Published in Cantor’s Paradise·PinnedMember-onlyHow to Derive the Fastest Series For e“My powers are ordinary. Only my application brings me success.” — Isaac Newton (1643–1727) Leonard Euler in his Introductio in analysin infinitorum of 1748 used the limit definition of e along with Newton’s generalized binomial expansion of (a + b)^n to derive the computationally efficient series approximation, While…Approximation6 min read

Published in Towards Data Science·PinnedMember-onlyFractal Music from RandomnessWell-chosen variables can make a randomized sequence of notes sound musical — “My life seemed to be a series of events and accidents. Yet when I look back, I see a pattern.” ― Benoit B. Mandelbrot (1924–2010) Motivation and Background Music is a rich source of multi-dimensional data. …Fractals7 min read

Published in Cantor’s Paradise·PinnedMember-onlyPascal’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) Background 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 Triangle5 min read

Published in Cantor’s Paradise·PinnedMember-onlyRamanujan, 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. …Mathematics2 min read

Published in Cantor’s Paradise·PinnedMember-onlyThe 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…Fractals4 min read

Aug 4Member-onlyBig 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 [1]. For reference, Equation 1 shows the classic series that was the starting point for these…E5 min read

Published in ILLUMINATION-Curated·Jul 23Member-onlyHistorySnap, crackle, pfishhh… Countless creaks and hisses, A pop… Then puff of steam. Whimsical whorls give way to a burst of micro pyrotechnics. Beneath, The low warm rumble of flames Lapping effortlessly up and around The tumble of logs. …Poetry2 min read

Published in Cantor’s Paradise·Jul 22Member-onlyBach 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) spoke in reference to the prospective contents of the Voyager spacecraft’s “Golden Record.” …Bach6 min read

Published in ILLUMINATION-Curated·Jul 10Member-onlyMy Friend, Benoit Mandelbrot“It is always the late-blooming flowers that smell the sweetest.” ― Benoit B. Mandelbrot (1924–2010) Benoit Mandelbrot was an iconoclastic visionary whose unique talents helped him to synthesize a century’s worth of disparate mathematical findings and explorations into a new paradigm he called fractal geometry. From 2001 to 2007, I…Mandelbrot5 min read

Published in Math Simplified·Jun 23Member-onlyA 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) At the Heart of Growth 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…E5 min read