Tom Veatch

Flames of inspiration often leave smoke signals behind.
From mine, these.

New Stuff

A Parallel Fourier Transform

Wow. This can't be true: an O(10) time algorithm for a a 1024 point fourier transform.

Let me set the stage. Now Gauss could be the greatest mathematician in history. I've always considered Tukey the greatest algorithm-making mind in history. How in the world could you wrap your mind around the FFT?! So they had the same idea (Gauss 1805, Tukey 1965) we call it the Fast Fourier Transform. The FFT is the most famous algorithm in history; almost incomprehensible to follow it, so much the more so it must have been to invent it. And the fastest FFT has always taken a hard limit of O(N*log(N)) time to calculate, with N the number of samples in the input.

The conclusion? My self-imposed, weekend Divacon homework exercise yielded a 12-symbol expression which computes a Parallel Fourier Transform in O(log(N)) time, ignoring communication: 1024x faster than Gauss and Tukey on 1024-sample data.

The story: My friend George Mou's 1990 PhD thesis was about how to think about parallel algorithms, and offered us a new orthogonal structure of analysis of computation, a formalism for algorithm expression, and a programming construct that imposes a particular way of thinking, which is actually an extremely general parallel decomposition structure onto your problem which yields world-beating results. If you can develop this skill, you can conquer the world.

George is what I'd call a world-historical genius, and I'm amazed to know him and lucky to be his friend. So I thought I'd document my education, so far, in Divacon, and maybe offer up some bits of tutorial explanation in the same, as well as in the calculations of, and the intuitions about, the Fourier transform, which I've worked on previously.

It was supposed to be a simple, outcome-uncertain homework exercise. George had told me he had a three line FFT written in Divacon which presumeably like all the others took him a half hour to write 30 years ago, but it wasn't in his thesis or published anywhere I could find, so I figured I would have to do honest homework to do an FFT in it. I figured that if I matched what he said he had done, I would surely know Divacon better, and believe me I'm impressed with the power of what that means. It was a Divacon exercise, not so much FFT algorithm research. So I puzzled on it for a few days, thinking about the geometry of the Fourier transform equation elements, going back and forth doing n'th partial re-reading of George's thesis, and finally just reading Numerical Recipes' section on FFT which used a 1942 version of the math, and it was like plug and play, pick out the divide and the combine from the usual suspects, the one that matched, use the regular base predicate everyone always uses for everything, ok, then discover that the base function was also the standard one which was a big surprise, but I'm okay with that. Then just looking at the equation to try to see the difference between one layer and the next, I could write the postadjustment function as a plain and simple, old, multiply-add, and by then it was essentially done.

A one line expression came out of this, and it says \(PFT\ \sim\ O(log(N)) is its time of computation.

Interested yet? I hope if you are a computer scientist, or even a mere programmer, you will find this super motivating and empowering to read about and then to learn yourself. Divacon is a big deal, and today it is stupid what a big secret it is.

The Infinite Cliff and the Logical Problem of Evolution

A species under functional pressure faces a logical problem. Solve it or die. A scientist understanding evolution faces a logical analysis problem namely to define and understand the logical bits in the problem, being dimensions of functional interactions.


the math of learning

Translation Graphs

A project for representing media and documents along with their translations.

Version 1.0 as of 5/29/2024: super-easy multilinear text display tables. Download code, how-to & examples.

Myers-Briggs Bad! Or is it?

Claims that there is no validity to Myers-Briggs are overstated.

Myers-Briggs vs Big 5, Round 2

Big 5 isn't better, nor even much different, from Myers-Briggs. Both can be understood as mixture models, a linear combination of two extreme cases or pure types, representing for each dimension. E/I = Extraversion; P/J = Openness x Conscientiousness; F/T = Agreeability. Then one step forward, one step back, Big5 helpfully adds a Neuroticism dimension, but unhelpfully removes N/S a.k.a. Universe, while adding an unnecessary layer of moral judgement. Big 5 methodology reflects the principle of Garbage In, Garbage Out. Further research is called for which is 1) capable of detecting the Universe people occupy, 2) less operational and instrumental in its approach to what humans actually are.


Read IPA First. Let the International Phonetic Alphabet bring basic phonetic literacy to individuals and whole languages.

Self-study method and material, offering a access to basic literacy on one sheet of paper.

Given pictures instead of Latinate vocabulary as labels for columns and rows, even illiterate learners can identify the sound for each symbol, thus overcoming the one insurmountable obstacle in acquiring literacy.

Neural Networks + Fuzzy Logic + Space

A careful, accessible introduction to neural networks assuming only high school algebra and a little geometry and differentiation. NNs are defined mathematically, along with how to run them, how to train them (by the usual gradient descent), how to train them better (so I suppose: using 'Newton-Raphson', which really ought to kill!). I also discuss how to understand the training algorithm's implicit reasoning about the adjustments it decides to make; I share an interpretation that backpropagation is like an Anti-Dunning-Kruger learning system (and therefore morally superior to most men?). Then I give a whole Fuzzy Logic re-interpretation of NNs, along with suggestions on how to enhance their logical reasoning capabilities. I tried the wikipedia page, and got so frustrated I wrote my own introduction. So yes, I suggest reading this if you want to really understand neural networks, and if your other resources have made it seem inscrutable. It's a few pages of actual math, yes, but all the steps are laid out: no leaps! It's not short, but you don't have to be a math major to follow along. I encourage your study here if you are interested in really knowing how neural nets work.

Also this adds Fuzzy Logic to neural networks, including how to train them. Finally this goes into Space Representing Neural Networks so robots can represent space, or humans' representation of space can be understood better. Three months of work is in here.

Taj Mahal Magic

The magical-perception inducing optical illusions of the Taj Mahal.


A logical decomposition of the Six Stages of Grief using minimal assumptions, but compatible with Bliss Theory.

Benefax (Progress)

Introducing the Benefax Flows Model, which generalizes economics from transactions to flows and from unidimensional, monetary valuation to many dimensions of value, and which focusses decision-making on downstream ripple effects. Much better!

Mean Thoughts

Thoughts about the Pythagorean means: the average, the geometric mean, and the hyperbolic mean. This is surprisingly deep for so simple a topic. Understanding at least 2/3 of it is part of your path to wealth and meaningfulness.

Life as Art

An essay, inspired by Paul Klee discussing art, discussing life. I was coding up a Paul Klee style colored fiber, from his Pedagogical Studies, and happened across a quote that was very inspiring and rather parallel to how I feel about the combined killing logic and living power of evolution. Have a look, maybe you'll feel it too. And you can also click through to see the 63 steps along the way from an idea to a working, artistically-useful (I hope) computer-drawing function.


If you don't know who the turkey is, it's you. A new kind of speech act, trolling, is seen to have a rich and peculiar structure, analysed here using five imputed participants including as follows: true speaker (the troll), acting speaker, true audience, straight audience, and shocked/reactive (trolled) audience. This collaboration elaborates the intentions and moral views of each, and the social consequences.

Staking Government

How to snake out the blocked pipes of government.

Foundations of Future AI

Towards a New Cognitive Science

Barnyard Mommy

An ode to motherhood. Cute, funny, short.

The Ugliness of Beauty

Beauty can also be ugly, an ugly business

The Chemistry Teacher's Son

A placeholder for autobiographical content. Nothing finished here.

Tom's Blog

A blog is a terrible idea, but I have one anyway. Lots of fun short topics here.

Bliss Theory and Thereabouts

How psychology must be that bliss is possible. Read about it here.

Tom's Diet and Supplements

Feel free to ignore my example, or to be stimulated to learn from it. But do read and share my 'Message to Black, Brown, and "White" People'. Also the important idea of the Optimum Healthy Allowances for nutrients, not just the RDA which is a limited if not bankrupt idea.

Logic from Space

A solution to the Crease of Logic.


On the Crease of Logic

Darwinian Selection Logic

Because although mutation is random, selection is functionally logical. Combined with the Crease of Logic, this resolves the conflict between the evolution-denying God-As-Designer folks and the evolution-affirming atheist folks. Have a look.

Bliss Theory: Emotion in General

On a mathematical represention of emotion, with decomposed functions including Identification which turns out to have a central role.

Gesture Learning

How to learn effective physical movement.
A research program for reinforcement learning by embodied systems, like people.

(Universal) Gravity is BS

An argument for Local, Quantum Gravity. Based on the winding problem in which galaxies remain spiralled or barred rather than a smoothly-blended coffee cup of stars, even after supposedly 20 rotations, I propose that gravitational attraction doesn't actually reach out forever, but only to a limited distance from each point of origin. Not only does this explain the mysterious maintenance of spiral galaxy arm coherence but it also explains the interaction of spiral arms with the galactic bar shape, and explains the process of formation of barred galaxies, and it predicts and explains systematic color variation within spiral galaxy arms (which are like tornadoes, spiralling around their centerlines, and detached gravitationally from the galactic center, according to this theory). Current alternative proposals in terms of dark matter and standing waves are demolished. Perhaps the merged projection of such local gravity quanta creates and constitutes space itself, thereby creating the expansion at and after the Big Bang.

On Hierarchy and Value

Hierarchy and its emotional opposite, Oneness, are explored. How Jordan Peterson agrees with Karl Marx. Why hierarchy is the compassionate, inclusive solution. The geometry of hierarchy. How dominance hierarchy transmutes into admiration hierarchy, creating communities of shared values. Emotion within social hierarchy.

Negative Dimensionality

Abstracting geometry.

Hilbert's axiomatization of geometry, full of redundancy, led me to a generalization which makes geometric dimensionality a characteristic that can be counted up (as in point to line to plane to space, etc.) and down (space to plane to line to point: etc.) Geometries, by intersecting, create lower-dimension geometries; for example two intersecting 2D planes create a 1D line. Geometries, by projecting, create higher-dimension geometries; for example, two 0D points project a 1D line.

But if there is no upper limit, perhaps there is also no lower limit. The idea that a geometry might have negative dimensionality seems absurd, considered within the assumptions of spatial thinking, yet it derives from the same less-redundant axiom set as the geometries we understand. Suggestions for intuition and use of this idea are also given.

House Color Scheme Tester

Something I made to see what color combinations look good together. Act like you know, click the buttons, it'll help you explore your own aesthetic sense.


Introduction to and its author: motivation, scope, style, audience, history.

Meaningfulness Itself

Where the sense of meaningfulness is understood as transactional transcendence, using bliss theory.

Value in our 21st Century

Tom explores fundamentals around money.

Price discovery; Market Equilibrium a false idea; Money being the metric of Value, what is the value of money?; Financial forces: Hydraulic Flow versus Smarts; Can we even bet?; The Turkey, The Butcher, and the Antifragile Investor; The Making of Money; Tech Deflation; Two Arms of Value; Real Value; Epilogue.


On the growth -- and decline -- of species like pine trees, or humans.

An essay toward a parametric model of species success and failure. A clear thought experiment showing how it proceeds, what factors influence it, and how to aim for a soft landing as resource limitations start to squeeze. Relevant to us? I think so.


The many things referred to by 'Self', with commentary. Concluding with emotionally disinhibitory goodness.

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Copyright © 2000-2021, Thomas C. Veatch. All rights reserved.
Modified: 12/20/2021