# Science Writing

Harlan Brothers in Science Spectrum

The Euclid-Euler theorem links the work of two of the greatest mathematical minds of the ages

“Mathematics is the art of giving the same name to different things.” – Henri Poincaré (1854–1912)

Mathematics 9 min read

Harlan Brothers in Science Spectrum

How the brilliant cognitive scientist Roger Shepard turned the tables on us all

“Reality is merely an illusion, albeit a very persistent one.” — Albert Einstein (1879-1955)

A few months ago, I wrote an article about an audio illusion known as Shepard Tones, first described in 1964 by renowned cognitive scientist and psychologist, Roger N. Shepard (see “If Escher Made Music”)...

Cognitive Science 7 min read

Harlan Brothers in Science Spectrum

Coltrane’s iconic solo is a wonderful example of a power-law in action

Fractal Music 9 min read

Harlan Brothers in Science Spectrum

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.”

Mathematics 6 min read

Harlan Brothers in Science Spectrum

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...

Fractal Music 11 min read

Harlan Brothers in Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

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 Science Spectrum

Fractal Structure in the Music of Duke Ellington and Art Tatum

Fractal Music 5 min read

Harlan Brothers in Science Spectrum

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

Bach & Fractals 8 min read

Harlan Brothers in Science Spectrum

Harlan Brothers in Science Spectrum

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 Science Spectrum

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

Fractal Music 13 min read

Harlan Brothers in Science Spectrum

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

Fractal Music 9 min read

Harlan Brothers in Science Spectrum

Harlan Brothers in Science Spectrum

Harlan Brothers in Science Spectrum

Harlan Brothers in Science Spectrum

Harlan Brothers in Science Spectrum

Number Theory 2 min read

Harlan Brothers in Science Spectrum

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 & Fractals 6 min read

Harlan Brothers in Science Spectrum

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...

Fractal Music 7 min read

Harlan Brothers in Science Spectrum

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…