 - c^2)^2)(((y - a)^2 + z^2 + w^2)^2 - 2 b^2 ((y - a)^2 + z^2 - w^2) - 2 c^2 ((y - a)^2 + z^2 + w^2) + (b^2 - c^2)^2)
(((x + a)^2 + z^2 + w^2)^2 - 2 b^2 ((x + a)^2 + z^2 - w^2) - 2 c^2 ((x + a)^2 + z^2 + w^2) + (b^2 - c^2)^2)(((x - a)^2 + z^2 + w^2)^2 - 2 b^2 ((x - a)^2 + z^2 - w^2) - 2 c^2 ((x - a)^2 + z^2 + w^2) + (b^2 - c^2)^2)
((x^2 + y^2 + w^2)^2 - 2 (a + c)^2 (x^2 + y^2 - w^2) - 2 b^2 (x^2 + y^2 + w^2) + ((a + c)^2 - b^2)^2)((x^2 + y^2 + w^2)^2 - 2 (a - c)^2 (x^2 + y^2 - w^2) - 2 b^2 (x^2 + y^2 + w^2) + ((a - c)^2 - b^2)^2)
((x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 - z^2) - 2 (b + c)^2 (x^2 + y^2 + z^2) + (a^2 - (b + c)^2)^2)((x^2 + y^2 + z^2)^2 - 2 a^2 (x^2 + y^2 - z^2) - 2 (b - c)^2 (x^2 + y^2 + z^2) + (a^2 - (b - c)^2)^2)
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Re: Polynomials for Toratopes

Postby Marek14 » Tue Oct 14, 2014 8:14 am

OK, finally consolidated the tiger. The full expanded and consolidated list looks like this:

+ x^8 + 4 x^6 y^2 + 4 x^6 z^2 + 4 x^6 w^2 + 6 x^4 y^4 + 12 x^4 y^2 z^2 + 12 x^4 y^2 w^2 + 6 x^4 z^4 + 12 x^4 z^2 w^2 + 6 x^4 w^4 + 4 x^2 y^6 + 12 x^2 y^4 z^2 + 12 x^2 y^4 w^2 + 12 x^2 y^2 z^4 + 12 x^2 y^2 w^4 + 4 x^2 z^6 + 12 x^2 z^4 w^2

+ 12 x^2 z^2 w^4 + 4 x^2 w^6 + y^8 + 4 y^6 z^2 + 4 y^6 w^2 + 6 y^4 z^4 + 12 y^4 z^2 w^2 + 6 y^4 w^4 + 4 y^2 z^6 + 12 y^2 z^4 w^2 + 12 y^2 z^2 w^4 + 4 y^2 w^6 + z^8 + 4 z^6 w^2 + 6 z^4 w^4 + 4 z^2 w^6 + w^8
- 4 a^2 x^6 - 12 a^2 x^4 y^2 - 4 a^4 x^4 z^2 - 4 a^2 x^4 w^2 - 12 a^2 x^2 y^4 - 8 a^2 x^2 y^2 z^2 - 8 a^2 x^2 y^2 w^2 + 4 a^2 x^2 z^4 + 8 a^2 x^2 z^2 w^2 + 4 a^2 x^2 w^4 - 4 a^2 y^6 - 4 a^2 y^4 z^2 - 4 a^2 y^4 w^2 + 4 a^2 y^2 z^4 + 8 a^2

y^2 z^2 w^2 + 4 a^2 y^2 w^4 + 4 a^2 z^6 + 12 a^2 z^4 w^2 + 12 a^2 z^2 w^4 + 4 a^2 w^6
+ 4 b^2 x^6 + 12 b^2 x^4 y^2 + 4 b^2 x^4 z^2 + 4 b^2 x^4 w^2 + 12 b^2 x^2 y^4 + 8 b^2 x^2 y^2 z^2 + 8 b^2 x^2 y^2 w^2 - 4 b^2 x^2 z^4 - 8 b^2 x^2 z^2 w^2 - 4 b^2 x^2 w^4 + 4 b^2 y^6 + 4 b^2 y^4 z^2 + 4 b^2 y^4 w^2 - 4 b^2 y^2 z^4 - 8 b^2

y^2 z^2 w^2 - 4 b^2 y^2 w^4 - 4 b^2 z^6 - 12 b^2 z^4 w^2 - 12 b^2 z^2 w^4 - 4 b^2 w^6
- 4 c^2 x^6 - 12 c^2 x^4 y^2 - 12 c^2 x^4 z^2 - 12 c^2 x^4 w^2 - 12 c^2 x^2 y^4 - 24 c^2 x^2 y^2 z^2 - 24 c^2 x^2 y^2 w^2 - 12 c^2 x^2 z^4 - 24 c^2 x^2 z^2 w^2 - 12 c^2 x^2 w^4 - 4 c^2 y^6 - 12 c^2 y^4 z^2 - 12 c^2 y^4 w^2 - 12 c^2 y^2

z^4 - 24 c^2 y^2 z^2 w^2 - 12 c^2 y^2 w^4 - 4 c^2 z^6 - 12 c^2 z^4 w^2 - 12 c^2 z^2 w^4 - 4 c^2 w^6
+ 6 a^4 x^4 + 12 a^4 x^2 y^2 - 4 a^4 x^2 z^2 - 4 a^4 x^2 w^2 + 6 a^4 y^4 - 4 a^4 y^2 z^2 - 4 a^4 y^2 w^2 + 6 a^4 z^4 + 12 a^4 z^2 w^2 + 6 a^4 w^4
- 4 a^2 b^2 x^4 - 8 a^2 b^2 x^2 y^2 - 40 a^2 b^2 x^2 z^2 - 40 a^2 b^2 x^2 w^2 - 4 a^2 b^2 y^4 - 40 a^2 b^2 y^2 z^2 - 40 a^2 b^2 y^2 w^2 - 4 a^2 b^2 z^4 - 8 a^2 b^2 z^2 w^2 - 4 a^2 b^2 w^4
+ 4 a^2 c^2 x^4 + 8 a^2 c^2 x^2 y^2 - 8 a^2 c^2 x^2 z^2 - 8 a^2 c^2 x^2 w^2 + 4 a^2 c^2 y^4 - 8 a^2 c^2 y^2 z^2 - 8 a^2 c^2 y^2 w^2 - 12 a^2 c^2 z^4 - 24 a^2 c^2 z^2 w^2 - 12 a^2 c^2 w^4
+ 6 b^4 x^4 + 12 b^4 x^2 y^2 - 4 b^4 x^2 z^2 - 4 b^4 x^2 w^2 + 6 b^4 y^4 - 4 b^4 y^2 z^2 - 4 b^4 y^2 w^2 + 6 b^4 z^4 + 12 b^4 z^2 w^2 + 6 b^4 w^4
- 12 b^2 c^2 x^4 - 24 b^2 c^2 x^2 y^2 - 8 b^2 c^2 x^2 z^2 - 8 b^2 c^2 x^2 w^2 - 12 b^2 c^2 y^4 - 8 b^2 c^2 y^2 z^2 - 8 b^2 c^2 y^2 w^2 + 4 b^2 c^2 z^4 + 8 b^2 c^2 z^2 w^2 + 4 b^2 c^2 w^4
+ 6 c^4 x^4 + 12 c^4 x^2 y^2 + 12 c^4 x^2 z^2 + 12 c^4 x^2 w^2 + 6 c^4 y^4 + 12 c^4 y^2 z^2 + 12 c^4 y^2 w^2 + 6 c^4 z^4 + 12 c^4 z^2 w^2 + 6 c^4 w^4
- 4 a^6 x^2 - 4 a^6 y^2 + 4 a^6 z^2 + 4 a^6 w^2
- 4 a^4 b^2 x^2 - 4 a^4 b^2 y^2 + 4 a^4 b^2 z^2 + 4 a^4 b^2 w^2
+ 4 a^4 c^2 x^2 + 4 a^4 c^2 y^2 - 12 a^4 c^2 z^2 - 12 a^4 c^2 w^2
+ 4 a^2 b^4 x^2 + 4 a^2 b^4 y^2 - 4 a^2 b^4 z^2 - 4 a^2 b^4 w^2
- 8 a^2 b^2 c^2 x^2 - 8 a^2 b^2 c^2 y^2 - 8 a^2 b^2 c^2 z^2 - 8 a^2 b^2 c^2 w^2
+ 4 a^2 c^4 x^2 + 4 a^2 c^4 y^2 + 12 a^2 c^4 z^2 + 12 a^2 c^4 w^2
+ 4 b^6 x^2 + 4 b^6 y^2 - 4 b^6 z^2 - 4 b^6 w^2
- 12 b^4 c^2 x^2 - 12 b^4 c^2 y^2 + 4 b^4 c^2 z^2 + 4 b^4 c^2 w^2
+ 12 b^2 c^4 x^2 + 12 b^2 c^4 y^2 + 4 b^2 c^4 z^2 + 4 b^2 c^4 w^2
- 4 c^6 x^2 - 4 c^6 y^2 - 4 c^6 z^2 - 4 c^6 w^2
+ a^8 + 4 a^6 b^2 + 6 a^4 b^4 + 4 a^2 b^6 + b^8 - 4 a^6 c^2 - 12 a^4 b^2 c^2 - 12 a^2 b^4 c^2 - 4 b^6 c^2 + 6 a^4 c^4 + 12 a^2 b^2 c^4 + 6 b^4 c^4 - 4 a^2 c^6 - 4 b^2 c^6 + c^8

Daunting, isn't it? OK, let's try to simplify it. We know (from symmetry of tiger) that x and y can be exchanged, z and w can be exchanged and you can also swamp these two pairs, provided you also swap a and b.

The first and last row are the easiest. The last row is simply (a^2 + b^2 - c^2)^4. This reminds us of (a^2 - b^2)^2 term of torus.
The first row is almost (x^2 + y^2 + z^2 + w^2)^4, except it's missing the term "+ 24 x^2 y^2 z^2 w^2". But that's okay - we can add it if we want. Since the term contains all four coordinates, it's reduced to zero by every cut, and therefore doesn't appear in any of them.

This, by the way, might prove problematic in the general case. The degree of the polynomial will be usually higher than the dimension. Either these terms have to be guessed or additional, oblique cuts would be needed (for example a cut that sets two coordinates to the same value).

So we have our degree-8 part and the degree-0 part. Now to resolve the degrees 6, 4 and 2.

Degree 2

- 4 a^6 x^2 - 4 a^6 y^2 + 4 a^6 z^2 + 4 a^6 w^2
- 4 a^4 b^2 x^2 - 4 a^4 b^2 y^2 + 4 a^4 b^2 z^2 + 4 a^4 b^2 w^2
+ 4 a^4 c^2 x^2 + 4 a^4 c^2 y^2 - 12 a^4 c^2 z^2 - 12 a^4 c^2 w^2
+ 4 a^2 b^4 x^2 + 4 a^2 b^4 y^2 - 4 a^2 b^4 z^2 - 4 a^2 b^4 w^2
- 8 a^2 b^2 c^2 x^2 - 8 a^2 b^2 c^2 y^2 - 8 a^2 b^2 c^2 z^2 - 8 a^2 b^2 c^2 w^2
+ 4 a^2 c^4 x^2 + 4 a^2 c^4 y^2 + 12 a^2 c^4 z^2 + 12 a^2 c^4 w^2
+ 4 b^6 x^2 + 4 b^6 y^2 - 4 b^6 z^2 - 4 b^6 w^2
- 12 b^4 c^2 x^2 - 12 b^4 c^2 y^2 + 4 b^4 c^2 z^2 + 4 b^4 c^2 w^2
+ 12 b^2 c^4 x^2 + 12 b^2 c^4 y^2 + 4 b^2 c^4 z^2 + 4 b^2 c^4 w^2
- 4 c^6 x^2 - 4 c^6 y^2 - 4 c^6 z^2 - 4 c^6 w^2

First we notice that some rows don't have equal coefficients for all terms. We can correct that by separation.

- 4 a^6 x^2 - 4 a^6 y^2 + 4 a^6 z^2 + 4 a^6 w^2
- 4 a^4 b^2 x^2 - 4 a^4 b^2 y^2 + 4 a^4 b^2 z^2 + 4 a^4 b^2 w^2
+ 4 a^4 c^2 x^2 + 4 a^4 c^2 y^2 - 4 a^4 c^2 z^2 - 4 a^4 c^2 w^2
+ 4 a^2 b^4 x^2 + 4 a^2 b^4 y^2 - 4 a^2 b^4 z^2 - 4 a^2 b^4 w^2
- 8 a^2 b^2 c^2 x^2 - 8 a^2 b^2 c^2 y^2 - 8 a^2 b^2 c^2 z^2 - 8 a^2 b^2 c^2 w^2
+ 4 a^2 c^4 x^2 + 4 a^2 c^4 y^2 + 4 a^2 c^4 z^2 + 4 a^2 c^4 w^2
+ 4 b^6 x^2 + 4 b^6 y^2 - 4 b^6 z^2 - 4 b^6 w^2
- 4 b^4 c^2 x^2 - 4 b^4 c^2 y^2 + 4 b^4 c^2 z^2 + 4 b^4 c^2 w^2
+ 4 b^2 c^4 x^2 + 4 b^2 c^4 y^2 + 4 b^2 c^4 z^2 + 4 b^2 c^4 w^2
- 4 c^6 x^2 - 4 c^6 y^2 - 4 c^6 z^2 - 4 c^6 w^2
---
- 8 a^4 c^2 z^2 - 8 a^4 c^2 w^2
+ 8 a^2 c^4 z^2 + 8 a^2 c^4 w^2
- 8 b^4 c^2 x^2 - 8 b^4 c^2 y^2
+ 8 b^2 c^4 x^2 + 8 b^2 ^4 y^2
Now, the rows fall in two distinctive patterns: either all coordinates have the same sign, or x and y have opposite sign from z and w. We'll put the rows together to reflect this:

- 4 (a^6 + a^4 b^2 - a^4 c^2 - a^2 b^4 - b^6 + b^4 c^2)(x^2 + y^2 - z^2 - w^2)
- 4 c^2 (2 a^2 b^2 - a^2 c^2 - b^2 c^2 + c^4)(x^2 + y^2 + z^2 + w^2)
- 8 b^2 c^2 (b^2 - c^2)(x^2 + y^2)
- 8 a^2 c^2 (a^2 - c^2)(z^2 + w^2)

Degree 4

Here, the general form of rows should be (x^2 + y^2 + z^2 + w^2)^2, with possible sign change at some of the coordinates.
The a^4 and b^4 row have minus signs at terms combining x or y with z or w, suggesting the ideal form (x^2 + y^2 - z^2 - w^2)^2
+ 6 a^4 (x^4 + 2 x^2 y^2 - 2/3 x^2 z^2 - 2/3 x^2 w^2 + y^4 - 2/3 y^2 z^2 - 2/3 y^2 w^2 + z^4 + 2 z^2 w^2 + w^4)
+ 6 a^4 (x^4 + 2 x^2 y^2 - 2 x^2 z^2 - 2 x^2 w^2 + y^4 - 2 y^2 z^2 - 2 y^2 w^2 + z^4 + 2 z^2 w^2 + w^4) + 4 a^4 (x^2 z^2 + x^2 w^2 + y^2 z^2 + y^2 w^2)
+ 6 a^4 (x^2 + y^2 - z^2 - w^2)^2 + 4 a^4 (x^2 + y^2)(z^2 + w^2)

And, analogically,
+ 6 b^4 (x^2 + y^2 - z^2 - w^2)^2 + 4 b^4 (x^2 + y^2)(z^2 + w^2)

The c^4 row is simply
+ 6 c^4 (x^2 + y^2 + z^2 + w^2)^2

Now for the a^2 b^2 row where unexpectedly high number 40 appears for mixed pairs of coordinates. All signs are equal, so we can reduce this to
- 4 a^2 b^2 x^4 - 8 a^2 b^2 x^2 y^2 - 40 a^2 b^2 x^2 z^2 - 40 a^2 b^2 x^2 w^2 - 4 a^2 b^2 y^4 - 40 a^2 b^2 y^2 z^2 - 40 a^2 b^2 y^2 w^2 - 4 a^2 b^2 z^4 - 8 a^2 b^2 z^2 w^2 - 4 a^2 b^2 w^4
- 4 a^2 b^2 (x^4 + 2 x^2 y^2 - 2 x^2 z^2 - 2 x^2 w^2 + y^4 - 2 y^2 z^2 - 2 y^2 w^2 + z^4 + 2 z^2 w^2 + w^4) - 32 a^2 b^2 (x^2 z^2 + x^2 w^2 + y^2 z^2 + y^2 w^2)
- 4 a^2 b^2 (x^2 + y^2 - z^2 - w^2)^2 - 32 a^2 b^2 (x^2 + y^2)(z^2 + w^2)

Now we only have the a^2 c^2 and b^2 c^2 rows left.
+ 4 a^2 c^2 x^4 + 8 a^2 c^2 x^2 y^2 - 8 a^2 c^2 x^2 z^2 - 8 a^2 c^2 x^2 w^2 + 4 a^2 c^2 y^4 - 8 a^2 c^2 y^2 z^2 - 8 a^2 c^2 y^2 w^2 - 12 a^2 c^2 z^4 - 24 a^2 c^2 z^2 w^2 - 12 a^2 c^2 w^4
+ 4 a^2 c^2 (x^4 + 2 x^2 y^2 - 2 x^2 z^2 - 2 x^2 w^2 + y^4 - 2 y^2 z^2 - 2 y^2 w^2 - 3 z^4 - 6 z^2 w^2 - 3 w^4)
+ 4 a^2 c^2 (x^4 + 2 x^2 y^2 - 2 x^2 z^2 - 2 x^2 w^2 + y^4 - 2 y^2 z^2 - 2 y^2 w^2 + z^4 + 2 z^2 w^2 + w^4) - 16 a^2 c^2 (z^4 + 2 z^2 w^2 + w^4)
+ 4 a^2 c^2 (x^2 + y^2 - z^2 - w^2)^2 - 16 a^2 c^2 (z^2 + w^2)^2

And analogically
+ 4 b^2 c^2 (x^2 + y^2 - z^2 - w^2)^2 - 16 b^2 c^2 (x^2 + y^2)^2

The full degree-4 part therefore looks like this:
+ 2 (3 a^4 - 2 a^2 b^2 + 2 a^2 c^2 + 3 b^4 + 2 b^2 c^2)(x^2 + y^2 - z^2 - w^2)^2 + 6 c^4 (x^2 + y^2 + z^2 + w^2)^2 + 4 (a^4 + 4 b^2 c^2 + b^4)(x^2 + y^2)(z^2 + w^2) - 16 b^2 c^2 (x^2 + y^2)^2 - 16 a^2 c^2 (z^2 + w^2)^2

Degree 6

Here, the general structure should be (x^2 + y^2 + z^2 + w^2)^3 or (x^2 + y^2 - z^2 - w^2)^3.
The a^2 row looks like this:
- 4 a^2 x^6 - 12 a^2 x^4 y^2 - 4 a^4 x^4 z^2 - 4 a^2 x^4 w^2 - 12 a^2 x^2 y^4 - 8 a^2 x^2 y^2 z^2 - 8 a^2 x^2 y^2 w^2 + 4 a^2 x^2 z^4 + 8 a^2 x^2 z^2 w^2 + 4 a^2 x^2 w^4 - 4 a^2 y^6 - 4 a^2 y^4 z^2 - 4 a^2 y^4 w^2 + 4 a^2 y^2 z^4 + 8 a^2 y^2 z^2 w^2 + 4 a^2 y^2 w^4 + 4 a^2 z^6 + 12 a^2 z^4 w^2 + 12 a^2 z^2 w^4 + 4 a^2 w^6
- 4 a^2 (x^6 + 3 x^4 y^2 + x^4 z^2 + x^4 w^2 + 3 x^2 y^4 + 2 x^2 y^2 z^2 + 2 x^2 y^2 w^2 - x^2 z^4 - 2 x^2 z^2 w^2 - x^2 w^4 + y^6 + y^4 z^2 + y^4 w^2 - y^2 z^4 - 2 y^2 z^2 w^2 - y^2 w^4 - z^6 - 3 z^4 w^2 - 3 z^2 w^4 - w^6)
- 4 a^2 (x^6 + 3 x^4 y^2 - 3 x^4 z^2 - 3 x^4 w^2 + 3 x^2 y^4 - 6 x^2 y^2 z^2 - 6 x^2 y^2 w^2 + 3 x^2 z^4 + 6 x^2 z^2 w^2 + 3 x^2 w^4 + y^6 - 3 y^4 z^2 - 3 y^4 w^2 + 3 y^2 z^4 + 6 y^2 z^2 w^2 + 3 y^2 w^4 - z^6 - 3 z^4 w^2 - 3 z^2 w^4 - w^6) - 16 a^2 (x^4 z^2 + x^4 w^2 - x^2 z^4 - x^2 w^4 + y^4 z^2 + y^4 w^2 - y^2 z^4 - y^2 w^4 + 2 x^2 y^2 z^2 + 2 x^2 y^2 w^2 - 2 x^2 z^2 w^2 - 2 y^2 z^2 w^2)
- 4 a^2 (x^2 + y^2 - z^2 - w^2)^3 - 16 a^2 (x^2 + y^2 - z^2 - w^2)(x^2 + y^2)(z^2 + w^2)

The b^2 row has always opposite signs to a^2, but is otherwise identical, leading to
+ 4 b^2 (x^2 + y^2 - z^2 - w^2)^3 + 16 b^2 (x^2 + y^2 - z^2 - w^2)(x^2 + y^2)(z^2 + w^2)

a^2 and b^2 can be therefore combined into
- 4 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)^3 - 16 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)(x^2 + y^2)(z^2 + w^2)

As for c^2 row, all signs are the same and it is, in fact, equal to
- 4 c^2 (x^2 + y^2 + z^2 + w^2)^3

So the complete degree 6 is
- 4 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)^3 - 16 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)(x^2 + y^2)(z^2 + w^2) - 4 c^2 (x^2 + y^2 + z^2 + w^2)^3

And we have the complete equation:

(x^2 + y^2 + z^2 + w^2)^4 - 4 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)^3 - 16 (a^2 - b^2)(x^2 + y^2 - z^2 - w^2)(x^2 + y^2)(z^2 + w^2) - 4 c^2 (x^2 + y^2 + z^2 + w^2)^3 + 2 (3 a^4 - 2 a^2 b^2 + 2 a^2 c^2 + 3 b^4 + 2 b^2 c^2)(x^2 + y^2 - z^2 - w^2)^2 + 6 c^4 (x^2 + y^2 + z^2 + w^2)^2 + 4 (a^4 + 4 b^2 c^2 + b^4)(x^2 + y^2)(z^2 + w^2) - 16 b^2 c^2 (x^2 + y^2)^2 - 16 a^2 c^2 (z^2 + w^2)^2 - 4 (a^6 + a^4 b^2 - a^4 c^2 - a^2 b^4 - b^6 + b^4 c^2)(x^2 + y^2 - z^2 - w^2) - 4 c^2 (2 a^2 b^2 - a^2 c^2 - b^2 c^2 + c^4)(x^2 + y^2 + z^2 + w^2) - 8 b^2 c^2 (b^2 - c^2)(x^2 + y^2) - 8 a^2 c^2 (a^2 - c^2)(z^2 + w^2) + (a^2 + b^2 - c^2)^4
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Re: Polynomials for Toratopes

Postby ICN5D » Fri Oct 17, 2014 3:24 am

Wow. That is a really insanely huge equation. Holy crap. Now I see why 5D shapes haven't been explored before. Those degree-16's must be beyond imagination, and unexplorable. But, everything seems like that, at first. Thank you guys for toratopic notation, it's quite useful. These polynomials, they're a bit more complicated than I was imagining. It seems like the only value it has, is being an interesting math problem, by itself. The generic 2-torus is simple, but the tiger can really make things complicated.


Starting from circles in six coordinate planes isn't any better, though. For starters, you can only use four planes (two have empty cuts).


Really? You wouldn't have to add in the empty cuts? I was thinking it did. They would be for the concentric circle pair of either torus intercept, above and below the 2D plane. If deriving the torus equation used all three 2D planes, then why wouldn't a tiger need all six 2D planes? Does the four circles intercept do it all?
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Re: Polynomials for Toratopes

Postby Marek14 » Fri Oct 17, 2014 3:39 am

ICN5D wrote:Wow. That is a really insanely huge equation. Holy crap. Now I see why 5D shapes haven't been explored before. Those degree-16's must be beyond imagination, and unexplorable. But, everything seems like that, at first. Thank you guys for toratopic notation, it's quite useful. These polynomials, they're a bit more complicated than I was imagining. It seems like the only value it has, is being an interesting math problem, by itself. The generic 2-torus is simple, but the tiger can really make things complicated.


Starting from circles in six coordinate planes isn't any better, though. For starters, you can only use four planes (two have empty cuts).


Really? You wouldn't have to add in the empty cuts? I was thinking it did. They would be for the concentric circle pair of either torus intercept, above and below the 2D plane. If deriving the torus equation used all three 2D planes, then why wouldn't a tiger need all six 2D planes? Does the four circles intercept do it all?


Well, that's exactly it. You WOULD have to add the empty cuts, but since they are empty, it is not at all obvious how their equation would look. From the full equation, we can make xy cut to be:

(x^2 + y^2)^4 - 4 (a^2 - b^2)(x^2 + y^2)^3 - 4 c^2 (x^2 + y^2)^3 + 2 (3 a^4 - 2 a^2 b^2 + 2 a^2 c^2 + 3 b^4 + 2 b^2 c^2)(x^2 + y^2)^2 + 6 c^4 (x^2 + y^2)^2 - 16 b^2 c^2 (x^2 + y^2)^2 - 4 (a^6 + a^4 b^2 - a^4 c^2 - a^2 b^4 - b^6 + b^4 c^2)(x^2 + y^2) - 4 c^2 (2 a^2 b^2 - a^2 c^2 - b^2 c^2 + c^4)(x^2 + y^2) - 8 b^2 c^2 (b^2 - c^2)(x^2 + y^2) + (a^2 + b^2 - c^2)^4
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Re: Polynomials for Toratopes

Postby ICN5D » Fri Oct 17, 2014 5:14 am

I've been thinking about what the equation would look like. Non empties are straightforward: we get a product of lower toratopes that are spaced apart by cut open diameter value. The empty cuts have to accommodate a spacing into an extra dimension, by use of a complex conjugate. Removing all dimensions out of a diameter will leave the radius behind, which is a negative number, somehow making a complex component. If cutting with plane xy , then the diameter b should be imaginary.

So, for non-empty cut of tiger, in 2D:


Tiger ((II)(II))

(sqrt(x^2 + y^2) -a)^2 + (sqrt(z^2 + w^2) -b)^2 - c^2



cut into ((I)(I))

(sqrt(x^2) -a)^2 + (sqrt(z^2) -b)^2 - c^2



Expands into

((x-a)^2 + (z-b)^2 - c^2)*((x+a)^2 + (z-b)^2 - c^2)*((x-a)^2 + (z+b)^2 - c^2)*((x+a)^2 + (z+b)^2 - c^2)

Four circles in a square array

For empty cut, we have two instances of a concentric pair of circles. By moving the 2D plane into 3D, to the value of b, we have maximum separation of concentric circles, by (a-c) and (a+c).

But, this doesn't come into view (become real) until a certain value in the complex plane (3D) has been reached: the second diameter b. When we see one pair of circles, the second pair is shifted by 2*b in the complex plane


cut into ((II)())

(sqrt(x^2 + y^2) -a)^2 + (-b)^2 - c^2

I'm not sure about this one. I thought the -b would get a square root, but I guess not. Somehow we get an imaginary parameter " bi " , that comes from setting z and w to zero.



So, maybe something like this?

(x^2 + y^2 - (a-c-bi)^2) * (x^2 + y^2 - (a+c-bi)^2) * (x^2 + y^2 - (a-c+bi)^2) * (x^2 + y^2 - (a+c+bi)^2)

and when we move out from center in the complex plane of 3D, to a concentric pair of 2 circles, we see:

(x^2 + y^2 - (a-c)^2) * (x^2 + y^2 - (a+c)^2) * (x^2 + y^2 - (a-c+2bi)^2) * (x^2 + y^2 - (a+c+2bi)^2)

With an equal zw version as well

This is my best guess as to how imaginary numbers define an empty intercept. I'm not 100% sure on where bi should be, but I do know the value of 2*bi comes about, in the 'move out from center' intercepts. By translating into 3D, we add the value of +bi to all four circle intercepts. The 2 circles spaced in -bi cancel out the imaginary component (and become real), and add to +bi , making 2bi for the second pair. Or, maybe the imaginary bi is translating the dimensions x and y, as in:

((x-bi)^2 + (y-bi)^2 - (a-c)^2) * ((x-bi)^2 + (y-bi)^2 - (a+c)^2) * ((x+bi)^2 + (y+bi)^2 - (a-c)^2) * ((x+bi)^2 + (y+bi)^2 - (a+c)^2)

and

(x^2 + y^2 - (a-c)^2) * (x^2 + y^2 - (a+c)^2) * ((x+2bi)^2 + (y+2bi)^2 - (a-c)^2) * ((x+2bi)^2 + (y+2bi)^2 - (a+c)^2)

Maybe its possible to use this half real/complex equation, for the tiger?
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Re: Polynomials for Toratopes

Postby ICN5D » Sun Oct 19, 2014 6:07 am

Thinking about it some more, I'm pretty sure the complex component goes with the diameter value, and not the dimensions. I also thought of the ditorus empty cut:


(((II)I)I)

(sqrt((sqrt(x^2 + y^2 ) - R1)^2 + z^2 ) - R2)^2 + w^2 - R3^2 = 0


Cut into empty hole in 2D: ((()I)I)

(sqrt(( -a)^2 + x^2 ) - b)^2 + y^2 - c^2 = 0



Which expands into



Full empty cut

((x-b-ai)^2 + y^2 - c^2) * ((x+b-ai)^2 + y^2 - c^2) * ((x-b+ai)^2 + y^2 - c^2) * ((x+b+ai)^2 + y^2 - c^2)



Moving out from center to torus intercept, bringing ((I)I) into view, with an imaginary counterpart

((x-b)^2 + y^2 - c^2) * ((x+b)^2 + y^2 - c^2) * ((x-b+2ai)^2 + y^2 - c^2) * ((x+b+2ai)^2 + y^2 - c^2)
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Re: Polynomials for Toratopes

Postby ICN5D » Thu Oct 23, 2014 7:18 pm

Today, I ran a little experiment on complex conjugates and expressing empty cuts.


Complex Conjugate Hypothesis:

Complex conjugates can be used to locate intercepts in a higher dimension, outside an empty cut. Using the empty 1D torus cut of (()I) as an example, we assign the major diameter R as an imaginary value, Ri. At this time, I believe the imaginary component goes with the diameter, and not the axis, as in xi.


The degree-4 equation of a torus. Expressed in terms of x, y, and z intercepts in 1D:

(x±r±R)^4 + (y±r±R)^4 + (z±r±Ri)^4 - A = 0

A = Consolidation Compliment to cancel out repeat terms


(x±r±R)^4
(y±r±R)^4
(z±r±Ri)^4


• Expands into

(x-r-R)(x+r-R)(x-r+R)(x+r+R)

(y-r-R)(y+r-R)(y-r+R)(y+r+R)

(z-r-Ri)(z+r-Ri)(z-r+Ri)(z+r+Ri)




• Multiplies into:

For
(x-r-R)(x+r-R)
Code: Select all
      x     -r     -R

x    x^2    -rx    -Rx         

+r   rx     -r^2   -Rr             

-R  -Rx      Rr     R^2         


x^2 - r^2 + R^2 + rx - rx -2Rx + Rr - Rr

simplifies into

x^2 - r^2 + R^2 -2Rx




For
(x-r+R)(x+r+R)
Code: Select all
     x     -r    +R

x   x^2    -rx   Rx         

r    rx   -r^2   Rr           

R    Rx    -Rr   R^2             


x^2 - r^2 + R^2 + rx - rx + 2Rx + Rr - Rr

simplifies into

x^2 - r^2 + R^2 + 2Rx




• Now take both quadratics:

(x^2 - r^2 + R^2 -2Rx)(x^2 - r^2 + R^2 + 2Rx)
Code: Select all
         x^2        -r^2       R^2       -2Rx

x^2      x^4      -r^2x^2     R^2x^2    -2Rx^3 

-r^2   -r^2x^2       r^4     -R^2r^2     2Rr^2x

R^2     R^2x^2    -R^2r^2      R^4      -2R^3x

2Rx     2Rx^3     -2Rr^2x     2R^3x     -4R^2x^2


x^4 + r^4 + R^4 + 2Rx^3 - 2Rx^3 - 2r^2x^2 + 2R^2x^2 + 2Rr^2x + 2R^3x - 2R^3x - 2Rr^2x - 2R^2r^2 - 2R^2r^2 - 4R^2x^2

simplifies into

x^4 + r^4 + R^4 - 2r^2x^2 - 2R^2x^2 - 2R^2r^2




• Using equation above, apply to axes Y and Z. Simply put in place of X, and swap R for Ri in Z :

x^4 + r^4 + R^4 - 2r^2x^2 - 2R^2x^2 - 2R^2r^2

y^4 + r^4 + R^4 - 2r^2y^2 - 2R^2y^2 - 2R^2r^2

z^4 + r^4 + Ri^4 - 2r^2z^2 - 2Ri^2z^2 - 2Ri^2r^2




• For the Z intercepts polynome, apply i^2 = -1 , i^4 = 1

z^4 + r^4 + R^4 - 2r^2z^2 + 2R^2z^2 + 2R^2r^2

-- Effectively, we get a sign change, depending on exponent. The imaginary component is what makes Z different.




• Consolidate all three into one, recording which terms repeat to derive the Consolidation Compliment of A :

x^4 + r^4 + R^4 - 2r^2x^2 - 2R^2x^2 - 2R^2r^2

y^4 + r^4 + R^4 - 2r^2y^2 - 2R^2y^2 - 2R^2r^2

z^4 + r^4 + R^4 - 2r^2z^2 + 2R^2z^2 + 2R^2r^2


x^4 + y^4 + z^4 + r^4 + R^4 - 2r^2x^2 - 2r^2y^2 - 2r^2z^2 - 2R^2x^2 - 2R^2y^2 + 2R^2z^2 - 2R^2r^2



• Terms that repeat :

r^4 twice
R^4 twice
- 2R^2r^2 will cancel out with the + 2R^2r^2 , leaving the other - 2R^2r^2 behind

A = 2(R^4 + r^4)

So, referring to step one before expansion, the quartic equation for a torus may be expressible as:

(x±r±R)^4 + (y±r±R)^4 + (z±r±Ri)^4 - 2(R^4 + r^4) = 0



But before getting further into that, let's try graphing this equation, and see what we get.

x^4 + y^4 + z^4 + r^4 + R^4 - 2r^2x^2 - 2r^2y^2 - 2r^2z^2 - 2R^2x^2 - 2R^2y^2 + 2R^2z^2 - 2R^2r^2

simplifies into

x^4 + y^4 + z^4 + r^4 + R^4 - 2(rx)^2 - 2(ry)^2 - 2(rz)^2 - 2(Rx)^2 - 2(Ry)^2 + 2(Rz)^2 - 2(Rr)^2

R = 3
r = 1

Set diameter values:

x^4 + y^4 + z^4 + 1^4 + 3^4 - 2(1x)^2 - 2(1y)^2 - 2(1z)^2 - 2(3x)^2 - 2(3y)^2 + 2(3z)^2 - 2(3)^2

Graphs into:


Image



Makes interesting surface, topologically identical to a torus, but oblate in the 45 degree angles. Neat to see how using 1D intercepts can get close. There's some missing terms needed to smooth out and define a fully circular diameter. Hmm. But, since the resulting shape has a hole along Z, the theory is sound. Complex conjugates can be used to define empty cuts.

Some more research is needed on this. I think cross-multiplying the x,y,z products in step one may yield the correct terms (xy,yz,xz) required. They seem to be the only ones missing. In studying the shape produced from the equation, you can see perfect circles sitting right on the axes. So, perhaps by cross multiplying, the oblique 1D intercepts are defined, well enough to smooth out the oblique blobs.


What I have:
x^4 + y^4 + z^4 + r^4 + R^4 - 2(rx)^2 - 2(ry)^2 - 2(rz)^2 - 2(Rx)^2 - 2(Ry)^2 + 2(Rz)^2 - 2(Rr)^2

What it should be:
x^4 + y^4 + z^4 + R^4 + r^4 + 2(xy)^2 + 2(xz)^2 + 2(yz)^2 - 2(rx)^2 - 2(ry)^2 - 2(rz)^2 - 2(Rx)^2 - 2(Ry)^2 + 2(Rz)^2 - 2(Rr)^2 = 0


So, somehow, the x,y,z terms need to get multiplied together at some point, no pun intended.




Now that I know the equation needs more terms, the first mentioned form at the top would need the Oblique Compliment of 2[(xy)^2 + (xz)^2 + (yz)^2],


(x±r±R)^4 + (y±r±R)^4 + (z±r±Ri)^4 + 2((xy)^2 + (xz)^2 + (yz)^2) - 2(R^4 + r^4) = 0

Which expands into


(x-r-R)(x+r-R)(x-r+R)(x+r+R) + (y-r-R)(y+r-R)(y-r+R)(y+r+R) + (z-r-Ri)(z+r-Ri)(z-r+Ri)(z+r+Ri) + 2((xy)^2 + (xz)^2 + (yz)^2) - 2(R^4 + r^4) = 0

R = 3
r = 1

(x-1-3)(x+1-3)(x-1+3)(x+1+3) + (y-1-3)(y+1-3)(y-1+3)(y+1+3) + (z-1-3i)(z+1-3i)(z-1+3i)(z+1+3i) + 2((xy)^2 + (xz)^2 + (yz)^2) - 2(3^4 + 1^4) = 0



Which .....


Image


Graphs correctly!

Wow. Extremely freaking cool. This was a very neat exercise in deriving an equation. I've never constructed one from scratch, using 1D slices. Also, in exploring other forms of polynomials.

By far the most important result was the use of complex conjugates to define empty cuts and holes. I'll need to look into the cross multiplying of x,y,z , and try to find a sound method to get the oblique compliment of 2((xy)^2 + (xz)^2 + (yz)^2) . Perhaps it's just as simple as assigning oblique 1D angles. I have a much better understanding of how this degree-4 equation works, and why the terms are the way they are.
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Re: Polynomials for Toratopes

Postby PWrong » Tue Oct 28, 2014 1:12 am

Cool, so could a method like this work for any kind of shape expressed by an implicit equation? If not, it would be interesting to see what conditions need to be placed on the equation for it to work.
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Re: Polynomials for Toratopes

Postby ICN5D » Tue Oct 28, 2014 4:33 am

I'm not sure about any shape, but most definitely toratopes. The 1D intercept definition gets 90% of the shape, where the oblique regions need to be defined to smooth it out. Some later experiments with torisphere ((III)I) partially confirmed my theory on the 4D oblique compliment. I think it's generic for all toratopes, per dimension. That is, each dimension has its own group of complimenting terms, that works universally for all shapes. If so, that would be very useful and awesome. One thing is for sure, this strange polynomial has a very high degree of symmetry, which may extrapolate easily to higher dimensions. Something I've been searching for since starting this thread. But, parsing out all the +/- is tedious. Is there a way to naturally get a ± without having to fully expand it into + , - ?


My observations on the oblique compliment:

For 3D : All possible 2D planes, xy, xz, yz, in the format:

2((xy)^2 + (xz)^2 + (yz)^2)


For 4D: All possible 2D planes, extended in the same format ( verified with torisphere only )

2((xy)^2 + (xz)^2 + (xw)^2 + (yz)^2 + (yw)^2 + (zw)^2)


For 5D : All possible 2D planes

2((xy)^2 + (xz)^2 + (xw)^2 + (xv)^2 + (yz)^2 + (yw)^2 + (yv)^2 + (zw)^2 + (zv)^2 + (wv)^2)

So it would seem. Haven't gotten to 3-torus or tiger, yet.


But, the fact that the 1D intercepts made almost the entire shape, holes and everything, makes me think it can be taken further. That's an interesting thought. One would have to figure out the extra terms needed to fully define it, though. What other shapes/surfaces did you have in mind?

For the implicit of a toratope, I found the following method useful:

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


Deriving single axis solutions,

((x)) - four real solutions
(sqrt(x^2 ) - a)^2 - b^2
x = ± b ± a

((y)) - four real solutions
(sqrt(y^2) - a)^2 - b^2
y = ± b ± a

(()z) - four complex solutions
( - a)^2 + z^2 - b^2
z = ± b ± ai

One thing I realized lately, which became very clear in my mind, was that in terms of 1D solutions, the ring is real and the hole is imaginary. Of course, all points outside the torus and hole are imaginary as well. But, the hole within the ring is an imaginary region contained by a real region. Imaginary numbers are used to locate intercepts across empty higher dimensional space. Which also helps explain the sign change with Z, as being opposite of X and Y . This also means two distinct 5D toratopes, (((II)I)(II)) and (((II)(II))I) will have multicomplex solutions, with two imaginary components.

(((II)I)(II))
(sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2

solve for Z :

((()Z)())
(sqrt(( - a)^2 + z^2) - b)^2 + ( - c)^2 - d^2
Z = ± d ± b ± ai ± cj

Sixteen bicomplex solutions. But, when shifting by + ai or - ai, and + cj or - cj, four of the 16 can be brought into the real plane, making Z = ± d ± b .


And, for toratiger (((II)(II))I) :

((sqrt(x^2 + y^2) - a)^2 + (sqrt(z^2 + w^2) - b)^2 - c)^2 + v^2 - d^2 = 0

solve for V:

((()())V)
(( - a)^2 + ( - b)^2 - c)^2 + v^2 - d^2 = 0
V = ± d ± (c ± a)i ± (c ± b)j

Another case of sixteen bicomplex solutions. Two of the 16 come into the real plane by satisfying the shift of ± (c ± a)i, and ± (c ± b)j , making V = ± d . And to go even further, we get tricomplex solutions in 7D, and tetracomplex solutions in 9D. That's pretty cool!
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Re: Polynomials for Toratopes

Postby PWrong » Tue Oct 28, 2014 5:17 am

What I'm thinking is, if we try surfaces that aren't toratopes at all, we can get a better idea of how this works without going into higher dimensions. Probably some symmetries are required. We can easily say a lot about the symmetry of each toratope. The torus for example has rotational symmetry about the z axis.

Say we have an arbitrary implicit equation f(x,y,z) = 0, where f is a radical function (I'm not sure if this has a precise meaning, but let's say it's like a polynomial except that square roots are also allowed).

Then we solve the equations
f(x,0,0) = 0,
f(0,y,0) = 0,
f(0,0,z) = 0.

Let the solutions be x1, x2, ..., xn. Now I think we can always say that f(x,0,0) = 0 is equivalent to
(x - x1)(x - x2)...(x - xn) = 0

Now we "consolidate" these expressions (I'm still not sure what this means precisely), to get a new function g(x,y,z). The question is, how similar is g(x,y,z) to f(x,y,z), and what can we do to make them equal?
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Re: Polynomials for Toratopes

Postby ICN5D » Tue Oct 28, 2014 6:26 am

It'll be very similar to the original f(xyz), but missing some terms. Just like the first shape I graphed, it was a lumpy torus with a hole. Compared to a smooth torus, the largest areas of deviation were focused in the 45 degree angles, between axes. All that was needed were the three oblique terms. By consolidating, you're adding unlike terms, and tossing out repeating terms from expanding the product of solutions.


x^4 + r^4 + R^4 - 2r^2x^2 - 2R^2x^2 - 2R^2r^2

y^4 + r^4 + R^4 - 2r^2y^2 - 2R^2y^2 - 2R^2r^2

z^4 + r^4 + R^4 - 2r^2z^2 + 2R^2z^2 + 2R^2r^2


x^4 + y^4 + z^4 + r^4 + R^4 - 2r^2x^2 - 2r^2y^2 - 2r^2z^2 - 2R^2x^2 - 2R^2y^2 + 2R^2z^2 - 2R^2r^2

which after fully expanding, lacks the 45 degree oblique region terms, 2x^2y^2 , 2x^2z^2 , and 2y^2z^2 .


Just like here, we're tossing out the 2x repeats of R^4 and r^4 , and combining everything into one. The " + 2R^2r^2 " will cancel out with " - 2R^2r^2 " , leaving the other minus sign version behind.
Last edited by ICN5D on Tue Oct 28, 2014 6:41 am, edited 4 times in total.
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