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2D Kepler Conjecture
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.…
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Sum of Reciprocals
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…
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Bertrand’s Postulate
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…
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Stegosaurus Tiling
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…
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Vampire Ein Stein
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…
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Behold, an Ein Stein!
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…
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5-Color Theorem
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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Alien Suns Reversing in Exoplanet Skies
Not only can suns stand still in the sky, from some exoplanets their motion can apparently reverse! Wooster physics-math double majors Xinchen (Ariel) Xie ’21 and Hwan (Michelle) Bae ’19 and I just published an article elucidating these apparent solar reversals. Michelle and I began studying the architecture of daylighting terrestrial buildings for energy efficiency as part…
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Slide Rule Examples
Slide rules were widely used in engineering, science, and mathematics until the early 1970s, including during the Gemini and Apollo space programs. Although rendered largely obsolete by the advent of inexpensive electronic calculators, their descendants continue to have specialized applications, such as backup flight computers. Buzz Aldrin and a floating slide rule during Gemini 12 on…
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Slide Rule Theory
Slide rules were the analog computers that ruled science and engineering for 400 years. Their brilliant innovation was using logarithms to convert multiplication and division to addition and subtraction, and Slide rules feature logarithmic scales that slide past each other. For straight slide rules, logarithms of the numbers are proportional to their distances along them.…
Category: Mathematics
Recent Comments
Nice post, John! Thanks for writing these. I always enjoy them.
David Smith’s ein stein is an awesome shape!
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Thanks, Mark! I enjoy reading your posts as well.