Polynomials for Toratopes

[ICN5D]

Well, maybe that is the oblique compliment, if not some form of it. The full expansion of the root product sum, is that a consolidated one? Probably so, but I have to ask. But, nonetheless, it is still some really good experimental evidence.

Here's a good one: Are you able to use that same method to derive the others we know of? Like the 3D and 4D quartic? Technically speaking, by subtracting out the consolidated root product sum from the fully expanded real equation, we should get just the oblique compliment. If you can, then this is a step in the right direction I'd do this myself, were it not for still having the broken computer, and a now expired M10 subscription.

[PWrong]

I can use the ExpandToratope function to get the full polynomial of any toratope you like. Later on I'll write something you can view using CDF player, which is free.

Maybe we should look at what the sum will look like in general. First a definition: I think we should say the "order" of a toratope is the number of pairs of brackets. So quartics are order 2, octics are order 3 and so on. The "depth" of a toratope is the maximum number of brackets surrounding any variable. So ((II)(II)) is depth 2 but (((II)I)I) is depth 3. These definitions should go on the wiki.


Your conjecture is essentially this: every toratope of dimension n and order m can be described by an equation of the following form:

Product[(x_1 + A_i), i=1, i=8] + ... + Product[(x_n + A_i), i=1, i=8]
+ oblique compliment, which depends only on the variables x_1, ..., x_n
+ constants, containing none of the variables x_1, ..., x_n.

The constants A_i depend only on the toratope constants (e.g. a, b, r for the tiger). In fact we expect that each A_i is the sum of the toratope constants, each multiplied by a fourth root of unity, i.e 1, i, -1, -i.

So far, the conjecture is only confirmed for the quartic toratopes, m = 2. Really we don't have much reason to expect it to hold for higher orders. But having it clearly spelled out might help us prove it.

[Marek14]

I can use the ExpandToratope function to get the full polynomial of any toratope you like. Later on I'll write something you can view using CDF player, which is free.

Maybe we should look at what the sum will look like in general. First a definition: I think we should say the "order" of a toratope is the number of pairs of brackets. So quartics are order 2, octics are order 3 and so on. The "depth" of a toratope is the maximum number of brackets surrounding any variable. So ((II)(II)) is depth 2 but (((II)I)I) is depth 3. These definitions should go on the wiki.


Your conjecture is essentially this: every toratope of dimension n and order m can be described by an equation of the following form:

Product[(x_1 + A_i), i=1, i=8] + ... + Product[(x_n + A_i), i=1, i=8]
+ oblique compliment, which depends only on the variables x_1, ..., x_n
+ constants, containing none of the variables x_1, ..., x_n.

The constants A_i depend only on the toratope constants (e.g. a, b, r for the tiger). In fact we expect that each A_i is the sum of the toratope constants, each multiplied by a fourth root of unity, i.e 1, i, -1, -i.

So far, the conjecture is only confirmed for the quartic toratopes, m = 2. Really we don't have much reason to expect it to hold for higher orders. But having it clearly spelled out might help us prove it.

The degree is also related to the trace of the toratope: degree = degree of the trace * number of toratopes in the trace. Degree of a point is taken as 1. Tiger has trace of four circles, so its degree is 4 x 2 = 8.

[PWrong]

What are trace and degree here?

[Marek14]

What are trace and degree here?

Degree - degree of polynomial, 2^order.

Trace is the lowest non-empty cut of toratope. For tiger, it's four circles in 2x2 array because tiger doesn't intersect any coordinate axes, but it does intersect some planes. For torus, torisphere, spheritorus and anything from that family, it's four points in a line because you will always have an axis that intersects the toratope in four points.

[ICN5D]

All cuts in fact will relate to the degree. Empty and non-empty cuts will still produce the correct number of intercepts, expressed as a product of lower degrees, which will add up to the full degree of the full toratope.

[PWrong]

I found a short list in another thread. Are there any toratopes with a trace that is more interesting than an array of identical n-spheres?

What's the trace of (tigertiger) = (((II)(II))((II)(II)))?
I can see that it's (((I)(I))((I)(I))), but what's that geometrically?

[ICN5D]

That one would be 16 tigers in a 2x2x2x2 tesseract array, which intercepts no lower than 4D. All 3D cuts slide into a hole. I think we call that one a duotiger tiger. To generate a toratope with a trace array of lower toratopes of your choice, simply replace all single dimensions with a circle (II).

For trace of 16 ditoruses (((II)I)I) in tesseract array, we get a ((((II)(II))(II))(II)), which cuts down to ((((I)(I))(I))(I)).
For a trace of 32 toritigers (((II)(II))I) in penteract array, we get the ((((II)(II))((II)(II)))(II)), which cuts down to ((((I)(I))((I)(I)))(I))
For a trace of 32 tiger toruses (((II)I)(II)) in penteract array, we get the ((((II)(II))(II))((II)(II))), which cuts down to ((((I)(I))(I))((I)(I)))

[PWrong]

OK I think I've got it. It seems easier if you visualise each toratope as a tree. The leaves are the variables and each node is a pair of parenthesis. To get the trace of a toratope T, first chop off all the leaves to get an expression T' (which is not yet a toratope). Then write the depth of each new leaf. This gives you a list of numbers d_1, d_2, ...
Now remove redundant brackets from T' to get a toratope T''. The trace of T is a 2^{d_1} x 2^{d_2} x... array of T''.

So if we take ( ((((II)I)II) ((II)(II))) ((II)II) (II)), the trace should be a 16x8x8x4x2 array of ((I(II))II)'s. Does that seem right?

Incidentally, I just realised that in Mathematica you can write TreeForm[{{1,1},{1,1}}] to express a tiger as a tree. There are a bunch of functions I could use to create a ToratopeTrace function.

[Marek14]

OK I think I've got it. It seems easier if you visualise each toratope as a tree. The leaves are the variables and each node is a pair of parenthesis. To get the trace of a toratope T, first chop off all the leaves to get an expression T' (which is not yet a toratope). Then write the depth of each new leaf. This gives you a list of numbers d_1, d_2, ...
Now remove redundant brackets from T' to get a toratope T''. The trace of T is a 2^{d_1} x 2^{d_2} x... array of T''.

So if we take ( ((((II)I)II) ((II)(II))) ((II)II) (II)), the trace should be a 16x8x8x4x2 array of ((I(II))II)'s. Does that seem right?

Incidentally, I just realised that in Mathematica you can write TreeForm[{{1,1},{1,1}}] to express a tiger as a tree. There are a bunch of functions I could use to create a ToratopeTrace function.

As for your example:
(((((II)I)II)((II)(II)))((II)II)(II))
First, we'll "deflate" it by omitting all I's except by leaving one in each innermost pair of parentheses:
(((((I)))((I)(I)))((I))(I))
Next, we reduce parentheses with only a single pair of parentheses as a term:
(((I x3)((I x1)(I x1)))(I x2)(I x1)) - x n means "was originally in n nested pairs"
By removing parentheses around single I's, we get the shape of the trace:
((I(II))II)
Now we can get the trace by putting back the multipliers. This will be a 8x2x2x4x2 array.

Yes, it's similar to a tree representation. Basically, the exact form of the trace determines the parentheses skelet, or "species" of toratope -- all toratopes of form (((x)y)z) are ditoruses and have the same trace of eight points on a line, for example. But then you can consider a class of all toratopes with the same toratope as their trace, no matter how many they are or how they are put (a genus), you get toratopes with the same shape of their tree that differ just in length of various branches.

For example, a double tiger (((II)(II))(II)) has a trace in the shape of torus. The basic trace is a 2x2x2 array, but you can have a look at more complicated members of this genus:
Torus double tiger: ((((II)I)(II))(II)) - 4x2x2 array
Tiger/torus tiger: (((II)(II))((II)I)) - 2x2x4 array
Tiger torus tiger: ((((II)(II))I)(II)) - 2x2x2 array of major pairs
Double tiger torus: ((((II)(II))(II))I) - 2x2x2 array of minor pairs

[PWrong]

So I was right about the shape but completely wrong about the array. I get it now, you count up from the leaf to the next nontrivial node.

Here is the tree for the toratope:


Here's the tree for the trace:


And here's the tree for the trace toratope:

[Marek14]

Seems good, yes.

[PWrong]

I just looked back at this conjecture:

Every toratope of dimension n and order m can be described by an equation of the following form:

Product[(x_1 + A_i), i=1, i=8] + ... + Product[(x_n + A_i), i=1, i=8]
+ oblique compliment, which depends only on the variables x_1, ..., x_n
+ constants, containing none of the variables x_1, ..., x_n.

and then at the fully expanded polynomial for the tiger:

Code: Select all

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

The conjecture is clearly impossible. There are terms in the expanded polynomial for the tiger, such as 12 b^2 w^2 z^4, which are a product of two different variables AND the constants. These terms aren't in any of the three components of the conjecture. There will have to be several other components.

[ICN5D]

Yeah, you're right. Look at that. Hmm. Well, I wonder why the quartics are so simple, and generalize easily into higher dimensions? Surely there can't be that much of a difference in the oblique terms for octics, just a new pattern. One that combines the dimensions with diameters.

Maybe we could take a stab at simplifying the terms, that were derived in your earlier post , to see if there's a relation?

Is that freeware you mentioned before able to work with the tiger polynome? I'd like to test various ideas on the oblique terms, if not more. I keep thinking, if we could find a good, easy to follow pattern here, with this polynomial, then it would lead to writing degree 16's in 5D, and deg-32 in 6D, which are insanely complex. It's kind of my intuition with it, but who knows.

Observation with the terms so far:

• Anything with a single dimension or diameter, alone or combined with one or more diameter terms, come from the root product sum, after consolidation , i.e. : x^8 , x^4 a^2 b^4 , a^8 , etc

• Anything with multiple dimensions with one or more diameter terms come from the oblique compliment . i.e. : x^4 y^6 , a^2 b^4 x^4 z^2 , x^6 y^2 w^4 , etc


So, this being the case, oblique terms for an octic surface will have diameter terms, and up to 3 dimension terms combined. Plus, the oblique terms are no higher than degree-8. This sets some rules to follow, when searching for it, which is good to know.

[PWrong]

Is that freeware you mentioned before able to work with the tiger polynome?

All that CDF player does is let you view mathematica files that someone else has made, without changing anything. I might be able to write something that lets the reader change parameters using sliders or something. But I'll be on holiday until the 4th of December.

Here's a good one: Are you able to use that same method to derive the others we know of? Like the 3D and 4D quartic?

I have an ExpandToratope function that was meant to do that. But it doesn't seem to work for (((II)I)I) or anything more complicated than a tiger. I had to expand the rest manually.

Ditorus is
-(x^2 + y^2) (4 a^2 c + 4 b^2 c - 4 c^3 - 4 c w^2 - 4 c x^2 - 4 c y^2 - 4 c z^2)^2 + (a^4 - 2 a^2 b^2 + b^4 - 2 a^2 c^2 - 2 b^2 c^2 + c^4 - 2 a^2 w^2 + 2 b^2 w^2 + 2 c^2 w^2 + w^4 - 2 a^2 x^2 - 2 b^2 x^2 + 6 c^2 x^2 + 2 w^2 x^2 + x^4 - 2 a^2 y^2 - 2 b^2 y^2 + 6 c^2 y^2 + 2 w^2 y^2 + 2 x^2 y^2 + y^4 - 2 a^2 z^2 - 2 b^2 z^2 + 2 c^2 z^2 + 2 w^2 z^2 + 2 x^2 z^2 + 2 y^2 z^2 + z^4)^2

[PWrong]

Here's the ditorus simplified.

-16 (x^2 + y^2) c^2 (x^2 + y^2 + z^2 + w^2 + c^2- a^2 - b^2)^2 + ((a - b - c) (a + b - c) (a - b + c) (a + b + c) - 2 (b^2 (x^2 + y^2 + z^2 - w^2) + a^2 (x^2 + y^2 + z^2 + w^2) - c^2 (x^2 + y^2 + z^2 + w^2) - 2 c^2 (x^2 + y^2)) + (x^2 + y^2 + z^2 + w^2)^2)^2

There are a few parallels with the tiger:
-64 b^2 r^2 (x^2 + y^2) (z^2 + w^2) + ((-a^2 + b^2 + r^2)^2 + 2 (r^2 - b^2 ) (x^2 + y^2 - z^2 - w^2) - 2 a^2 (x^2 + y^2 + z^2 + w^2) + (x^2 + y^2 + z^2 + w^2)^2)^2.


Here's ((xy)(zw)v). There's definitely some patterns here.
-64 b^2 r^2 (x^2 + y^2) (z^2 + w^2) + ((-a^2 + b^2 + r^2)^2 + 2( r^2 - b^2) (x^2 + y^2 - z^2 - w^2) - 2 a^2 (v^2 + w^2 + x^2 + y^2 + z^2) + 2 ( b^2 + r^2) v^2 + (v^2 + x^2 + y^2 + z^2 + w^2)^2)^2

[PWrong]

EDIT: I mixed up some of the radii in probably every post so far. I've corrected it in this one, but the previous posts are wrong. They had 'a' as the minor radius, 'b' and 'r' as the two major radii. In this post, 'r' is the minor radius.

I've got all the toratopes of the form ((m)(n)k), such as ((IIIII)(III)II).

Let
X = x_1^2 + ... + x_m^2
Y = x_{m+1}^2 + ... + x_{m+n}^2
Z = x_{m+n+1}^2 + ... + x_{m+n+k}^2.

Then the expanded polynomial is
-64 a^2 b^2 X Y + ((-r^2 + b^2 + a^2)^2 + 2 (a^2 (-X + Y + Z) + b^2 (X - Y + Z) - r^2 (X + Y + Z)) + (X + Y + Z)^2)^2.

[PWrong]

There are two categories of octic toratope: tiger type ((m)(n)k), and ditorus type (((m)n)k).

Tiger type is covered above, and I'll do ditorus type here.

Let
X = x_1^2 + ... + x_m^2
Y = y_1^2 + ... + x_n^2
Z = z_1^2 + ... + z_n^2

Then (((m)n)k) is described by:
(sqrt((sqrt(X) -a)^2 + Y) - b)^2 + Z = c

The expanded polynomial is:
-16 a^2 X (a^2 - b^2 - c^2 + X + Y + Z)^2 + ((a - b - c) (a + b - c) (a - b + c) (a + b + c) + 2 (a^2 (3 X + Y + Z) - b^2 (X + Y - Z) - c^2 (X + Y + Z)) + (X + Y + Z)^2)^2

[PWrong]

Applying the same strategy to quartic toratopes, we would get
-4 a^2 X + (-a^2 + r^2 - X - Y)^2.

How many types of order 4 toratopes are there? I can think of these:

((a)(b)(c)d), e.g. ((II)(II)(II)I)
(((a)b)(c)d), e.g. (((II)I)(II)I)
(((a)(b)c)d), e.g. (((II)(II)I)I)
((((a)b)c)d), e.g. ((((II)I)I)I)

[Marek14]

Well, an order 4 toratope will be created by replacing a I in order 3 toratope with (II) (remembering that extra I's can always be added), so...

Order 1 - (II)
Order 2 - ((II)I)
Order 3 - (((II)I)I) or ((II)(II))
Order 4 - ((((II)I)I)I), (((II)(II))I), (((II)I)(II)) or ((II)(II)(II))
Order 5 - (((((II)I)I)I)I), ((((II)(II))I)I), ((((II)I)(II))I), ((((II)I)I)(II)), (((II)(II)(II))I), (((II)(II))(II)), (((II)I)((II)I)), (((II)I)(II)(II)) or ((II)(II)(II)(II))
Order 6 - ((((((II)I)I)I)I)I), (((((II)(II))I)I)I), (((((II)I)(II))I)I), (((((II)I)I)(II))I), (((((II)I)I)I)(II)), ((((II)(II)(II))I)I), ((((II)(II))(II))I), ((((II)(II))I)(II)), ((((II)I)((II)I))I), ((((II)I)(II)(II))I), ((((II)I)(II))(II)), ((((II)I)I)((II)I)), ((((II)I)I)(II)(II)), (((II)(II)(II)(II))I), (((II)(II)(II))(II)), (((II)(II))((II)I)), (((II)(II))(II)(II)), (((II)I)((II)I)(II)), (((II)I)(II)(II)(II)) or ((II)(II)(II)(II)(II))

[PWrong]

Cool, so it's this sequence. http://mathworld.wolfram.com/RootedTree.html

I was visualising these toratopes as a tree with n nodes and each node having lots of leaves, with the number of leaves on each node being a, b, c...

[Marek14]

Cool, so it's this sequence. http://mathworld.wolfram.com/RootedTree.html

I was visualising these toratopes as a tree with n nodes and each node having lots of leaves, with the number of leaves on each node being a, b, c...

Yes, rooted trees, since those show possible nestings of parentheses.

[ICN5D]

Excellent work, awesome. I've been curious about ditorus polynomial. Also, nice symmetry in these Both degree-8's have a similar looking layout.

So, ExpandToratope function won't go beyond tiger? Well, we could consolidate all 3D cuts of (((II)I)(II)), which should make full expanded polynomial.


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

4D
(((x)z)(wv)) = (sqrt((sqrt(x^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(((y)z)(wv)) = (sqrt((sqrt(y^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(((xy))(wv)) = (sqrt((sqrt(x^2 + y^2) - a)^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(((xy)z)(w)) = (sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2) - c)^2 - d^2
(((xy)z)(v)) = (sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - b)^2 + (sqrt(v^2) - c)^2 - d^2


The cut algorithm will tell us which dimensions are stacked along, and which diameters are paired, for a torus

3D : torus (x^2 + y^2 + z^2 + a^2 - b^2)^2 - 4a^2(x^2 + y^2)

1. ((()z)(wv)) = (sqrt((- a)^2 + z^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(w^2 + v^2 + (z±b±ai)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2)

2. (((x))(wv)) = (sqrt((sqrt(x^2) - a)^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(w^2 + v^2 + (x±a±b)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2)

3. (((y))(wv)) = (sqrt((sqrt(y^2) - a)^2) - b)^2 + (sqrt(w^2 + v^2) - c)^2 - d^2
(w^2 + v^2 + (y±a±b)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2)

4. (((x)z)(w)) = (sqrt((sqrt(x^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2) - c)^2 - d^2
((x±a)^2 + z^2 + (w±c)^2 + b^2 - d^2)^2 - 4b^2((x±a)^2 + z^2)

5. (((x)z)(v)) = (sqrt((sqrt(x^2) - a)^2 + z^2) - b)^2 + (sqrt(v^2) - c)^2 - d^2
((x±a)^2 + z^2 + (v±c)^2 + b^2 - d^2)^2 - 4b^2((x±a)^2 + z^2)

6. (((y)z)(w)) = (sqrt((sqrt(y^2) - a)^2 + z^2) - b)^2 + (sqrt(w^2) - c)^2 - d^2
((y±a)^2 + z^2 + (w±c)^2 + b^2 - d^2)^2 - 4b^2((y±a)^2 + z^2)

7. (((y)z)(v)) = (sqrt((sqrt(y^2) - a)^2 + z^2) - b)^2 + (sqrt(v^2) - c)^2 - d^2
((y±a)^2 + z^2 + (v±c)^2 + b^2 - d^2)^2 - 4b^2((y±a)^2 + z^2)

8. (((xy))(v)) = (sqrt((sqrt(x^2 + y^2) - a)^2) - b)^2 + (sqrt(v^2) - c)^2 - d^2
(x^2 + y^2 + (v±c)^2 + (a±b)^2 - d^2)^2 - 4(a±b)^2(x^2 + y^2)

9. (((xy))(w)) = (sqrt((sqrt(x^2 + y^2) - a)^2) - b)^2 + (sqrt(w^2) - c)^2 - d^2
(x^2 + y^2 + (w±c)^2 + (a±b)^2 - d^2)^2 - 4(a±b)^2(x^2 + y^2)

10. (((xy)z)()) = (sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - b)^2 + (- c)^2 - d^2
(x^2 + y^2 + z^2 + a^2 - (b±d±ci)^2)^2 - 4a^2(x^2 + y^2)


(((II)I)(II)) will have 10 solutions in 3D, eight are real and two are complex. All solutions factor out into four torus intercepts:

((w^2 + v^2 + (z±b±ai)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2))^4
((w^2 + v^2 + (x±a±b)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2))^4
((w^2 + v^2 + (y±a±b)^2 + c^2 - d^2)^2 - 4c^2(w^2 + v^2))^4
(((x±a)^2 + z^2 + (w±c)^2 + b^2 - d^2)^2 - 4b^2((x±a)^2 + z^2))^4
(((x±a)^2 + z^2 + (v±c)^2 + b^2 - d^2)^2 - 4b^2((x±a)^2 + z^2))^4
(((y±a)^2 + z^2 + (w±c)^2 + b^2 - d^2)^2 - 4b^2((y±a)^2 + z^2))^4
(((y±a)^2 + z^2 + (v±c)^2 + b^2 - d^2)^2 - 4b^2((y±a)^2 + z^2))^4
((x^2 + y^2 + (v±c)^2 + (a±b)^2 - d^2)^2 - 4(a±b)^2(x^2 + y^2))^4
((x^2 + y^2 + (w±c)^2 + (a±b)^2 - d^2)^2 - 4(a±b)^2(x^2 + y^2))^4
((x^2 + y^2 + z^2 + a^2 - (b±d±ci)^2)^2 - 4a^2(x^2 + y^2))^4


Expanding and consolidating all 10 possible 3D intercepts of (((II)I)(II)) should produce full polynomial, however frightening that might be ....

[PWrong]

I have some ideas on how to improve the ExpandToratope function so that it will work for any polynomial featuring square roots. I was trying ((II)(II)(II)) yesterday. It's easy enough to get the polynomial, the hard part is simplifying it to something reasonable.

[PWrong]

I just had an idea, but it might be hard to explain in my phone. Let T be a toratope described by

T = T(x1,x2,..., a1,a2,...) = r.

Then the equation for (T1) is
(T-r)^2 + y^2 = R^2.

Rearrange to
T = r + sqrt(R^2 - y^2).

So if we take the expanded polynomial for T, which will have lots of r's floating about, and replace all of those r's with r + sqrt(R^2 - y^2), then that will describe (T1). Then we just do one expanding operation to remove that one square root.

[ICN5D]

Hmm, maybe make something like a SimplifyToratope function, that will recognize a symmetric subgroup, then organize accordingly? Easier said than done, probably. Or, did you use a systematic ( and programmable ) method , for ditorus and tiger?

[PWrong]

There are some very powerful and useful built in functions like Simplify, Expand and Collect. I'll use a combination of those to fix the ToratopeExpand function.

It seems like we want a function that separates an expression into three parts: a function of a,b,... only, a function of x,y,... only, and a function of both. That might not be that hard to write.

[PWrong]

I briefly tried the sedectic (16th power) toratopes. The only one that I could get down to a reasonable size is ((II)(II)(II)). It looks like having extra levels makes things much more complicated.

Following this page, we define

Code: Select all

RootExpand[{a_, b_}, x_] := (x^2 - SymmetricPolynomial[1, {a, b}])^2 - 4 SymmetricPolynomial[2, {a, b}]
RootExpand[{a_, b_, c_},  x_] := ((x^2 - SymmetricPolynomial[1, {a, b, c}])^2 - 4 SymmetricPolynomial[2, {a, b, c}])^2 - 64 SymmetricPolynomial[3, {a, b, c}] x^2
RootExpand[{a_, b_, c_, d_},  x_] :=
(((x^2 - S[1, 4])^2 - 4 S[2, 4])^2 - 64 S[2, 4] x^2)^2 - S[4, 4] (64 (x^2 - S[1, 4]) x^2 - 16 (x^2 - S[1, 4])^2 + 64 S[2, 4])^2


This function RootExpand solves equations of the form sqrt(a) + sqrt(b) + sqrt(c) = x. Sorry the variables are a bit misleading.

Now ((II)(II)(II)) has function

Code: Select all

In[30]:= ToratopeFunctionVar[{{1, 1}, {1, 1}, {1, 1}}][r, a, b, c][x1, x2, y1, y2, z1, z2]

Out[30]= -r + Sqrt[(-a + Sqrt[x1^2 + x2^2])^2 + (-b + Sqrt[y1^2 + y2^2])^2 + (-c + Sqrt[z1^2 + z2^2])^2]

We can quickly expand and rearrange this to
a^2 + b^2 + c^2 - r^2 + X + Y + Z == 2 a Sqrt[X] + 2 b Sqrt[Y] + 2 c Sqrt[Z].

Now we can use the RootExpand function to get rid of the square roots:

Code: Select all

In[34]:= RootExpand[{4 a^2 X, 4 b^2 Y, 4 c^2 Z},
a^2 + b^2 + c^2 - r^2 + X + Y + Z]

Out[34]= -4096 a^2 b^2 c^2 X Y Z (a^2 + b^2 + c^2 - r^2 + X + Y +
    Z)^2 + (-4 (16 a^2 b^2 X Y + 16 a^2 c^2 X Z +
      16 b^2 c^2 Y Z) + (-4 a^2 X - 4 b^2 Y -
     4 c^2 Z + (a^2 + b^2 + c^2 - r^2 + X + Y + Z)^2)^2)^2

This basic method will work for any toratope of depth 2. For any more depth, we'll need to apply RootExpand multiple times.

[ICN5D]

Whoa, look at that thing. Minor rearrangement, and establish X, Y, Z,

{[(x²+y²+z²+w²+v²+u²+a²+b²+c²-r²)² -4a²(x²+y²) -4b²(z²+w²) -4c²(v²+u²)]² - 4[16a²b²(x²+y²)(z²+w²) + 16a²c²(x²+y²)(v²+u²) + 16b²c²(z²+w²)(v²+u²)]}² - 4096a²b²c²(x²+y²)(z²+w²)(v²+u²)(x²+y²+z²+w²+v²+u²+a²+b²+c²-r²)²

I'll have to admit, my mind really likes these patterns. Such awesome symmetry. And just think, it describes a six dimensional, smoothly curving, hypertoric ring.

Solving for the 3-plane xzv will cancel out ywu, making a real solution. Setting major diameters a,b,c to 3, and minor diameter r to 1. Setting y -> a , w -> b , u -> c , we get

(((x^2+a^2+y^2+b^2+z^2+c^2+3^2+3^2+3^2-1^2)^2 -4*3^2*(x^2+a^2) -4*3^2*(y^2+b^2) -4*3^2*(z^2+c^2))^2 - 4(16*3^2*3^2*(x^2+a^2)(y^2+b^2) + 16*3^2*3^2*(x^2+a^2)(z^2+c^2) + 16*3^2*3^2*(y^2+b^2)(z^2+c^2)))^2 - 4096*3^2*3^2*3^2*(x^2+a^2)(y^2+b^2)(z^2+c^2)(x^2+a^2+y^2+b^2+z^2+c^2+3^2+3^2+3^2-1^2)^2


And graphing such monstrosities,




Makes eight spheres in a 2x2x2 cube array. Awesome, good to see it's working!

[PWrong]

Here's a more general RootExpand function defined inductively, using the second last equation from the sums of square roots page. Not that the square roots will always cancel out.

Code: Select all

RootExpand[{a_}, x_] := x^2 - a

RootExpand[l_, x_] :=
   Sum[
      Sum[
         CoefficientList[RootExpand[Most[l], x], x][[i + 1]] CoefficientList[RootExpand[Most[l], x], x][[j + 1]] (x + Sqrt[Last[l]])^i (x - Sqrt[Last[l]])^j,
     {j, 0, 2^(Length[l] - 1)}],
   {i, 0, 2^(Length[l] - 1)}]