Science Writing
Harlan Brothers in Cantor’s Paradise
The Golomb ruler is a mysterious and elegant combinatorial object with many real-life applications
“Measure what is measurable, and make measurable what is not so.”
— Galileo Galilei (1564–1642)
When I first heard of the Golomb ruler, it immediately brought to mind the golem — a mythical clay creature of Jewish folklore, mysteriously endowed with the power of life. As I learned, the Golomb ruler, too, has power and ample mystery of its own...
Mathematics 6 min read
Harlan Brothers in Cantor’s Paradise
Bach’s famous cello suites are a rich source of fractal structure
“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)
Like countless others lovers of music through the ages, I am a huge fan of Bach. My first serious appreciation of his music came when studying counterpoint at the Berklee College of Music. However...
Fractals 11 min read
Harlan Brothers in Cantor’s Paradise
Audioactive sequences and the playful genius of John Conway
“You don’t educate people by telling them useful things; you educate people by telling them interesting things” — John Horton Conway (1937–2020)
I was first exposed to the genius of John Conway through my work at the NSF-funded Yale Fractal Geometry Workshops...
Mathematics 8 min read
Harlan Brothers in Cantor’s Paradise
Subtract if you can, add if you can’t — a simple rule leads to wild behavior and musical structure
“Simplicity is the ultimate sophistication.” — Leonardo da Vinci (1452–1519)
For many years, now, I’ve been fascinated by the intersection of music and mathematics. Ever since reading the cover story of Scientific American’s 1985 article on the Mandelbrot set, I’ve been thinking about ways to represent mathematics as music...
Mathematics 8 min read
Harlan Brothers in Cantor’s Paradise
Considering I’ve been using it for decades, I suddenly found my ignorance alarming.
"Why does everyone think the future is space helmets, silver foil, and talking like computers, like a bad episode of Star Trek?" - Tracey Ullman
My fascination with tin foil began in childhood with tin foil boats, light reflectors, and electronic circuit explorations...
Science 11 min read
Harlan Brothers in Cantor’s Paradise
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
This mesmerizing spectacle of nature has been considered both a good omen and a harbinger of danger. The underlying physics help explain why.
“It was a beautiful sight, that electric fire playing around the mastheads and yardarms, and illuminating the whole ship with its pale light. It was like a fairy scene..."
Harlan Brothers in Cantor’s Paradise
Researchers Discover a Fractal Protein
Amazingly, cyanobacterium Synechococcus elongatus spontaneously forms Sierpinski triangles
It was nearly half-century ago that Benoit Mandelbrot first coined the term fractal to describe the broad family of self-similar shapes that repeat their pattern in either exactly or roughly the same fashion over a wide range of scales....
Science 7 min read
Harlan Brothers in Cantor’s Paradise
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)
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
Technology 7 min read
Harlan Brothers in Cantor’s Paradise
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...
Harlan Brothers in Cantor’s Paradise
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 John Coltrane’s Countdown
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...
Harlan Brothers in Cantor’s Paradise
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...
Harlan Brothers in Cantor’s Paradise
Alice: “Give me as many digits of π as you want. I will demonstrate how you can turn them into digits of e.” …
Harlan Brothers in Towards Data Science
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...
Music & Mathematics: Cosmic Twins
The connections between mathematics and music run deep. The subjects can be thought of as cosmic twins, born out of a tantalizing mixture of…
