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)

When I first heard of the Euclid-Euler theorem, I wondered what the connection might be between these giants of mathematics — a connection that spans two millennia...

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 (seeIf Escher Made Music”)...

Cognitive Science  7 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..."

Science 6 min read

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

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

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  & Fractals 6 min read

Harlan Brothers in Science Spectrum

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