visual ruminant

The Hofstadter Transcendental Triangle Variety

Owen Maresh — Thu, 23 Feb 2012 23:41:54 +0000

The Hofstadter point is a transcendental triangle center. So this gets me thinking: well, if I have three noncollinear complex numbers, that’s a triangle, right? And for three complex numbers I can treat them as elements of

So here’s the recipe: the Hofstadter Transcendental Triangle Variety is the set of points in such that given some point , the “Hofstadter mean” of those three points is .

Challenge of the day

Owen Maresh — Fri, 10 Feb 2012 11:55:06 +0000

Evaluate

complex solutions of diophantine polynomials and trees

Owen Maresh — Thu, 02 Feb 2012 00:41:14 +0000

Chaitin has implemented an exponential polynomial that more or less is a lisp implementation, and Yuri Matiyasevich implemented one in the seventies that was used in his proof to answer Hilbert’s 10th problem in the negative.

In Baez’s This Week’s Finds, week 202, an isomorphism between trees and seven tuples of trees is noted.

This makes me wonder: what could we do with complex solutions of Chaitin’s and Matiyasevich’s polynomials?

logarithmic trickery

Owen Maresh — Fri, 13 Jan 2012 15:35:00 +0000

Consider that since , and therefore, somewhat unexpectly:
we can use it to construct new identities thusly

Let

Let

Therefore, since we know that we can use that
to explicitly calculate a new identity thusly:

1. Compute the logarithm of , :

2. Compute the logarithm of , :

3. Combine them:

>>> A = fp.nprod(lambda n: sqrt(1+1/(n*n)), [1,inf])
>>>
>>> A
mpf(’1.9109509100512501′)
>>> B = fp.nprod(lambda m: (1+1/(m**3))**(1/exp(m)), [1,inf])
>>>
>>> B
mpf(’1.3140291251423164′)
>>> A**log(B)
mpf(’1.1934623097049237′)
>>> B**log(A)
mpf(’1.1934623097049237′)

dreaming about reading mathematics books

Owen Maresh — Tue, 10 Jan 2012 11:53:29 +0000

one: It is a thick, yellow, Springer-Verlag dealie, illustrated and about fractals. There is a color plate on nearly every page. I dreamt of being in a bookshop and considering buying it.

two: has lots of equations at the beginning, and they change as I read them. last page has a distorting/metamorphosing picture of einstein/feynman in green and blue hues.

three: also a text that changes, I remember a diagram of helix against a point scatter. Also changed.

last night: I remember something about the Euler-Mascheroni constant . Impenetrable references section. Also picture of fractals.

Have you dreamt of reading mathematics books? If so, leave a comment about them.

Sprott decoder (now with quartic and quintic transformations)

Owen Maresh — Tue, 10 Jan 2012 08:53:30 +0000

here is the source, compile with
gcc syllegy.c -o syllegy -lm

zeta theta hybrideque

Owen Maresh — Thu, 05 Jan 2012 22:01:08 +0000

Define . Note that while resembling the Riemann zeta function , the exponential term suppresses the pole at

the banshee

Owen Maresh — Sun, 01 Jan 2012 13:04:16 +0000

I had this thought in the shower, and it descends from my “symmetry may be a long term red herring” idea, and it goes like this:

Imagine that you have something quite like the Monster group , except that there is a fixed pair of elements in this thing like — let’s call it for banshee. Every time you take two elements from and calculate their product, there is a chance that you’ll be given something which is not an element of , like a real number or some other object.

Let’s say it transpires that the boogeyman actually has some utility: if you happen to have a hangup on symmetry, your apprehension of the Monster group is going to obscure the existence of the banshee to you.

Interesting, because of the sheer size of the order of the Monster, it is easier to confuse this banshee with a group than if you were to take Klein’s viergruppe or a cyclic group and replace one of it’s elements. The banshee is not a group

interpolating the primorials

Owen Maresh — Sun, 01 Jan 2012 12:48:22 +0000

Here’s a puzzle: consider that the Gamma function is the interpolation of the factorials. What about interpolating the primorials. If is the $n$-th prime, then what is the correct analogue of the Gamma function if we define the primorial as:

mockup of special functions video

Owen Maresh — Sun, 01 Jan 2012 07:07:26 +0000

This is something I’m working on: