New Views of the Divisor Plot

(This post was originally published in October of 2013 in I moved it here because this is a better home for it.)

Recently, a fellow number pattern explorer (Gregory Bryant) showed me some patterns he discovered, which he calls Factor Map:


It’s a variation of something I discovered years ago, which I call the Divisor Plot – a special way of seeing composite numbers that reveals some amazing patterns – in particular…parabolas!

Screen Shot 2013-10-09 at 9.47.12 PM

The basic plot had been explored by several people in the past (including Wolfram), but until recently, only small numbers were plotted.


Gregory found a way to create the pattern out of parabolas only. He plots the odd-numbered parabolas as red and the even-numbered ones as blue. (His plot is also rotated by 90 degrees)…


Robert Wannamaker has applied similar ideas to musical polyrhythms:


I discuss these patterns more fully in my blog post, “Barking up the Prime Number Tree“:


I believe that we number pattern explorers are contributing to a new revolution in mathematics – a way of understanding very large numbers which behave more like physical models of the universe than they do traditional math.

This revolution in math is also visually stimulating.



Why is it a Color “Wheel” and Not a Color “Line”?

This article has been moved to Nature, Brain, Technology.


Is There Such a Thing As Math Without Numbers?

One reason I started this blog was because I had asked myself a question: Is there such a thing as Visual Math? I’m talking about a totally visual form of math – where there are no numbers involved (and no verbal language). Is this possible? If there are no numbers (or quantities, or any symbols that stand for numbers) can we still call it math?

Here are a few definitions of math:

Merriam-Webster defines it as “the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations”.

Wikipedia defines mathematics as “the study of quantity, structure, space, and change”.

These references to “space”, “structure”, and “change” certainly ground the notion of math in the context

of physical experience.

But math is genuinely abstract, and whether or not it is ultimately derived from earthly experience, it may

or may not refer to the physical world (as in many problems in number theory, for instance).

At any rate, is number really the essence of math?

This is not a new question, as explored in a book by Geoffrey Hellman.

I will not try to answer the question of whether number is indeed the essence of math (as opposed to some other concept such as logic, sets, etc.).

I will instead ask whether math can be experienced, expressed, created, performed, manipulated, or acted-upon, using ONLY visual language, and WITHOUT numbers.

Is visual math simply a different way of thinking about numbers than reading and writing equations?

As you can see – I have lots of questions, and no answers!

I look forward to hearing what you think.

Barking Up the Prime Number Tree

You’ve heard it before:

“Prime numbers are the building blocks of all numbers”.

“Prime numbers are like atoms”.

I think prime numbers are more like HOLES  – peeking out from behind the patterns created by the composite numbers. Yea, prime numbers may be atoms when it comes to multiplication. But atoms in themselves are not so interesting. It might be more productive to study the molecules. The composite numbers generate a wealth of patterns – including the prime number sequence. Consider:

“Prime numbers are what is left when you have taken all the patterns away.”
― Mark HaddonThe Curious Incident of the Dog in the Night-time


We often hear the question, “Why have we never found a formula that generates only prime numbers?”

Answer: There probably is no formula. Stop looking at the prime numbers. Mathematician Gregory Chaitin suggests that there is a lot more to be learned from studying the composite numbers.

It’s like this. What if you were in a dense forest with the sun shining high above. You set out on a quest to figure out a way to re-create the exact pattern of light speckles on the forest floor. These speckles of light are caused by the sun passing through leaves and branches. You will have a hard time finding any algorithm, any 3D computer graphic program…any way…to represent that pattern of light. That is because you are looking in the wrong place. Look at the trees. They are making shadows. Represent the trees (and the way the light passes over them and lands on the ground) and you will have a way to generate the shadows and therefore the light speckles. It’s like using the Sieve of Eratosthenes to cast shadows.

Primes are leftover cracks between the composites. They are the empty, pattern-void, gaps hiding in the crevices of something wonderful indeed: The collection of composite numbers, with all their symmetries, family relationships, and overlapping patterns. They form some of the most beautiful structure I have ever seen.

Primes are cool, of course. Don’t get me wrong. In fact, Cicadas use them. Cicadas have mating cycles in which they all emerge from pubescence en masse, engage in a huge orgy, and then die. The “Periodical Cicadas” do this every 7, 13, or 17 years. Why did they evolve to have prime number mating cycles? Why of course: to HIDE from their predators. These predators would love nothing more than to descend on an orgy of cicadas en masse, for a periodic feast. But in order to do this, the predators would have to evolve foraging cycles that are synchronized to the mating cycles of the cicadas.

My point: These little buzzing critters have hit upon the value of primes as a way to evade structure – as a way to hide behind the usual polyrhythms of the biosphere. Indeed, primes are what is left when the patterns are removed (which is why they are so useful in cryptography).

As cool as primes are, it’s a waste of time trying to identify the heartbeat of the primes. The heartbeat has been fibrillating since the beginning of time, and it shows no sign of stopping.

In fact, the way I see it, the sequence of primes is NOT a heartbeat. It is all the silent gaps between an infinity of heartbeats – all beating at different frequencies. These are the composites.

So check out the composite numbers. I’m talking numbers like 12! (12 factorial). Among the big composites you will find symmetries within symmetries. Maybe even some metaphors or hints to the structure of the universe. There’s lots of material for visual patterning when visualizing the Sieve of Eratosthenes for big numbers like nine billion. And if you’re like me, you’ll gobble it up – and maybe some mathematical insights will pop out in the process.

If you can’t see the forest for the trees, look at the trees.

Very Large Numbers are Not Numbers, and Infinity Doesn’t Exist

Remember Nietzsche’s famous announcement, “God is dead“?

In the domain of mathematics, Nietzsche’s announcement could just as well refer to infinity. And for that matter, the whole notion that numbers exist … before they are thought, uttered, or used.

Yes, I am questioning the very foundations of mathematical truth. This might annoy you, or worse, make you stop reading this blog post. How can anyone doubt the perfect, always-been-there/foreverness of numbers?

There are some philosophers who are putting up a major challenge to the Platonic stronghold on math: Brian Rotman, author of Ad Infinitum, is one of them. I am currently reading his book. I thought of waiting until I was finished with the book before writing this blog post, but I decided to go ahead and splurt out my thoughts – hey – that’s what blogs are for!

Charles Petzold gives a good review of Rotman’s book here.


Is Math a Human Activity or Eternal Truth?

My thoughts at the moment are this: You (reader) and I (writer) have brains that are almost identical as far as objects in the universe. We share common genes, language, and we are vehicles that carry human culture. We cannot think without language.  “Language speaks man” – Heidegger.

Since we have not encountered any aliens, it is not possible for us to have an alien’s brain planted into our skulls so that we can experience what “logic”, “reality” or “mathematical truth” feels like to that alien (yes, I used the word, “feel”). Indeed, that alien brain might harbor the same concept as our brains do that 2+2=4….but it might not. In fact, who is to say that the notion of “adding” means anything to the alien? Or the concepts of “equality”? And who is to say that the alien uses language by putting symbols together into a one-dimensional string?

More to the point: would that alien brain have the same concept of infinity as our brains?

It is quite possible that we can never know the answers to these questions because we cannot leave our brains, we can not escape the structure of our langage, which defines our process of thinking. We cannot see “our” math from outside the box. That is why we cannot believe in any other math.

So, to answer the question: “Is math a human activity or eternal truth?” – I don’t know. Neither do you. No one can know the answer, unless or until we encounter a non-human intelligence that speaks an identical mathematical truth.


Big Numbers are Patterns

My book, Divisor Drips and Square Root Waves, explores the notion of really large numbers as more about pattern than size (the size of the number referring to where it sits in the countable ordering of other numbers). In this book, I explore the patterns of the neighborhoods of large numbers in terms of their divisors. This is a decidedly visual attitude of number, whereby number-theoretical ideas emerge from the contemplation of the spatial patterning.

The number:


doesn’t seem to have much meaning. But when you consider that it is the number of ways in which you can arrange a single deck of cards, it suddenly has a short expression. In fact it can be expressed simply as 52 factorial, or “52!”.

So, by expressing this number with only three symbols: “5”, “2”, and “!”, we have a way to think about this really big-ass number in an elegant, meaningful way.

We are still a LONG way from infinity.

Now, one argument in favor of infinity goes like this: you can always add 1 to any number. So, you could add 1 to 52! making it 80658175170943878571660636856403766975289505440883277824000000000001.

Indeed, you can add 1 to the estimated number of atoms in the universe to generate the number 1080 + 1. But the countability of that number is still in question. Sure you can always add 1 to a number, but can you add enough 1’s to 1080 to each 10800?


Are we getting closer to infinity? No my dear. Long way to go.

Long way to “go”?  What does “go” mean?

Bigger numbers require more exponents (or whatever notational schemes are used to express bigness with few symbols – Rotman refers to hyper-exponents, and hyper-hyper-exponents, and further symbolic manipulations that become increasingly hard to think about or use).

These contraptions are looking less and less like everyday numbers. In building such contraptions in hopes to approach some vantage point to sniff infinity, one finds a dissipative effect – the landscape becomes ever more choppy.

No surprise: infinity is not a number.

Infinity is an idea. Really really big numbers – beyond Rotman’s “realizable” limit – are not countable or cognizable. The bigger the number, the less number-like it is. There’s no absolute cut-off point. There is just a gradual dissipation of realizability, countability, and utility.


Where Mathematics Comes From

Rotman suggests taking God out out mathematics and putting the body back in. The body (and the brain and mind that emerged from it) constitute the origins of math. While math requires abstractions, there can be no abstraction without some concrete embodiment that provides the origin of that abstraction. Math did not come from “out there”.

That is the challenge that some thinkers, such as Rotman, are proposing. People trained in mathematics, and especially people who do a lot of math, are guaranteed to have a hard time with this. Platonic truth is built in to their belief structure. The more math they do, the more they believe that mathematical truth is discovered, not generated.

Now, do I really believe that mathematics is a purely human invention? I do have some sympathy with Roger Penrose: when I explore the Mandelbrot Set, I have to ask myself, “who the hell made this thing!” Certainly no mathematician!

After all, the Mandelbrot Set has an infinite amount of fractal detail.

At least thats what they say.

What is Visual Mathematics?

First of all, what is “math”?

For many people, “math” means equations, numbers, symbols, theorems and proofs, scrawled on pieces of paper. About ten inches above these scrawls is a wrinkled forehead, and just below that: two glazed eyes.

Because of my terrible grades in math, never getting beyond Algebra II, and then failing the lowest possible math class in college, I concluded that I was a complete math failure.

But I now know that mathematics encompasses a much broader set of activities than what most of us have been lead to believe. The curriculum is what failed. Thanks to my skills in visual thinking, drawing, and spatial logic (talents often associated with dyslexia) I was able to leap over that crufty heap of equations and explore mathematical concepts with my visual brain. And I didn’t even know I was doing math…until I got accepted into MIT, all expenses paid, to earn a master’s degree. MIT doesn’t pay full tuition and stipend to math flunkies. So, how do you explain this?

The purpose of bringing this up is to tell all you math flunkies out there that math (Real Math) may not be what you think it is. If you have a knack for music, puzzles, perspective drawing, a love of architecture, or are good at building things, you may have a natural talent or mathematical concepts, even if you can’t stand to look at symbols piled onto each other. You are a bundle of potential. The science and engineering industries NEED you. Too bad they’ll probably never find you.

Brett Victor takes the extreme view in Kill Math, claiming that we think of math as “assigning meaning to a set of symbols, blindly shuffling around these symbols according to arcane rules.” And therein lies the main culprit – the thing that used to snag me all throughout school:

RULES (without explanation)

I used to constantly ask my teachers, Why? Why? Why? The response from my teachers can be best summed up by a quote, apparently from the poet W. H. Auden:

“Minus times minus equals plus. The reason for this we need not discuss.”

Exploring animated patterns using simple computer programming can open the minds (not just the eyes) of young visual thinkers (or anyone who is turned off by equations). The reason is that the “why” is built-in to the activity. My own personal trajectory towards understanding and appreciating mathematical equations came only after years of programming computer animations, at first only for the sake of aesthetics. I came up with my own distilled bits, which often expressed a concept in an elegant way.

The point is: only after having had the experience of creation did these distilled bits mean anything to me. That is what math is about.

And THAT is the philosophical basis for the revolution in math that needs to take place in our schools. Ken Fan, who founded a math club for girls in Cambridge Mass, says: “I am concerned that the way math is typically taught in school in the US often fails to acknowledge the extraordinary differences in the way we think – too often, schools demand that students solve their assignments in very specific, overly rigid ways.”

I would even go a step further and ask, why must we always “solve” something? Why must there always be a “problem”? Is that the essence of mathematical activity? Can’t math be based upon a paradigm of creation?

This is a new blog and this my first blog post! I am planning on touching upon many topics. I am a Math Flunky who discovered Real Math despite my education. I have a story to tell. And I have reason to believe that my story is not at all original.

Stay tuned for posts about fractals, Buffon’s Needle, the golden ratio, locusts, the geometry of sex, multihomuncularity, and why we love trees.

But mostly, I’ll just be showing you a bunch of pictures.