Writing

Harlan Brothers in Cantor’s Paradise

A Solution to the Problem of Deepfakes

Public key encryption holds an answer to the ever-expanding epidemic of disinformation and outright fakery

As long as I can remember, it bothered me that our leaders could lie with impunity and that in much of the world, the law of the land is “might makes right,” primarily at the expense of the less fortunate.  The modern reality that governments can make up “facts” and marauding militias can murder, rape, pillage, and burn without fear of consequence makes me angry. …

Technology 7 min read

Harlan Brothers in Cantor’s Paradise

If Escher Made Music

Psychoacoustics, Shepard tones, and the logarithmic perception of pitch

 “We adore chaos because we love to produce order.” — Maurits Cornelis Escher (1898 — 1972) 

I recall having a blacklight poster of M.C. Escher’s “Other World” as a teenager. At the time, things couldn’t get too much cooler than the combination of blacklights AND Escher. If you’re unfamiliar with the piece, here’s the original woodcut print:

Science 10 min read

Harlan Brothers in Cantor’s Paradise

Yes, There is a Fractal Invisibility Cloak

A serendipitous discovery by radio astronomer Nathan Cohen led to a new class of metamaterials 

 “The invisible and the non-existent look very much alike.” — Steven Weinberg (1933–2021) 

I remember Nathan Cohen speaking at the Yale memorial service for Benoit Mandelbrot. He was talking about a serendipitous discovery of his which happened soon after starting his first full-time job teaching at Boston University. Six months…

Science 4 min read

Harlan Brothers in Cantor’s Paradise

A Gentle Introduction to Box-Counting

This fundamental technique for estimating fractal dimension can help deepen our intuition for understanding the natural world 

Box-counting is an empirical technique we use to estimate the fractal dimension of an object, image, or set. It is based on a simple idea — cover the object with increasingly smaller boxes and then...

Mathematics 10 min read

Harlan Brothers in Cantor’s Paradise

A Simple Proof of Wilson’s Theorem

This elegant eighteenth century theorem gives a necessary and sufficient condition for proving primality

John Wilson was an English mathematician and judge who is best known for his formula for identifying prime numbers...

Prime Numbers  5 min read

Harlan Brothers in Cantor’s Paradise

Fractal Structure in the Music of Duke Ellington and Art Tatum

Duration Scaling: Part 3

This is the third part of our discussion about the fractal characteristic of duration scaling in music. In the first installment, we explained the phenomena of duration scaling and described a method for mathematical analysis and measurement...

Fractals  5 min read

Harlan Brothers in Cantor’s Paradise

The Dawn of Fractal Music

Composers have been doing the math for over six centuries

“The breeze at dawn has secrets to tell you. Don’t go back to sleep.” — Rumi (1207 — 1273)

What is fractal music? The term “fractal music” can mean different things to different people. For some, it simply refers to music that was generated by mapping a fractal image to pitches, durations, dynamics, and…

Fractals  6 min read

Harlan Brothers in Cantor’s Paradise

Bach and Cantor: A Fractal in Cello Suite No. 3

“It’s easy to play any musical instrument: all you have to do is touch the right key at the right time and the instrument will play itself.” — Johann Sebastian Bach (1685–1750) 

Johann Sebastian Bach is widely acknowledged for his exceptional skill in combining aesthetic sensibilities with mathematically derived musical…

Bach & Fractals 8 min read

Harlan Brothers in Cantor’s Paradise

Fun, Fun, Fun with the Number One

How to get from e to 1 

Known as the multiplicative identity and star of Benford’s Law, the number “1” holds a unique place in number theory. Sometimes referred to as unity, 1 is the basis for the unit measure and is not generally thought of as being composed of smaller elements. There are certainly well-known series…

Number Theory  4 min read

Harlan Brothers

“A Ph.D. is Highly Overrated”

…or so I’m told. 

Make no mistake, I would love to have a Ph.D. The discipline it requires and the depth of knowledge it affords, for me, makes it a worthy goal. Years ago, when it might have been practical to pursue an advanced degree in mathematics, I had spoken with several of my…

Human Parts  4 min read

Harlan Brothers in Cantor’s Paradise

Fractal Structure in John Coltrane’s Countdown

“All a musician can do is to get closer to the sources of nature, and so feel that he is in communion with the natural laws.” — John Coltrane (1926–1967)

John Coltrane’s album Giant Steps, first recorded in 1959, stands out as one of the seminal recordings in jazz history. While it is best known for its title track “Giant Steps”, the third track, “Countdown,” is a similarly virtuosic performance over a challenging chord progression at a blistering pace...

Fractals  7 min read

Harlan Brothers in Cantor’s Paradise

The Nature of Fractal Music: Part II (Music & Space)

In Part I of this series, we discussed how we can look at music as a fractal phenomenon that unfolds in time. Here, we’ll look at fractal analysis related to characteristics of music as generally represented in a two-dimensional score or piano roll type of transcription...

Fractals  13 min read

Harlan Brothers in Cantor’s Paradise

The Nature of Fractal Music: Part I (Music & Time)

Benoit Mandelbrot had a strong intuition that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this understanding? In this series, we will discuss the mathematics of precisely what it means to say that a piece of music is fractal...

Fractals  9 min read

Harlan Brothers in Cantor’s Paradise

The Beauty of Iterated Function Systems: A Primer

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the perfection such as only the greatest art can show.” — Bertrand Russell (1872–1970)

Iterated function systems (IFS), were first popularized in the 1980s by Michael Barnsley, a pioneer in the field...

Fractals  8 min read

Harlan Brothers in Cantor’s Paradise

Johann Sebastian Bach and the Mathematical Mind

“If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.” — Albert Einstein (1879–1955)

In the Western world, since the time of Pythagoras there has been a sort of common wisdom that music and mathematics are intimately connected. This is certainly true in the physical sense...

Bach   4 min read

Harlan Brothers in Math Simplified

A Simple Introduction to Euler’s Number, e

“All the truths of mathematics are linked to each other, and all means of discovering them are equally admissible.” — Adrien-Marie Legendre (1752–1833)

The number e, also known as Euler’s number or the base of the natural logarithm, pops up whenever we examine continuous rates of growth (or decay) that…

e   5 min read

Harlan Brothers in Towards Data Science

Giants Steps: The Fractal Structure of John Coltrane’s Iconic Solo

“When you begin to see the possibilities of music, you desire to do something good for people, to help humanity free itself from its hang-ups.” — John Coltrane (1926–1967) 

John Coltrane's improvisation on his composition Giant Steps, first recorded in 1959, stands out as one of the seminal recordings in…

Coltrane  7 min read

Harlan Brothers in Cantor’s Paradise

Pascal’s Triangle: A Secret Unveiled

“All of humanity’s problems stem from man’s inability to sit quietly in a room alone.” ― Blaise Pascal (1623–1662)

For 1500 years, mathematicians from many cultures have explored the patterns and relationships found in what we now, in the West, refer to as Pascal’s triangle. Named posthumously for the French…

Pascal's Triangle   5 min read

Harlan Brothers in Cantor’s Paradise

The Truth About Fractals

“In science, all important ideas need names and stories to fix them in the memory.” ― Benoit B. Mandelbrot (1924–2010)

Major milestones in mathematics are generally measured in centuries. It is serendipitous then to come of age in the wake of a fundamentally new perspective in mathematical thought. In 1975…

Fractals   4 min read

Harlan Brothers in Cantor’s Paradise

Ramanujan, 𝑒, π and 𝜙

“An equation means nothing to me unless it expresses a thought of God.” — Srinivasa Ramanujan (1887–1920)

The brilliant young Indian mathematician Srinivasa Ramanujan, was a prodigy whose deep mathematical insights continue to thrill number theorists a century later. His first letter to English mathematician G. H. Hardy contained interesting…

Mathematics   2 min read

Harlan Brothers in Cantor’s Paradise

Bach Meets Mandelbrot

“I think we should send all of Bach; but of course we would be bragging, but it is surely excusable to put the best possible face on at the beginning of such an acquaintance.” ―Lewis Thomas (1913–1993)

More than two centuries separate The Art of Fugue, written by Johann Sebastian Bach, and the primitive computer printout of the first graphical representation of what is arguably the most famous fractal...

Bach 6 min read

Harlan Brothers in Cantor’s Paradise

π in, e out

“Surely it is not knowledge, but learning; not owning but earning; not being there, but getting there; that gives us the greatest pleasure.” ―Carl Friedrich Gauss (1777–1855)

Alice: “Give me as many digits of π as you want. I will demonstrate how you can turn them into digits of e.” …

Pi   4 min read

Harlan Brothers in Towards Data Science

Fractal Music from Randomness

“My life seemed to be a series of events and accidents. Yet when I look back, I see a pattern.” ― Benoit B. Mandelbrot (1924–2010)

Music is a rich source of multi-dimensional data. This data can take the symbolic form of a score, usually represented by MIDI values, or the form of an audio signal. For many years...

Fractals   7 min read

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Music & Mathematics: Cosmic Twins

“Music is the pleasure the human mind experiences from counting without being aware that it is counting.” — Gottfried Wilhelm von Leibniz (1646–1716) (co-discoverer of calculus)

The connections between mathematics and music run deep. The subjects can be thought of as cosmic twins, born out of a tantalizing mixture of…

Music   7 min read

Harlan Brothers in Cantor’s Paradise

Big List of Fast New Series for e

“The essence of mathematics is in its freedom.” ― Georg Cantor (1845–1918)

The following list of series for e was derived using the methods described here, and originally published in the College Mathematics Journal.

For reference, Equation 1 shows the classic series that was the starting point for these derivations. It was first published by Issac Newton in 1665...

e   4 min read