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Dandelin Spheres
In 1609, Johannes Kepler first described how planets orbit the sun in ellipses. Kepler understood an ellipse as both the locus of points whose distances from two foci sum to a constant and as the intersection of a cone and a plane. But how are these familiar definitions equivalent? In 1822, Germinal Dandelin discovered a beautiful construction that proves this…
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Spinors
Fermions like electrons, protons, and neutrons inhabit a 720° world: 360° rotations negate their quantum states, but 720° rotations restore them. A simple macroscopic model of such spinors is an arrow translating on a Möbius strip: as the center circle rotates, the attached arrow flips after 360° but flips back after 720°. In Dirac notation but…
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Squares & Cubes
Marvelously, the square of the sum of natural numbers is the sum of their cubes! Equivalently, the sum of their cubes is the square of their sum. This mathematical gem is attributed to Nicomachus of Gerasa who lived almost 2000 years ago. For example, More generally, or The accompanying animation illustrates the identity, where the cubes…
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Archimedes & Euler
A complex function that is its own derivative normalized to one at zero implicitly defines the famous Archimedean and Euler constants of circular motion and exponential growth. Even in a world of strong gravity, where the ratio of a circle’s circumference to its diameter noticeably varied from place to place, this exponential function and the axioms of mathematics…
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Geographic Tongue
The improbable email was from a pre-dental math major asking about physics research projects combining math and dentistry, but my reaction was, “Yes — only at Wooster!”. Like animated tattoos, the surface patterns of benign migratory glossitis slowly move on the human tongue. I knew my colleague Niklas Manz was working with my dentist to model…
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Novel Math, Nobel Physics
When I was a kid I used to read Scientific American at the local library. I loved Martin Gardner‘s Mathematical Games column, and I vividly remember his description of Roger Penrose‘s then recent discovery of two shapes that force a nonperiodic tiling of the plane, an aperiodic tiling, a kind of visual music. Only later…
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Higgs Without Molasses
Although almost all ordinary mass effectively arises from the kinetic and binding energy of quarks and gluons bound to protons and neutrons in atomic nuclei, the Higgs mechanism does endow some particles like quarks and weakons with intrinsic masses. Here I gently introduce the Higgs mechanism without using loose analogies like vacuum field viscosity. Stationary…
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Relaxing Fermat
In 1637, while reading a copy of Diophantus’s Arithmetica, Pierre de Fermat famously scribbled “Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.” which roughly translates to “It…
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Wooster Epicycles
A vector is the sum of its components, a mechanical vibration is a combination of its normal mode motions, a quantum state is a superposition of its eigenstates, and any “nice” function is a Fourier sum of real or complex sinusoids, . The animation below traces the Wooster W in epicycles of 100 circles-moving-on-circles in the…
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720° untangles 360° tangles
Despite growing up in three dimensions, as a kid I did not recognize one of 3D’s deep and subtle properties: full rotations tangle, but double rotations untangle! Following physicist Paul Dirac, twist a belt one full turn about its length. The 360° single twist cannot be undone without changing the belt buckles’ orientations, although the twist…
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.