The Global Roots of Exploding Dots

[This is the text of a presentation I made on October 7, 2017 at the kick-off event for Global Math Week, held at the Courant Institute of Mathematical Sciences in New York City. Earlier in the day, James Tanton gave his usual brilliant presentation on Exploding Dots, so in my talk I was able to assume that the audience knew what Exploding Dots is about; they also recognized my riff on Tanton’s signature line “I’m going to tell you a story that isn’t true”, as well as the significance of the word “Kapow!” (and its variants) in the Exploding Dots story. You might want to visit YouTube and sample Tanton’s Exploding Dots videos to get a feel for what it’s all about. For the full Exploding Dots spiel, try There are two videos of this talk: there’s (the one made by the Global Math Week folks back on October 7) and there’s (my PowerPoint slides, narrated by me). The latter has better visuals, and it has captions, but it’s missing the audience reactions.]

I’m going to tell you a story that’s as true as I know how to make it. I don’t have a background in ethnomathematics, or the history of mathematics, or the history of math education, so please forgive any mistakes, omissions, distortions, or mispronunciations (and let me know about them, but not now!). The story I’m going to tell is as old as civilization, or at least as old as money — because as long as currency has existed in different denominations, there’s always been a need to make change, and to find systems for making change efficiently and accurately. The story I’m about to tell involves many parts of the world over the course of many centuries. And in many ways it’s a story about sand.


The first mathematical popularizer was Archimedes, with a work called “The Sand Reckoner”. He wanted to show that the mind could stretch itself to encompass things that might seem too vast for mere humans to contemplate, such as the number of grains of sand it would take to fill the universe. He came up with a number system that could be considered a version of base ten-thousand. Ten thousand is a pretty big base to use. The Babylonians used base sixty, which is still on the big side, but better; they used place value but they didn’t have a zero, which made their system a bit awkward.

And yet, even if zero wasn’t present in the way people wrote numbers, in Babylonia and elsewhere, it was latent in the devices people used for calculating with numbers. The word “calculus” refers to a small pebble, of the kind found in versions of the abacus used all over the world that involved moving pebbles in grooves rather than sliding beads along rods. An empty groove is in effect a zero, even if one doesn’t create a name for the absence of pebbles there.

A Roman abacus.

We can thank Brahmagupta and his fellow Indian mathematicians for creating the modern system of place value with zeroes, and we can thank Al-Khwarizmi and other mathematicians of the Arab world, in northern Africa and western Asia, for adopting and adapting the system, and making possible its importation to Europe, by way of the Italian thinker Leonardo of Pisa.

I don’t know how many of you heard Keith Devlin’s recent talk at MoMath, but he argued that Leonardo of Pisa, better known as Fibonacci, was sort of the Steve Jobs of the decimal system: he didn’t invent it, but he perfected, packaged, and promulgated it in a way that greatly benefited society.

So, when the Scientific Revolution got underway in Europe, our system of representing numbers was ready for it. Thanks to Leibniz and Newton, the early insights of Archimedes were codified in the differential and integral calculus. “Calculus”: those little stones again. Isaac Newton once wrote:

“I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

Leibniz had his own approach to the ocean of truth: he imagined that all truths could be known if we only had the right language. As part of his search for the right ways to express concepts of all kinds, including numbers, he invented the binary system, while acknowledging its roots in the I Ching, a millennia-old Chinese system of divination.

Fast forward a few centuries past the invention of the calculus, and we find the countries of Newton and Leibniz engaged in a war of unprecedented scale. Ironically, one of the tools enlisted in that war was a marriage of the ballistic equations of Newton and the binary logic of Leibniz: the ENIAC computer, whose original purpose was to calculate artillery firing tables for the U.S. Army. The war ended, but the computer stayed with us, for good and ill.


In the post-war era, a Belgian educator named Frederique Papy was inspired by the manipulatives he saw in use — things like the base ten blocks popularized by Hungarian mathematician Zoltan Dienes — but he felt that something was missing from existing approaches to making students’ number-work concrete. The geometrical specificity of the shapes of these objects disguises the uniformity of the place-value system, from one place to the next. Papy devised a pedagogical binary computer made of paper and counters, in the hope that playful exploration of base two would give students a deeper understanding of positional representation.

Papy in turn inspired another educator, but before I get to that part of the story, let’s jump back to the era right before Newton and Leibniz, when the study of probability was initiated by Pascal and Fermat. In the ancient China that gave us the I Ching, randomness was seen as a message from divine powers; in late Renaissance Europe, randomness played a less exalted role, in games of chance. Pascal had been asked by a nobleman gambler to do a present-value analysis of an interrupted game of chance, and the theory of probability began to take shape.

Now zoom forward to the 20th century again, past Papy, into the 1970s, as German mathematician Arthur Engel, teaching probability theory to fourth graders in the U.S., seeks a concrete way to help children make sense of random processes. The Papy computer is in Engel’s mind as an example of the kind of concreteness he seeks, and he comes up with a device of his own, his probabilistic abacus, which he later renamed the stochastic abacus, but which I think should be called the Engel machine. You can read about Engel machines in the August 2017 issue of my blog, or from videos on my Barefoot Math YouTube channel.

I became aware of Engel machines a couple of decades ago, and gave a talk about them at a Math Circle in Boston, where a brilliant teacher from Down Under named James Tanton already had a head full of ideas about how to enliven the K through 12 curriculum. I gave a talk about crazy stuff like base three-halves, which you can read about in the September 2017 issue of my blog. Base three-halves was just one idea too many to reside in James’ head without causing all his ideas to explode outward.  Kapow!

The binary system, as represented by Exploding Dots.


You may think you know the rest of my tale, but I haven’t told you the full backstory of my Math Circle talk. Part of what excited me about Engel machines was their connection with something called the abelian sandpile model. In the 1980s the Danish physicist Per Bak, the Chinese physicist Chao Tang, and the American physicist Kurt Wiesenfeld had come up with the abelian sandpile model as a way of beginning to think about the phenomenon of self-organization in physical systems.

Interestingly, something extremely close to the sandpile model was independently invented by mathematicians at about the same time. The American mathematician Joel Spencer, working right here at the Courant Institute, was studying a class of mathematical games he called Pusher-Chooser games, and this led Swedish mathematician Anders Björner, Hungarian mathematician Lászlo Lovász, and American mathematician Peter Shor to investigate what are now called chip-firing games. Spencer, Lovász and Shor went on to write a really fun article with Richard Anderson, Éva Tardos, and Shmuel Winograd that I recall reading with great pleasure when it appeared.

Engel machines, the sandpile model, and chip-firing games are basically three different ways of looking at the same kind of process.

The Indian physicist Deepak Dhar brought his own insights to the study of sandpiles. He was among the first researchers to explore the idea that sand, or at least an idealized mathematical version of sand, was latently a computational medium. There’s a wonderful saying from the early computer age: “We have tamed lightning, and we are using it to teach sand to think.” But one might ask, What does sand already know, even before we turn it into silicon and teach it our mathematics? It knows how to flow, but not just steadily: it knows how to flow in fits and starts, just as the boxes in Exploding Dots fill and explode and fill again, and just as neuronal membranes polarize and discharge and polarize again. Comparing sandpiles to the brain is pretty speculative stuff. I think it’s too soon for us to know how much there is to it.

In any case, leaving aside possible applications to physical systems, the mathematics of sandpiles is at an exciting stage. The model in its simplest form is just as simple as can be: we have an infinite array of squares with one square that we call the center, and we pile grains of sand on the center square, adding one grain at a time, and another, and another, subject to the rule that when any square contains four or more grains, kapow!, we send a single grain to each of the four neighboring squares, and we let that keep on happening until every square has three or fewer grains, before we add any more grains to the center square. We do this again and again and again.

Mathematically speaking, that’s all there is to the process. To make it more easily absorbed by our brains, we use colors: each square gets one of four colors, according to whether the number of grains there is 0, 1, 2, or 3.

The abelian sandpile model, after 17 grains have been added at the center.

In fact, we can get rid of the dots entirely, and just use colors. Different researchers use different colors; it’s just a matter of esthetic preference.

Such a simple rule, but such complex behavior! From this seed, a world grows. [This video of a growing sandpile was posted on YouTube as ; I don’t know who made it. I showed just the first 30 seconds of the video; after about 40 seconds, the video stops showing the whole growing sandpile.]

Once the picture grows large enough, you have to choose whether to focus on the big picture or on close-ups that show intricate structures that get more and more intricate over time. There are patterns, and there are defects in the patterns that are governed by patterns of their own.


Our computers do huge calculations and show us beautiful pictures of the way the model behaves when the number of grains is in the millions, billions, or trillions — numbers we humans reckon as being large. Yet for the most part our mathematics doesn’t let us prove that what we see in our pictures is what goes on in really big sandpiles.

Image courtesy of Jordan Ellenberg.

But you shouldn’t think that we’re not making progress; we are! We have some theorems now that tell us that our pictures aren’t completely lying to us. What’s strange and wonderful is where those theorems come from. It turns out that all sorts of different branches of mathematics, seemingly treating entirely different sorts of mathematical objects, feed into the study of the sandpile model. Some of the foundational work was done here at NYU by former Courant postdocs Charles Smart and Wes Pegden, in collaboration with Lionel Levine. The fractal structure of sandpiles is governed by the fractal structure of what are called Apollonian circle packings, from a far-off field of mathematics.

Image from the article “Apollonian structure in the Abelian sandpile” by Lionel Levine, Wesley Pegden, and Charles Smart.

This is a story that’s still being written, and your students and their students may help write it.

We especially need more women writing this story. I’m sorry that there haven’t been more women in this talk, but it’s not just my fault. It’s a global problem. An alien intelligence eavesdropping on our world’s scientific progress is likely to conclude that we’re not very serious about answering important questions, because to a large extent we’re still only using half of the available intellectual workforce. Actually, much less than half: think of all the other underrepresented groups who aren’t included in the story I’ve told you. Overlooking the people who are missing from a story is unfortunately as easy as overlooking the pebbles that are missing from a groove. We need to fix that. Naming the problem is a start.


Exploding Dots uses classic, powerful ideas like chip-trading and weds them to the power of modern educational technologies (whiteboards, laptops, and smartphones) without leaving behind the millions of students who only have access to the oldest mathematical technology we have: moving things around, whether we call them dots or pebbles or chips or tokens or counters.

And what do the students get out of it? I’ll quote Keith Devlin: “Exploding Dots does nothing for the expert because it represents on a page what the expert has in their mind. But that is why it can be so effective in assisting a learner arrive at that level of understanding!” I think Keith might say that James Tanton is a worthy follower of Leonardo of Pisa.

One thing I love about Exploding Dots is that it’s not an algorithm. You don’t have a list of steps to follow in a prescribed order; what you have instead are options and opportunities. It’s a garden of forking and merging paths that all miraculously converge to the same final answer. The human spirit requires that kind of freedom, especially the spirits of young, learning humans.

I’ve tried to show that Exploding Dots has roots in the ancient idea of making change, woven together with newer ideas from all around the world about computation and chance, and that these themes have come together in a beautiful mathematical tapestry that is still being woven.

And speaking of making change: The Global Math Project is about making a change in the way teachers and students think about math, by sharing an exalting and open-ended vision of mathematics as a growing subject whose most exciting years are still ahead of us. I hope that Global Math Week 2018, 2019, and beyond, share other beautiful mathematical stories with millions of kids. Newton’s great ocean of truth isn’t just great, it’s fantastic! And it’s time everyone got invited to jump in.

Thanks to Deepak Dhar, Sandi Gubin, Brian Hayes, Henri Picciotto, and James Tanton.

Next month: Who Knows Two? Impaled on a Fencepost


#1. Was “The Sand Reckoner” really the first piece of popular mathematical writing? Might there have been other works (in China, perhaps) that predate it?

#2. Technically, neither ENIAC nor Papy’s minicomputer was a full-fledged binary computer; both used binary-coded decimal.

#3. One thing I didn’t mention (a link between gambling and self-organized criticality) is a machine I’ve seen in casinos, but whose name I don’t know. You stick a quarter into a slot and the quarter goes down into a big pile of quarters sitting on a shelf, and a long arm pushes the pile just a bit closer to the edge of the shelf. You’d think that eventually the quarters sticking over the edge would fall down, and frequently a small number of quarters do, but the big avalanche of quarters that always looks like it’s imminent takes a surprisingly long time to happen. The stack of quarters organizes itself into a structure that’s more stable than it looks. Can anyone tell me what this casino game is called?


Richard Anderson, Lászlo Lovász, Peter Shor, Joel Spencer, Eva Tardos, and Shmuel Winograd, “Disks, balls and walls: Analysis of a combinatorial game.” American Mathematical Monthly, volume 96 (1989), pages 481–493.

Archimedes, The Sand Reckoner. Full text available at

Per Bak, How Nature Works: The science of self-organized criticality. Published 1996.

Anders Bjorner, Lászlo Lovász, and Peter Shor, “Chip-firing games on graphs”. European Journal of Combinatorics, volume 12 (1991), pages 283–291.

Keith Devlin, Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years. Published 2011.

Keith Devlin, “The Power of Simple Representations.”

Deepak Dhar, “The Abelian Sandpile and Related Models”.

Deepak Dhar, “States of matter”.

Jordan Ellenberg, “The Amazing, Auto-tuning Sandpile”. Published in Nautilus, April 2, 2015.

Lionel Levine and James Propp, “What is a Sandpile?”. Notices of the American Mathematical Society, volume 57 (2013), pages 976–979. Available at

Lionel Levine, Wesley Pegden, and Charles Smart, “Apollonian structure in the abelian sandpile”. Geometric And Functional Analysis, volume 26 (2016), pages 306–336. Available at

Jennifer Oulette, “Sand Pile Model of the Mind Grows in Popularity”, Quanta Magazine, April 7, 2014; available at

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