Well, we’ve got full image, essential image, and from optics a whole slew of machinery for dealing with images of things. Virtual images, focal depth, whatnot.

* Do functors have focal depth?

* Are there things which are not quite categories in which a functor could be diffracted?

* Think about imaging technologies: MRI, cat scans — anything which we use to construct visual representations of things might have a functorial analogue. Functorial holography? (and no, I don’t mean this: “holographic principle of higher category theory”

* Just how many of these w: optical phenomena have functorial analogues?

A. it would be good if it could Q or SL(2,Z) highlight images.

B. It would be nice to have a mathematician orientated special functions/fractal platform running over cuda, one that could make things as easy as Adobe Illustrator in some ways.

2. You’ve got a copy of Wegert’s Visual Complex Functions and NIST Handbook of Mathematical Functions, right? Also worth looking at is the Bateman project, Abramowitz and Stegun, Gasper’s book about hypergeometric series, and the Tata lectures on theta functions. It would be nice to start working through that material.

3. You’ve read Jim Belk and Bradley Forrest’s A Thompson Group for the Basilica… it would be nice if we could make videos of these transformations for arbitrary things.

4. Do not trust one renderer. That having been said, we need to agree on a hue model (which is mostly settled at the present?) Why? So we can encode sheaf-theoretic information in it! (Nevanlinna theoretic information as well)

5. If you write a paper about some transcendental meromorphic function which varies on a parameter, make a movie and post it to youtube along with the paper, or on your site.

6. This ‘dip it in formaldehyde, move like insects’ is dangerously slow.

7. Moar pictures. And don’t list points. Make a picture.

8. It would be nice if we could figure out a way for the nonprofessional mathematicians (Burner/Fractaltribe types) to contribute in a meaningful way. But at the moment everything is so Balkanized that there are a lot of disparate communities making very very slow progress on one thing or another.

(I’m sort of at the saddle point between Burners and complex analysts, which makes talking to either somewhat aggravating)

]]>(just imagine this extends to infinity…)

probably has some meaning in Teichmuller theory, but I’m a complex analyst, and want to string lacunaries across those circles, and insist that we can only link the pants when the lacunaries match.

Since the limit starting from a nontrivial root going parallel to the real axis of the zeta function is 1, we may subtract that off so that it converges.

144 evaluations, text file, done with mpmath

What’s amazing, is that when the values of this integral corresponding to nontrivial roots of the zeta function are plotted, some interesting structure revealed:

]]>S = f(z/s)

s = f(S/z)

It doesn’t actually have to be division, just some inverse process, and because I am fond of giving bright and strange names to things, I propose that we call these functions *hā.ʻukē.ʻukē*

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Suppose we have a $latex ${}_{5}F_{4}$ hypergeometric function:

And we started looking at monstrosities like this:

]]>Nature is zebra stripes: westerners will adjoin ‘irregular’ alongside ‘pathological’ or ‘malign’, and seek to impose fences: structures which are exactly regular and satisfy a group law of a sort. I have some contentions about the zebra stripes of nature:

* I suspect they have interesting long-range correlations which are kind of absent in fences.

* I think that fences are local. Even on the grand cosmic scale — the little fear that the universe might repeat exactly that lurks in the consciousness and science and religion occasionally do little to ameliorate — if you’re arguing that reality is mathematics (a la Frenkel and Tegmark), then the recipe-ish nature, a la put in the same things and get the same results back, whether applied to physical phenomena (where it works) or people (where it is foul and doesn’t). profoundly disregard the foamy and frothy and irrepetetive nature of the (whatever local context-spanning word you want to use).

* ‘fukinsei‘ is probably the best word for what I’m going after here, because all English words appear to carry connotations of problematical evil and horror, in a way.

And, just watch the first few seconds of this video:

I know that these can be accurately modeled by reaction diffusion systems, but the fingering here is strongly reminiscent of the fingering (I prefer ‘phalanges’)

Sometimes there are features which merely quantitative descriptions (here is a list of the roots) miss.

]]>So far I’ve made videos demonstrating the Borwein cubic theta functions identity:

And two showing how the j-function is invariant under SL(2,Z):

And you look at papers about the Borwein Cubic theta functions, or libraries of modular forms, and it’s all oily, stringy, two dimensional symbol manipulation. That creates an enormous barrier of entry to the material. Here is what I’m proposing:

We translate the content of the DLMF into realtime interactives http://www.visual.wegert.com/ phase portraits</a> on the complex plane that a **five year old could interact with on an ipad or other tablet with suitable fast grap****hics processor**.

Looking at something like this is worthwhile:

https://www-m10.ma.tum.de/bin/view/Lehrstuhl/StefanKranichPickIt#pickit

(insert a I am not a reformer or revolutionary tag here.I’m just your friendly transcendental kuroko

There is the absurdly barbaric suggestion that mathematics education should start with drilling addition, subtraction, multiplication, and division, or that all mathematics necessarily *starts* there. Imagine that being a Ramanujan could be common if only we got around to making a realtime user interface for all this special functions material.

(the longer term goal is to construct a new user interface for mathematics that doesn’t involve computer simulacra of two dimensional symbol manipulation with a cursor or mouse doing the input. I’d like to use a didgeridoo to drive real time special functions fractal generation and have that be the arena in which I do mathematics in rather than dealing with messy collections of symbol-spaghetti which are prone to error, but hey, I get this nice cultural syncretism between mathematics and a 40,000 year old instrument, and lots of rainbows, so potential win)

additionally, I know people usually kvetch about colored equations, but so many q-hypergeometric identities seem to look like wallpaper without colors:

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But the Euler-Mascheorni constant can also be expressed as an infinite series involving the Riemann zeta function:

and poking around google seems to indicate there are a few different ways of q-deforming the Riemann zeta function. Now, this suggests the question… “where do these different q-deformations agree”. I’m using the Maori expression *uru mai* to denote points where two different q-deformations are indistinguishable. For instance, there are three Jackson q-Bessel functions, and I can certainly make phase portraits of their differences and look for zeros — points uru mai with respect to a number of q-defromations.

(‘uru mai’ is Maori for consistent, just because the word ‘consistent’ has an overuse problem)

]]>Say this is a relief of the Gamma function in the third Jacobian theta function. Gamma could be replaced with any function whose limit as z approaches 1 is 1. Places where the relief is not one are places where the numerator differs from the denominator, thus taking a very fine microscope to the way the gamma function behaves.

Similarly, we can do the same for infinite sums. All you need is a function which goes to zero as its argument goes to zero. Sometimes you can even do self-reliefs:

So the self-relief of sine in sine would be:

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