Intellectual Property & Creative 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 Brothers has obtained five patents and worked as a design consultant.

His research in the mid-nineties into the problem of creating and authenticating tamper-proof digital recordings led to a patent for The Event Verification System™ (EVS).  EVS offers a broad solution that fulfills the ever-growing need for irrefutable authentication of digital information. The patent was sold to a well-established intellectual property firm.

Current projects range from novel consumer devices to commercial encryption techniques and educational tools.

In the area of pure research, Brothers has a long-standing 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 JournalHere 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].

The following links reference early research on one of the fundamental constants of Nature, the base of the natural logarithm, e:

NASA (Serendipit-e, John Knox)
Mathematical Association of America (Science News Online, Ivars Peterson)
Science Magazine (Random Samples, Dana Mackenzie)
Wolfram Research (Eric Weisstein)
UAB Magazine (Dan Willson)

Here are links to more information on e.


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). (Supplementary material)

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 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 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 can be found here.)

The following files are in Adobe PDF format.

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 is now 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]