As I transition to emeritus status tomorrow, I reflect on 33 years at Wooster. I am thankful for the freedom I’ve had to design my own courses, including eight first-year seminars; for the flexibility to explore a wide range of research topics, from celestial mechanics to biophysics to dancing in reduced gravity; and for six wonderful yearlong sabbaticals in Atlanta, Portland, Honolulu, and Raleigh. I remember most the many wonderful undergraduates I’ve worked with, including the 73 senior thesis students in the photo mosaic below. Of the 61 peer-reviewed science articles I wrote while at Wooster, 30 of them include 77 undergraduate co-authors (so far).
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Transition
73 yearlong senior-thesis co-adventurers -
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 would generate these same transcendental numbers.
Assuming the Taylor series expansion
f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} z^n,where z = x + i y \in \mathbb{C} with x,y \in \mathbb{R} and i = \sqrt{-1}, the derivative condition
f^\prime(z) = f^{(1)}(z) = \sum_{n=1}^\infty \frac{f^{(n)}(0)}{(n-1)!} z^{n-1} = \sum_{n=0}^\infty \frac{f^{(n+1)}(0)}{n!} z^n \equiv f(z) = f^{(0)}(z)coupled with the normalization condition
f^{(n+1)}(0) = f^{(n)}(0) = f^{(n-1)}(0) = \cdots = f^{(1)}(0) =f^{(0)}(0) = f(0) \equiv 1implies
f(z) = \sum_{n=0}^\infty \frac{z^n}{n!} \equiv \exp z.
Complex exponential repeatedly maps horizontal strips (left) to the entire complex plane (right), thereby defining the Archimedes and Euler constants. Click for a better view. Numerically plot this expression to discover two jewels. As in the figure, the exponential function maps the imaginary axis to the unit circle f(i\mathbb{R}) \rightarrow \mathbb{S}, with negative real parts forcing complex numbers inside (red) and positive real parts forcing complex numbers outside (blue). The function is exponential on the real axis with e-folding time 1, so f(1)=e, but periodic on the imaginary axis with period \tau, so f(z+i\tau)=f(z). Specifically, as it maps 1 to e, it maps horizontal strips of height \tau = 2\pi onto the entire complex plane, where
\pi =3.14159265358979323846264338327950\ldotsand
e = 2.71828182845904523536028747135266\ldotsare associated with Archimedes and Euler.
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Free-Fall Spinning Tunnels
Jump into an evacuated hole drilled straight through a uniform, static Earth-like sphere. Accelerate to 7.9 km/s (or 18 000 m.p.h.) at the center, then decelerate back to zero at the antipodes 42 minutes later! Step out of the hole upside down — or return 84 minutes after you left.
Last fall, as part of his senior thesis, Yuchen Gan ’21 and I used computer simulations to generalize this famous result to uniform spinning planets, where Coriolis and centrifugal effects force the tunnels into arcs curving away from the center and intersecting the surface in multiple places. We discovered many families of periodic tunnel networks that connect multiple surface locations even at non-equatorial latitudes, as in the animation. Such tunnels could ideally provide energy-free communication and transportation for the planets’ inhabitants.
But in January, in a wonderful aha! moment, we were surprised and delighted by a dramatic perspective change: the motion of an object or passenger (a “terranaut”) freely falling through the tunnel system is both spiky concave arcs with respect to the planet and a smooth convex ellipse with respect to inertial space! We subsequently proved mathematically that the inertial motion is that of a two-dimensional harmonic oscillator, and the ellipses are centered (not focused) on the planet.
Download a higher-resolution QuickTime MOV version of the animation with or without the red elliptic trace.
A mass continuously free falls through a tunnel connecting 6 surface points in a spinning planet, simultaneously executing concave spiky motion (with respect to the planet) and convex elliptic motion (with respect to inertial space) -
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 them with reaction-diffusion differential equations. Eventually, the math major Margaret McGuire ’20, along with physics major Chase Fuller ’19, Niklas, and I, succeeded in modeling “geographic tongue” in an article just now appearing in the journal Chaos.
Reaction–diffusion propagation speed depends on both the curvature of the wavefront in the surface and the curvature of the surface itself. Geographic tongue promises to be a new exemplar of reaction–diffusion on curved surfaces. Our computer simulations connect reaction–diffusion processes on curved surfaces with moving patterns on the human body.
Exemplary geographic tongue (Martanopue, Wikimedia Commons, 2012 CC BY-SA 3.0) and our Mathematica finite-element reaction-diffusion computer simulation -
Mars Sky Crane
At the NASA press conference today, chief engineer Adam Steltzner presented three iconic images of the space age: Armstrong’s photo of Aldrin on the lunar surface, Voyager 1’s photo of Saturn and its rings from above the ecliptic, the Hubble Space Telescope’s photo of the Eagle Nebula’s “Pillars of Creation” star-forming region. And then he added a new one.
The car-sized Perseverance rover hangs just above the surface, suspended by a bridle of three 7.5-meter nylon tethers from its powered descent stage, seconds before touchdown. The curly electrical umbilical that transported the 1s and 0s encoding the photo also dangles from the descent stage to the rover as the rockets blast regolith outward. Moments later pyrotechnics severed the tethers and umbilical, and the descent stage flew away to crash at a safe distance, leaving the rover six-wheels-on-Mars.
The car-sized Perseverance rover, suspended by tethers from its powered descent stage, a couple of meters above the surface, just seconds before touchdown, on Mars -
Nightfall
NASA’s Transiting Exoplanet Survey Satellite (TESS) has discovered a sextuply-eclipsing sextuple star system. I think of “Nightfall”.
The six stars of TCY 7037-89-I orbit each other in three binary pairs, as in the schematic. The primaries are slightly larger and hotter than our sun and the secondaries are about a half as large and a third as hot. All stars eclipse each other as seen from Earth, and a neural network helped identify them from the TESS light curves.
A sextuply-eclipsing sextuple star system, discovered with TESS, January 2021 In 1941, a Columbia University chemistry graduate student published one of the most famous short stories of science fiction’s Golden Age. Isaac Asimov‘s “Nightfall” imagines a planet orbiting six suns with a civilization evolved in perpetual daylight. Only once every 2000 years does an undiscovered moon eclipse a sun when it is alone in the sky plunging the civilization into darkness — and revealing tens of thousands of newly visible stars!
Isaac Asimov’s first cover story, “Nightfall”, Astounding Science Fiction, September 1941 -
Chemical Clock
Wooster’s summer 2019 Sherman-Fairchild group just published, “Disruption and recovery of reaction–diffusion wavefronts interacting with concave, fractal, and soft obstacles”, in Physica A. Working with Fish Yu ’21, Chase Fuller ’19, Margaret McGuire ’20, and Niklas Manz (Physics) was wonderful.
Sherman-Fairchild group, summer 2019 Extending previous work with Reba Glaser (SUNY Geneseo)’19 and Nate Smith ’18, we wrote computer simulations to document the distinct recovery of reaction–diffusion wavefronts disrupted by a variety of obstacles. Curvature dependent wavefront velocities ultimately restore the wavefronts, with perturbations that decay as power-law functions of time. But concave, spiral, and fractal obstacles can sustain wavefronts locally for long times. Soft obstacles with variable diffusivity, either intrinsically or due to light sensitivity, can enforce one-way propagation and, appropriately configured, can locally and indefinitely sustain incident wavefronts, creating clocks or repeaters, beating hearts for these excitable systems.
Soft obstacle with variable diffusivity (red gradient) allows wavefronts to circulate only counter clockwise inside the square channel, creating a kind of clock -
Dragon Eye
“Resilience rises! Not even gravity contains humanity when we explore as one for all.”
My eyes were glued to NASA-TV last weekend as I followed the flight of the SpaceX Dragon “Resilience” to the International Space Station. Ferrying a diverse Crew 1 of Mike Hopkins, Victor Glover, Shannon Walker, and Soichi Noguchi for a six-month stay on the ISS, this was the first commercial FAA certified human spaceflight.
Window view from the SpaceX Dragon “Resilience” during the first-ever FAA certified crewed spaceflight -
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 did I learn of Penrose’s important contributions to General Relativity, including topological arguments to demonstrate the inevitably of gravitational collapse leading to astrophysical black holes. And earlier this month I was excited to hear that Penrose — in his 90th year — shares the 2020 Nobel Prize in physics!
So I got out my Penrose tiles (thanks Woody) and assembled a small pattern. It’s not easy, but a combination of local edge and vertex rules (or a global inflation rule) can extend the aperiodic pattern to infinity.
I build a Penrose Tiling -
Cookie Cutter
Cookie dough in a cookie factory moves on a conveyor belt at a constant relativistic speed. A circular cutter stamps out cookies as the dough rushes by beneath it. In the factory frame, the dough is length contracted along the conveyor belt, and when the conveyor belt stops and the cookies are packaged for eating, they are stretched elliptically along the belt.
But the cookie dough riding the belt observes the circular stamp moving toward it and length contracted along the belt. How can the cutter contracted along the belt cut cookies stretched along the belt?
The cut is simultaneous in the factory frame but not simultaneous in the cookie dough frame! Clocks synchronized in their own frame are not synchronized in a moving frame, and the rear clock is ahead of the front clock. In the looping animation, the top panel illustrates the cut in the factory frame, where the cutter moves up & down simultaneously, while the bottom panel illustrates the cut in the dough frame, where the cutter rear moves up & down first and the cutter front moves up & down later. The wave of up-&-down motion spreads the cut over time lengthening it.
The relativistic effects of time dilation, length contraction, clock desynchronization, and the relativity of rigidity are significant only near the billion-kilometers-per-hour invariant speed of light and gravitational waves.
Relativistic cookie cutter in factory reference frame (top) and cookie dough frame (bottom)
Thanks, Mark! I enjoy reading your posts as well.