Science Writing

The follow is a list of free links to my articles on a wide range of topics related to math, science, and music.

Harlan Brothers in Science Spectrum

Proof of Perfection

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

The Greatest Optical Illusion?

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

John Coltrane’s Giant Steps Solo: Fractal Geometry in Action

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

“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…

Fractal Music  9 min read

Harlan Brothers in Science Spectrum

A King Among Rulers

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 Science Spectrum

The Bach Cello Suites: A Fractal Wonderland

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

The Unlikely Existence of The Cosmological Theorem

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

The Ultimate Guide to Recamán’s Sequence

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

Why is Aluminum Foil Only Shiny on One Side?

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

Seeing the Invisible

Abraham Wald and the power of the mind’s eye.

I have always been fascinated by stories of those who could see simple truths that were only obvious after they were pointed out. Abraham Wald was one such person...

Science 5 min read

Harlan Brothers in Science Spectrum

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 Science Spectrum

St. Elmo’s Fire

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

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 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

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

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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

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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

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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...

Fractal Music   5 min read

Harlan Brothers in Science Spectrum

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…

Fractal Music  6 min read

Harlan Brothers in Science Spectrum

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 Science Spectrum

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...

Fractal Music  7 min read

Harlan Brothers in Science Spectrum

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...

Fractal Music  13 min read

Harlan Brothers in Science Spectrum

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...

Fractal Music  9 min read

Harlan Brothers in Science Spectrum

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 Science Spectrum

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 Science Spectrum

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…

Fractal Music  5 min read

Harlan Brothers in Science Spectrum

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 Science Spectrum

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 Science Spectrum

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…

Number Theory   2 min read

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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

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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 Science Spectrum

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...

Mathematics   4 min read