Harlan J. Brothers

My Work:

Data Lead, Consultant
Bright Power, Inc.

February 2019 - January 2020

Data Scientist
BrightForge Technology Holdings

October 2017 - June 2018

Consulting Director
Forensic Mathematics Services

Mathematical Analysis for Malpractice
January 2014 - January 2018

Director of Research & Development
Brothers Technology, LLC

Intellectual Property, Research & Development
July 1994 - Present

Senior Consultant, Application Systems
Penguin Random House

July 2014 - June 2015

Guest Editor
The Real Life Survival Guide (NPR)

Media Industry
August 2011 - August 2014

Scholar at Large
Citizen Scientists League

July 2012 - February 2014

Director of Technology
The Country School

Primary/Secondary Education Industry
July 2002 - June 2011

Yale Mathematics Department

Higher Education Industry
June 2001 - May 2007


Bio         In the News
Patents         Articles
Publications       Professional Service
Presentations         References
Music         Related Links



A technology developer and inventor with successful designs that run the gamut from an encryption-based Event Verification System (bought by a pre-eminent IP firm) to an automatic household water alarm (licensed to Salton, Inc.). A published mathematician and researcher with numerous journal articles related to number theory and the fractal geometry. Experienced Director of Technology, project manager, inventor, educator, researcher, writer, programmer, and technical design consultant. Possesses a unique, cross-disciplinary background as a problem solver, creative thinker, and effective communicator.


Data-Conditioned Encryption Method
(Provably secure encryption system. Patent No. 8,712,040)
Event Verification System
(EVS™/Vericam™, Patent No. 5,799,083)
Interior Mailbox Light
(Postlite™, Patent No. 5,975,713)
Portable Ashtray
(SmokeMate™, Patent No. 5,673,709)
Luminous Pull-Cord
(Opticord™, Patent No. 5,454,056)
Household Safety Receptacle
(Digisafe™, Patent No. 5,320,545)


(Reprints and preprints available upon request)

H. J. Brothers, "The Nature of Fractal Music," in Benoit Mandelbrot - A Life in Many Dimensions, edited by Michael Frame, World Scientific Publishing (May, 2015).

N. Neger and H. J. Brothers, "Benoit Mandelbrot: Educator," in Benoit Mandelbrot - A Life in Many Dimensions, edited by Michael Frame, World Scientific Publishing (May, 2015).

M. F. Barnsley, M. Berry, M. Frame, I. Stewart, D. Mumford, K. Falconer, R. Eglash, H. J. Brothers, N. Lesmoir-Gordon, J. Barrallo, Glimpses of Benoît Mandelbrot (1924-2010). Notices of the American Mathematical Society, Vol. 8, No. 59, 2012; pages 1056-1063.

H. J. Brothers, Pascal's prism. The Mathematical Gazette, Vol. 96, No. 536, 2012; pages 213-220. (Supplementary material)

H. J. Brothers, Pascal's triangle: The hidden stor-e. The Mathematical Gazette, Vol. 96, No. 535, 2012; pages 145-148.

H. J. Brothers, Finding e in Pascal's triangle. Mathematics Magazine, Vol. 85, No. 1, 2012; page 51.

H. J. Brothers, Mandel-Bach Journey: A marriage of musical and visual fractals. Proceedings of Bridges Pecs, 2010; pages 475-478.

H. J. Brothers, Intervallic scaling in the Bach cello suites. Fractals, Vol. 17, No. 4, 2009; pages 537-545.  (Supplementary material)

H. J. Brothers, How to design your own pi to e converter. The AMATYC Review, Vol. 30, No. 1, 2008; pages 29-35.

H. J. Brothers, Structural scaling in Bach's cello suite no. 3. Fractals, Vol. 15, No. 1, 2007; pages 89-95.  (Supplementary material)

H. J. Brothers, Improving the convergence of Newton's series approximation for e. College Mathematics Journal, Vol. 35, No. 1, 2004; pages 34-39.   [723KB]

(The above article appears with permission of CMJ. Supplementary material can be found here.)

J. A. Knox and H. J. Brothers, Novel series-based approximations to e. College Mathematics Journal, Vol. 30, No. 4, 1999; pages 269-275.   [126KB]

(NOTE: The above paper was selected by mathematicians Ron Larson, Robert P. Hostetler, and Bruce H. Edwards as one of the fifty best articles on calculus from MAA periodicals. It appears as a supplement to their textbook, Calculus with Analytic Geometry, Seventh Edition.)

H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. The Mathematical Intelligencer, Vol. 20, No. 4, 1998; pages 25-29.   [1,143KB]


Mandel-Bach Journey: A Marriage of Musical and Visual Fractals, Bridges Conference, Pecs, Hungary, July 28, 2010.

Fractals and Music, The MathScience Innovation Center, Richmond, VA, August 16, 2007.

Fractal Music Lab: A Modern Approach to the Mensuration Canon, Norwalk Community College, Norwalk, CT, March 28, 2006.

Fractal Music, Mathematical Association of Two-Year Colleges in Connecticut, North Haven, CT, April 29, 2005.

Rubbing Shoulders With Newton: A New Look at a Fundamental Constant of Nature, Society for Amateur Scientists, 3rd Annual Meeting, Las Vegas, NV, January 14, 2005.

Fractal Music Lab, Yale University Workshop on Fractal Geometry, New Haven, CT, August 17, 2004.

Exponential Explorations: Inspiring Students With Their Own Contributions to Number Theory, Joint Meeting of the Mathematical Association of Two-Year Colleges in Connecticut and the Northeast Section of the Mathematical Association of America, Danielson, CT, May 5, 2000.


A sample of original compositions. Styles include Jazz, Brazilian, Pop, and Fractal.

Fractal music CD in honor of Benoit Mandelbrot for ScienceWriters2010, hosted at the Yale Peabody Museum for the National Association of Science Writers, November 6, 2010.

A selection of fractal music examples that illustrate specific power-law relations.

Jazz guitar performance with 12 piece Latin band Sonido Unidad.


Math music by Harlan Brothers
"Log-a-Rhythmic" is a futuristic, yet timeless, cross-cultural dance groove. It's based on Robert Schneider's non-Pythagorean scales, and uses an original algorithm to map harmonic and rhythmic structure to prime numbers.
Fractal music by Harlan Brothers
"100 Seconds of Pi" is an animation generated from the original algorithmic composition of the same name.  All pitches, durations, and dynamics in each of the six voices of the composition are derived from the digits of pi.
Fractal music by Harlan Brothers
"Brecker 8," a fractal music composition with accompanying video by Teja Krasek, featured on IBM's website.  (See selection number 6 in the section entitled "Fractals in culture and society.")
iKandl Birthday App, an eco-friendly, interactive, multimedia greeting app for the iPhone and iPad, now at the iTunes Store.

Mandelbrot Zoom, a fractal animation produced for the feature length documentary BACH & friends, January, 2010.

The Bach Project interview with filmmaker Michael Lawrence for feature length documentary on Johann Sebastian Bach's pervasive influence in 21st century music, December 5, 2008.

Fractal Music Composer is a Java app that enables you to create original prolation canons.

Mandelbrot's World of Fractals, Technical consultant for DVD by Nigel Lesmoir-Gordon, Gordon Films UK, in conjunction with Yale University and the National Science Foundation, December, 2003.

Liquid Crystal Glass House, Cellular automata programming for Michael Silver's design and exhibit at "Super-ficial: The Surfaces of Architecture in a Digital Age," New Museum of Contemporary Art, Zenith Media Lounge, New York City, January 31 - April 6, 2003.

In the News:

Fractal Music
Real Life Survival Guide with Bruce Barber on NPR
Putting the App in Happy Birthday  (Cindy Gerber)
The Mathematical Tourist  (Ivars Peterson)
Harlan J. Brothers in the Spotlight  (Forrest Mimms)
Science News  (Ivars Peterson)
UAB Magazine  (Dan Willson)
Science Magazine  (Dana Mackenzie)
NASA Goddard Institute for Space Studies  (John Knox)


Music & Mathematics: Cosmic Twins (The International Child Art Foundation)
Turn Off, Tune In, Chill Out (Real Life Survival Guide)
"Glimpses of Benoît Mandelbrot (1924-2010)." Notices of the American Mathematical Society, Vol. 59, No. 8, 2012; pages 1056-1063. With: M. F. Barnsley, M. Berry, M. Frame, I. Stewart, D. Mumford, K. Falconer, R. Eglash, N. Lesmoir-Gordon, and J. Barrallo.
Rubbing Shoulders with Newton: How I became a Mathematician (Citizen Scientists League)
Technology in Education  (The Hoot)

Professional Service:

Referee for The American Mathematical Monthly
Referee for The College Mathematics Journal


According to My Peers...


Brothers Technology
Research on Fractal Music
Fractal Music Workshops
On-Line Encyclopedia of Integer Sequences: Pascal's prism
OEIS Profile
Wolfram Research  (Eric Weisstein)
Fractal Geometry at Yale: Panorama of Uses  (Michael Frame)
Yale Workshop on Fractal Music (mirror)
Introduction to Fractal Geometry

Contact Information


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