PWrong wrote: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.
PWrong wrote:What are trace and degree here?
PWrong wrote: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.
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.
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.
Is that freeware you mentioned before able to work with the tiger polynome?
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?
PWrong wrote: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...
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
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]
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
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)}]
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