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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.…
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4D Unknot
In four dimensions, you can’t tie your shoelaces — because 4D knots don’t work. Any 1D curve in 4D space can be continuously deformed to the unit circle, which is an unknot. The looping animation below demonstrates how to undo a trefoil knot in 4D, where rainbow colors code the 4th dimension. The animation pauses when…
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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°. As the circle rotates,…
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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…
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.