Research
Harlan Brothers, founder of Brothers Technology, had his first success as an inventor in the early nineties with his sale of the Bathtub Buddy™ to a major manufacturer of small appliances, Salton Inc. Salton incorporated this unique water alarm into its popular Wet Tunes line of products. Since then Harlan has obtained five patents and worked as a design consultant.
His research into the problem of creating and authenticating tamperproof digital recordings led to a patent for The Event Verification System™ (EVS). EVS offers a broad solution that fulfills the evergrowing need for irrefutable authentication of digital information. The patent was sold to a wellestablished intellectual property firm.
Current projects range from novel consumer devices to commercial encryption techniques and educational tools.
In the area of pure research, Harlan has a longstanding interest in number theory and its applications. He has discovered formulas and relationships relating to the constants e, pi, and Euler's gamma. His paper entitled "Improving the Convergence of Newton's Series Approximation for e" includes the fastest known methods for computing this fundamental constant of nature. The article appears in the January 2004 issue of The College Mathematics Journal. Here is a presentation on the subject from the Third Annual Citizen Science Conference.
For six years he worked with Michael Frame and Benoit Mandelbrot at Yale University to explore the use of fractals in mathematics education. Projects at Yale included a lecture and workshop on the subject of fractal music composition and analysis. Here is a brief introduction to fractals in PDF format [1.7MB].
Working with Benoit inspired Harlan to explore new places to apply fractal geometry, both in the natural world and in number theory. Harlan has published and lectured on fractal geometry in music, has used it to explore an original 3D extension of Pascal's triangle, and used Iterated Function Systems to uncover biases in the distribution of prime numbers.
The following links reference early research on one of the fundamental constants of Nature, the base of the natural logarithm, e:

NASA (Serendipite, John Knox)

Here are
links to more information on
e.
Publications

H. J. Brothers, "Using Iterated Function Systems to Reveal Biases in the Distribution of Prime Numbers." arXiv eprints, arXiv:1701.00698, December 2016.


M. F. Barnsley, M. Berry, M. Frame, I. Stewart, D. Mumford, K. Falconer, R. Eglash, H. J. Brothers, N. LesmoirGordon, J. Barrallo, Glimpses of Benoit Mandelbrot (19242010). Notices of the American Mathematical Society, Vol. 8, No. 59, 2012; pages 10561063.


H. J. Brothers, "Pascal's triangle: The hidden store ." The Mathematical Gazette, Vol. 96, No. 535, 2012; pages 145148.


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


H. J. Brothers, "Intervallic scaling in the Bach cello suites." Fractals, Vol. 17, No. 4, 2009; pages 537545.
(Supplementary material can be found here.)


H. J. Brothers, "How to design your own pi to e converter." The AMATYC Review, Vol. 30, No. 1, 2008; pages 2935.


H. J. Brothers, "Structural scaling in Bach's cello suite no. 3." Fractals, Vol. 15, No. 1, 2007; pages 8995.
(Supplementary material can be found here.)

The following files are in Adobe PDF format.

J. A. Knox and H. J. Brothers, Novel seriesbased approximations to e. College Mathematics Journal, Vol. 30, No. 4, 1999; pages 269275. [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 is now a supplement to their textbook, Calculus with Analytic Geometry, Seventh Edition.)
