Number Portraits

I designed a set of number portraits for the integers from 1 to 36.

Each number is either a prime or a composite. If it is a prime number, then it has two divisors: 1 and itself. This is visualized as the gray-colored half circle, where the top represents 1 and the bottom represents the number itself. Composite numbers have other pairs of divisors, and these are visualized as the smaller, colored arcs.

The perfect squares (4, 9, 16, 25, and 36) each have a line segment located at the square root.

The numbers 6 and 8 each have one pair of divisors (besides 1 and themselves); they are (2,3) and (2,4), respectfully. Since the first number in each pair is 2, these arcs are colored green. Divisor pairs in which the first divisor is 3 are colored blue; divisor pairs in which the first divisor is 4 are colored red. And divisor pairs in which the first divisor is 5 are colored yellow. These colors visualize divisibility by these first 5 positive integers.

The highly-composite numbers 12, 24, and 36 are shown enlarged below.

The number 36 has the most divisors in this set. It’s divisor pairs are

(1, 36)  (2, 18) (3, 12) (4, 9) (6, 6).

These are in a similar spirit to the Divisor Plot images I created in 2010:


Thoughts on the Evolution of Communication

My dog and I engage in a lot of signaling. But it is not always deliberate, and it is not always conscious, and it is not always a two-way process.

In the morning, Otto licks my bald head. He can probably smell what I have been dreaming. I hold him and we have a nice cuddle. Just one of our many routines. He looks at me and I look at him. He is always checking me out. In the process of getting to know each other over several years we have come to read each other’s signals – our body language, interactions, responses, vocalizations…and smells.

image from

Semiosis emerges in the process. If there is a coupling of signals – a mutually-reinforcing signaling loop – two-way communication emerges. It is not always conscious – for either of us. Sometimes, a mutually-reinforcing signaling process which I was previously unaware of becomes apparent to me. When this happens, I become an active agent in that semiosis.

Otto is so intensely attentive to me – my routines (and deviations from them). He probably tunes-in to many more of my signals than I do to his. But then again, I am a human: I generate a lot of signal. Does he see this as “communication?” It is not clear: his brain is a dog brain, and mine is a human brain. We don’t share the same word for this experience (he only knows a few English words, and “communication” isn’t one of them).

I can be sure of one thing: we share a lot of signaling. And, as members of two highly-social species, we both like that.

I would conclude from this that communication among organisms in general (the biosemiosis that has emerged on Earth over the last few billion years) came about pretty much the same way that Otto and I established our own little world of emergent semiosis. As life evolved, trillions of coupled signaling channels reinforced each other over time and became more elaborate. Eventually, this signaling became conscious and intentional.

And so here we are: human communication has reached a level of sophistication such that I can type these words – and you can read them. And we can share the experience – across time and space.

We are always dreaming

Take a large pot of water and leave it out in sub-freezing temperatures for a few days. It will turn into a block of ice.

Now take that pot of water and put it on the stove and crank up the flame. Before long, it will start to boil.

Let it cool for a few hours at room temperature and it will resume its familiar liquid form.

If you drop a live fish into liquid water it will swim around and do fishy things.

Things would not go so well if you drop a fish onto a block of ice. Fish are not good skaters.

And if you drop a fish into boiling water…well, the fish will not be very happy.

Think about these states of water as metaphors for how your brain works. A block of ice is a dead brain. A pot of boiling water is a brain having a seizure. Water at room temperature is a normal brain.

The fish represents consciousness.


Liquid brain

There is a constant low level of electrical activity among neurons (like water molecules bouncing off of each other, doing the Brownian dance). Intrinsic random neuronal activity is the norm – it keeps a low fire burning all the time. In a sense, the brain has a pilot light.

A bit of randomness is helpful for keeping the mind creative and open to new ways of thinking – consciously and unconsciously. Like the ever-present force of natural selection that curates random mutation in genetic evolution, there are dynamical structures in the brain that permit more meaningful, useful energy to percolate from the random background.

Command and control

The majority of the brain’s activity is unconscious. At every second of your life a vast army of dynamical structures are buzzing around, managing the low-level mechanisms of multi-sensory input, attention, memory, and intent. These structures are vast, short-lived, and small. And they are entirely inaccessible to the conscious mind.

The command and control area of the brain is located at the front-top of the neocortex. The signature of consciousness is a network of relatively stable, large-scale dynamical structures, with fractal fingers branching down into the vast network of unconscious structures. The buzz of the unconscious mind percolates and fuses into something usable to the conscious mind. It offers up to the conscious mind a set of data-compressed packets. When the command and control center relaxes, we experience wandering thoughts. And those thoughts wander because the brain’s pilot light provides constant movement.

These ideas are derived from Dehaene’s Consciousness and the Brain.

Surrender to dreaming

When we start falling asleep, the command and control center begins to lose its grip. The backdrop of randomness sometimes makes its way past the fuzzy boundary of our consciousness – creating a half-dreaming state. Eventually, when consciousness loses out, all that is left is this random, low-level buzz of neural activity.

But dreaming is obviously not totally random. Recent memories have an effect…and of course so do old but powerful memories. The physical structure of the brain does not permit total randomness to stay random for very long. Original randomness is immediately filtered by the innate structure of the brain. And that structure is permeated with the leftovers from a lifetime of experience.

So here’s a takeaway from recent neuroscience, inspired by the findings of Stanislas Dehaene: WE ARE ALWAYS DREAMING. That is because the unconscious brain is continually in flux. What we recognize as dreaming is merely the result of lifting the constraints imposed by the conscious mind – revealing an ocean – flowing in many directions.

The unconscious brain can contribute to a more creative life. And a good night’s sleep keeps the conscious mind out of the way while the stuff gathered in wakefulness is given a chance to float around in the unconscious ocean. While in the ocean, it either dissolves away or settles into functional memory – kicking out an occasional dream in the process.


Hummingbird on a wire

hummingbirdI looked out the window this morning and I thought I saw a speck on the window pane. Upon closer look, I realized that the speck was a hummingbird perched high on a wire spanning two telephone poles.

I became the bird’s dedicated audience for about three minutes. I watched closely as the tiny bee-like creature surveyed the surroundings from its high vantage point.

What was the bird thinking? And can I use the word “thinking” to describe the activities in this bird’s mind? For that matter, does the bird have a mind? It certainly has a brain. And that brain has a special feature: its hippocampus is five times larger than that of song birds, seabirds, and woodpeckers. According to this article, “The birds can remember where every flower in their territory is and how long it takes to refill with nectar after they have fed.”

Thinking is a by-product of an animal body, which is a member of a species with specific needs, skills, and adaptations to a particular environment.

Fear (and Love) of Heights

If I were perched on a wire as high as the hummingbird, I would be terrified: “Get me down from here!” On the other hand, a bird feels perfectly at home at such high altitudes.

Consider a hawk sliding across the horizon above a vast valley. Looking down from its vantage point, the hawk may experience inner-peace – possibly moments of boredom (if you will permit me to apply these human-oriented emotion labels to a hawk’s subjective experience). A human hang-glider would experience exhilaration, and moments of fear. And maybe…moments of that same inner-peace that the hawk experiences.

Above image from:

When I have joyful flying dreams, my brain is not triggering the fear network. I am experiencing a peaceful freedom from gravity – with touches of exhilaration.

I wish I could become as light and deft (and fearless) as a bird, and watch the world from the tallest treetops in my neighborhood.

Thelonius Monk’s Shapeshifting Chord


One of my part-time hobbies is being a Monk interpreter. A Monk interpreter not only learns how to play Monk’s compositions, but also makes a point of getting into the head of this eccentric man. The reason to do this is that Monk was an improvisor – and he was driven by an inner vision. If you can tap that inner vision, then you can generate Monk-like music – and improvise on it…even while playing Beatles songs.

I wrote a piece in 2013 about Monk as a mathematician.


Math can be about patterns (visual or sonic). Math does not always have to be expressed in numbers. Monk once said,“All musicians are subconsciously mathematicians”.

A Symmetrical Chord

The chord I’m talking about has four notes. It is typically used as a dominant chord – which naturally resolves to the tonic. Unlike the classical dominant-seventh, this chord has a flatted fifth – which makes it slip into a symmetrical regime – as shown in the picture above – inscribed in the circle of fifths.



According to Wikipedia, this chord is called the “Dominant Seventh Flat-Five Chord“. The cool trick about this chord is that it can resolve to either of two different tonics – each being a tri-tone apart.

So for instance, a chord with these notes:     Eb   F   A   B      can resolve to either Bb or E as the home key.

This chord also happens to contain 4 of the 6 tones in a whole tone scale, which Monk famously used (often as a dominant arpeggio).

If you are not familiar with music theory, you may still appreciate the beauty of sonic geometry and how it can generate such variety. If you apply similar concepts to rhythm as to harmony then you have a wonderfully rich canvas for endless musical expression. I like the way Monk wove these geometries together in a way that makes the foot tap and the ear twinge – and the brain tweak.

Monk was of course not the only one to apply these ideas – but he did accomplish something remarkable: the application of embodied math. If you have spent as much time as I have learning his language, listening to him improvise can cause a smile – or the occasional giggle – to pop out. Like an inside joke.

There is plenty of material on the internet about Monk. Here’s one voice among the many who have acquired an appreciation for Monk:  How to Listen to Thelonius Monk – by George H. Jensen, Jr.