More thoughts on floretion algebra program
creigh at o2online.de
creigh at o2online.de
Sat Sep 18 12:25:24 CEST 2004
Hello,
This is my guess:
If we could get the FAMP program to run "in reverse", we would have
an extremely handy, perhaps even unique kind of tool at our
disposal.
By this is meant:
One enters in a sequence, say the Fibonacci numbers. Then
"FAMP-Superieur" would search for four sets
JES(Fib) = {j1, j2, j3, ....}
LES(Fib) = {l1, l2, l3, ....}
TES(Fib) = {t1, t2, t3, ....}
VES(Fib) = {v1, v2, v3, ....} subset floretion (real) algebra
where
jes( (j1)^n ) = jes( (j2)^n ) = jes( (j3)^n ) = ... = Fib(n)
les( (l1)^n ) = les( (l2)^n ) = les( (l3)^n ) = ... = Fib(n)
tes( (t1)^n ) = tes( (t2)^n ) = tes( (t3)^n ) = ... = Fib(n)
les( (t1)^n ) = les( (t2)^n ) = les( (t3)^n ) = ... = Fib(n)
Specifically, this would mean solving a system a polynomial
equations. My guess at a brute force method- i.e. just plugging in
numbers and hoping these lead to the sequence at hand, is on the order
of magnitude of 60^(16) checks for "interesting" sequences; this is a
number which could be either greatly reduced or increased for any
number of reasons. One also has the option of a "smart bruce
force" method which only checks those elements which
have some sort of symmetric representation on either
Floret's cube or star (basically, this is the only course I've
followed to now).
For a variety of reasons, it would be nice to know if the above sets were
"closed" in some sense. Example:
Let sig(x) be the swap operator, which reverses all "arrows" of a
floretion (if x = i' + 'j, then sig(x) = 'i + j' )
Fib(n) = ves( (Ex)^(n+1) ) = ves( (E(sig(x)))^(n+1) )
E = ( - i' - i' - 'ii' + 'jj' + 'kk' + 'jk' + kj' - 1 )/4
x = 'i + 'i + 'ji' + 'ki' + 1
sig(x) = i' + i' + 'ij' + 'ik' + 1
[ Note: sig(E) = E ]
It follows that Ex and E(sig(x)) in VES(Fib).
Then surely (Ex)E(sig(x)) should also have something
to do with the Fibonacci numbers. Indeed, we have
ves( ( (Ex)E(sig(x)) )^(n+1) ) = ves((ExEx)^(n+1))
= ves((Ex)^(2n+2)) = = Fib(2n+1)
In addition,
ves( ( Ex + E(sig(x)) )^n ) = ves( ( E(x + sig(x)) )^n ) =
= http://www.research.att.com/projects/OEIS?Anum=A077020
(disregarding signs)
Additional commentary:
-FAMP has also led to a nonabelian, countably infinite group.
Check out my rhyme at
http://mathforum.org/discuss/sci.math/m/633678/634274
-The program was also updated yesterday to include the
swap operator "sig", discussed above. In addition,
A055997(n+1) + A053141(n+1) = A001541(n+1) + A001109(n+1)
was also found.
Sincerely,
Creighton
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