Category: Mathematics

  • Growing Neural Networks

    John F. Lindner

    Artificial neural networks are increasingly important in society, technology, and science, and they are increasingly large and energy hungry. Indeed, the escalating energy footprint of large-scale computing is a growing economic and societal burden. Must we always use brute force, or can we get by with less? I just co-authored an article in Proceedings of…

  • Logic with Nonlinear Maps

    John F. Lindner

    In 1999, Bryan Prusha ’98 and I wrote an article for Physics Letters A illustrating why logic requires nonlinearity. Recently, with Bill Ditto, I revisited this theme by demonstrating how to encode all 16 binary boolean (true-false) functions in single iterations of a unimodal map, just published in Physica D. These encodings may facilitate the…

  • Weak Prime Number Theorem

    John F. Lindner

    As a child, I was inspired by Arthur C. Clarke‘s 1956 science fiction novel The City and the Stars to search for patterns in prime numbers. Chapter 6 begins: Jeserac sat motionless within a whirlpool of numbers. The first thousand primes, expressed in the binary scale that had been used for all arithmetical operations since…

  • 2D Kepler Conjecture

    John F. Lindner

    Johannes Kepler asserted in 1611 that no packing of identical balls has density greater than the hexagonal close-packed “cannonball” packing of oranges at a grocery store’s fruit stand. But the gulf between intuition and proof was so large that the latter was not achieved until the work of Thomas Hales and Samuel Ferguson in 1998.…

  • Sum of Reciprocals

    John F. Lindner

    The sum of the reciprocals of the natural numbers diverges, but slowly, like the logarithm of the number of terms. The sum of the reciprocals of the prime numbers also diverges, but even more slowly, like the logarithm of the logarithm of the number of terms, as the primes are sparse in the naturals! Here…

  • Bertrand’s Postulate

    John F. Lindner

    When searching for prime numbers, the next prime number is no larger than twice the current number. Postulated by Joseph Bertrand, first proved by Pafnuty Chebyshev, I present an elementary proof based on one by the teenage Paul Erdős. Erdős was one of the most prolific twentieth century mathematicians, publishing about 1500 articles with more…

  • Stegosaurus Tiling

    John F. Lindner

    John Chase, the head of the Walter Johnson High School Math Department, in Maryland, near Washington DC, liked my Stegosaurus variation of the Spectre monotile so much that he had his students paint it on the wall of their math office! Attached are a couple of photos he shared. Smith, Myers, Kaplan, and Goodman-Strauss recently discovered an infinite…

  • Vampire Ein Stein

    John F. Lindner

    Just a couple of months after announcing the remarkable discovery of a single shape that forces a non-periodic tiling of the plane, Smith, Myers, Kaplan, and Goodman-Strauss have announced an improved aperiodic monotile or ein stein. (Ein stein is “one stone” in German.) The hat and turtle shapes tile the plane only non-periodically, but with their…

  • Behold, an Ein Stein!

    John F. Lindner

    This academic year has been thrilling: first nuclear fusion breakeven, now an ein stein! Last week, a preprint at arxiv.org by David Smith et al. announced an “ein stein”, or one stone, a shape that forces a non periodic tiling of the plane, ending a half-century quest by many researchers, including me. A retiree and…

  • 5-Color Theorem

    John F. Lindner

    On 1852 October 23, Francis Guthrie noticed that he needed only 4 colors to color the counties of England so no two bordering counties shared the same color. This works for any map, but only in 1976, and with the aid of a computer, did Kenneth Appel and Wolfgang Haken finally prove the 4-color theorem. Here I outline…

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