• Where Are the Stars?

    John F. Lindner

    When viewing space photography, such as Apollo or International Space Station photos, people often ask, “Where are the stars?” Typically such photos properly expose the bright lunar or space station surfaces and consequently underexpose the dim background stars, rendering space as featureless black.

    Current ISS astronaut Matthew Dominick has been experimenting with photography, and his photo below, of a docked SpaceX Dragon taken from a docked Boeing Starliner, just after orbital sunset and just as Earth’s moon rises, does show stars from our Milky Way galaxy, with the spacecraft dimly illuminated by moonlight. Note the face in the Dragon window.

    Spacecraft and stars
    Dragon spacecraft with Milky Way stars illuminated faintly by moonlight, 2024 June 29. A 1s, f1.4, ISO 5000, 28mm photo by NASA astronaut Matthew Dominick. Click for a larger version.

  • 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 than 500 coauthors. (Indeed, my Erdős number, or collaboration distance, is 5.) Reportedly, Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems. Like Aigner & Ziegler’s presentation in their Proofs from The Book, my “illustrated” version is a modest attempt at such a proof, a deep result proved by elementary means: bounding the central binomial coefficient (2n)! / (n!n!) above and below exposes the necessity of primes p for all n < p \le 2n. Enjoy!

    Some Notation

    In analogy with the factorial function

    n! = \prod_{m\le n} m

    as the product of all positive integers m not greater than n, define the primorial function

    n\# = \prod_{p\le n} p

    as the product of all primes p not greater than n. Recall the binomial function

    \binom{n}{m} = \frac{n!}{m!(n-m)!}

    and the floor function

    \lfloor x\rfloor =\max\{n\in \mathbb {Z} \mid n\leq x\}.

    Primorial Function Upper Bound

    The primorial function is upper bounded by the exponential

    x\# = \prod_{p\le x}p \le 4^{x-1} < 4^x.

    Proving this for the largest prime q < x is sufficient, as the substitution unchanges the left side and lowers the right side. For x = 2, the bound 2 < 4 is correct. For induction, assume it’s true for primes x < 2m and for x = 2m+1 split the product

    (2m+1)\# = \prod_{p\le 2m+1} \hspace{-0.7em}p = \prod_{p\le m+1} \hspace{-0.5em}p\ \prod_{m+1 < p\le 2m+1} \hspace{-1.7em}p\hspace{1em}.

    The first factor is bounded by the induction hypothesis. For the second factor, consider the binomial expansion

    2^{2m+1}=(1+1)^{2m+1}=\sum_{k=1}^{2m+1}\binom{2m+1}{k} =\cdots + \binom{2m+1}{m} + \binom{2m+1}{m+1} + \cdots \ge \binom{2m+1}{m} + \binom{2m+1}{m+1} = 2\binom{2m+1}{m},

    where the two central binomial coefficients are equal (as in Pascal’s triangle). But the integer

    \binom{2m+1}{m} = \frac{(2m+1)!}{m!(m+1)!} = M \prod_{m+1 < p\le 2m+1} \hspace{-1.6em}p \hspace{1em}\ge \prod_{m+1 < p\le 2m+1} \hspace{-1.6em}p\hspace{1.2em},

    where the integer M>1, as the bounded primes p divide the numerator but not the denominator. Combine these results to get

    (2m+1)\# \le4^m \cdot 2^{2m} = 4^{2m}

    as desired.

    primorial
    Primorial function x\# (blue) and its upper bound (red).

    Central Binomial Prime Factors

    Consider the one central binomial coefficient

    \binom{2n}{n} = \frac{(2n)!}{n!n!} = \frac{(2n)!}{(n!)^2}.

    Since \lfloor n/p \rfloor factors of n! are divisible by p, and \lfloor n/p^2 \rfloor factors of n! are divisible by p^2, and so on, n! contains the prime p exactly \sum_{k\ge1} \lfloor n/p^k\rfloor times. Thus,

    n!=\prod_p p^{\sum_k \lfloor n/p^k\rfloor}

    and

    \binom{2n}{n}=\prod_p p^{\sum_k \left( \lfloor 2n/p^k\rfloor-2\lfloor n/p^k\rfloor \right) }.

    Since

    x-1<\lfloor x \rfloor \le x,

    the integer summands difference

    \left\lfloor \frac{2n}{p^k} \right\rfloor -2\left\lfloor \frac{n}{p^k}\right\rfloor < \frac{2n}{p^k}-2\left(\frac{n}{p^k}-1\right) = 2

    and thus must be either 0 or 1.

    If p^k > 2n, \lfloor 2n / p^k \rfloor = 0 and \lfloor n / p^k \rfloor = 0 and no power of p divides (2n)!/(n!)^2. If p^k \le 2n, then the divisor’s highest power k \le \log 2n / \log p, but if p > \sqrt{2n}, then \log 2n / \log p < 2, and the power must be 0 or 1.

    For n \ge 3, if 2n/3 < p \le n, then p \le n < 2p \le 2n < 3 p, which implies that (2n)! contains p and 2p and not 3p while n! contains p and not 2p, so the powers of p in (2n)!/(n!)^2 cancel.

    Largest prime powers dividing the central binomial coefficient.
    Largest prime powers dividing the central binomial coefficient. Only the “smallest” prime powers can divide the binomial multiple times.

    Central Binomial Upper Bound

    Split the central binomial coefficient into products of successive ranges of primes and generously bound the factors from above by the previous results to get

    \binom{2n}{n}=\prod_{\smash{p}} p^{k_p} =\prod_{\smash{p \le \sqrt{2n}}} \hspace{-0.5em} p^{k_p} \prod_{\smash{\sqrt{2n} < p \le 2n/3}} \hspace{-1.5em} p^{k_p}\ \prod_{\smash{2n/3 < p \le n}} \hspace{-1.1em} p^{k_p} \prod_{\smash{n < p \le 2n}} \hspace{-0.7em}p^{k_p} \le\prod_{\smash{p \le \sqrt{2n}}} \hspace{-0.5em}2n \prod_{\smash{p \le 2n/3}} \hspace{-0.4em}p \prod_{\smash{2n/3 < p \le n}} \hspace{-0.9em}p^{0} \prod_{\smash{n < p \le 2n}} \hspace{-0.6em}p < (2n)^{\sqrt{2n}} \cdot 4^{2n/3} \cdot 1 \cdot (2n)^N,

    where N is the number of primes between n and 2n, if any.

    Central Binomial Lower Bound

    Because the central binomial coefficient is the largest,

    2^{2n}=(1+1)^{2n}=\sum_{m=0}^{2n}\binom{2n}{m} = 2 + \sum_{m=1}^{2n-1}\binom{2n}{m} < 2n\binom{2n}{n},

    and so

    \frac{4^n}{2n} < \binom{2n}{n}.

    Central Binomial Squeeze

    Combine the central binomial coefficient upper and lower bounds to get

    \frac{4^n}{2n} < \binom{2n}{n} < (2n)^{\sqrt{2n}} \cdot 4^{2n/3}(2n)^N,

    which simplifies to

    4^{n/3} < (2n)^{\sqrt{2n}+N},

    and so the number of primes in n < p \le 2n is

    N > \frac{2n}{3 \log_2(2n)}-\sqrt{2n}-1.

    Evaluate the right side to find N>1 for all n > 507. For n \le 507, the sequence of primes

    2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 521,

    where each is smaller than twice its predecessor, then suffices to prove Bertrand’s postulate for all n \ge 1.

    Number of primes
    Number of primes N in n < p \le 2n (blue) and its generous lower bound (red).

  • Aero thermo dynamics

    John F. Lindner

    Up early this morning to watch the spectacular fourth integrated flight test of SpaceX’s Superheavy Starship, the largest rocket ever built. Each IFT has greatly improved on the previous one, and the fourth was no exception. For the first time, both the booster and the ship softly splashed down in the ocean!

    Especially impressive was watching live onboard views from the ship as it reentered the atmosphere from orbital speed. Adiabatic heating (not friction) ionized the surrounding air. The resulting plasma sheath would have caused a customary communication blackout, but starship’s large size and SpaceX’s low-Earth-orbit Starlink satellite internet constellation (operating “up” instead of “down”) enabled nearly continuous live video of the descent.

    One camera was pointed toward a forward control flap, which did suffer some heating damage, but continued to function, controlling the descent attitude and enabling the final flip-and-burn deceleration maneuver. Forward flaps on near-future versions of starship may be moved leeward to improve reliability and ease manufacturing.

    Starship 29 re-enters Earth’s atmosphere, 2024 June 6. Adiabatic heated and ionized gas wraps around the ship, like a meteoroid creating a meteor in its wake. S29 splashed down softly in the ocean, the largest object ever to survive reentry intact. (SpaceX)

  • 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 continuum of aperiodic monotiles, of which the stegosaurus is a specially simple equilateral example with only ±60° and ±90° turns (and a single 0° turn).

    Stegosaurus mural in the Walter Johnson High School Math Department office. (John Chase)
    Mural detail with a few upright stegosauruses near the middle. (John Chase)

  • A Better Alphabet

    John F. Lindner

    I still retain the episodic memory of my first encounter with the spelling of people. I was learning to read, and I got cat, mat, pat; I got lot, pot, dot; but I did not get people. Why the o, and why the le instead of el? Soon after I balked at Wednesday; surely that should be something like Wensday (or even Wenzday)? I had discovered the Latin alphabet’s historical irregularity (which may have contributed to my later focus on the regularity of natural laws studied by physics).

    Shavian is a better English alphabet created in the mid twentieth century by Kingsley Read for a competition funded by the will of Nobel-Prize-winning Irish playwright George Bernard Shaw. In contrast to Latin’s 2 × 26 = 52 letters, Shavian’s 48 letters uniquely represent 48 English sounds. The letters can be drawn with single gestures and are rotations or reflections or compounds of companion letters. Often the shapes of the letters suggest their pronunciations; for example, most unvoiced letters are tall, and their voiced counterparts are deep. Different letter pronunciations can accommodate different English accents with the same spelling. Shavian has no capital letters, but a leading center dot denotes proper names.

  • Chemical Black Hole Horizons and Light-Matter Interactions at the APS EGLS Spring Meeting

    Cody Leary

    I had a blast this weekend traveling with three Wooster students to the spring meeting of the Eastern Great Lakes section of the American Physical Society, at Kettering University in Flint, Michigan.  Two students (Junior Tali Lansing and Senior Kelsey McEwen) presented research there performed by them while at Wooster.  Tali presented her work done with professor Niklas Manz using a chemical wave system to model phenomena that occur near the event horizon of a black hole.  Kelsey, my senior independent study student, presented her work modeling the trajectory of small transparent particles illuminated by strong laser beams. Yohannes Abateneh also attended, and is making good progress on his own research extending Kelsey’s work, which I expect he may present at the next EGLS meeting this fall!  

    Here are Tali and Kelsey (left to right), presenting their posters at the meeting.

    We snapped a group photo with some branches we found, while we were feeling in a goofy mood.

    A highlight of the meeting for me was seeing Wooster Physics alum Joseph Smith ’15, who is now a professor at Marietta College, receive a region-wide American Physical Society award: the Doc Brown young investigator award. Professor Smith brought four undergraduate presenters to the conference from Marietta!

    Our students also reported back to me with their own highlights:

    Tali:  I chose to attend the talks about biological, chemical, and medical physics. It was interesting to hear about all of the different research going on, but I was most interested in Mahsa Servati’s presentation. She discussed the importance of diagnosing every mutation that can be present in one GBM because it impacts the chemotherapy and radiation treatment plan.

    I appreciated meeting undergraduate students from all over our region. There was a great sense of community and support within our group.

    Kelsey:  I really liked the talk about lasers damaging dielectrics. The speaker explained it really well, so I understood what he was talking about despite knowing nothing about the subject.  [Note from Dr. Leary –  that was Prof. Smith ’15’s talk!]

    Yohannes: I mostly attended the Nuclear and Particle Physics talks. The first one was concerned with Deriving Doppler’s effect using Maxwell Equations by assuming a relative velocity between light and an observer. The second talk was concerned with correcting measurements made on the terms on the emission sources from the collision of nuclei material. Last from this section was concerned with correcting deviations between predictions made by the standard model and measurements in hadronic B decays. I wasn’t really able to follow much of this talk since I haven’t taken particle physics. The last talk I attended was concerned with a condensed matter representation of the solar core. It was concerned with trying to explain things like the perihelion precession through a model of a lattice structure that was asymmetric.

    If you are a Wooster student reading this and are interested in attending a future physics conference, just reach out to us and we will work to make it happen!

  • Measuring the Solar System

    John F. Lindner

    Thousands of years ago, ancient astronomers like Aristarchus and Eratosthenes combined careful observations with simple mathematics to measure the solar system, especially the diameters D of Earth, Luna (Earth’s moon), Sol (Earth’s star, the sun), and the radii r of their orbits. You too can do this, but it helps to observe an eclipse or two.

    Step 1: Diameter of Earth

    Measure how shadow lengths vary with latitude. No need to pace the distance between Alexandria and Syene, just use your favorite map software! Assume Sol is far from Earth (and check in Step 3), so Sol’s rays are nearly parallel.

    Step 2:  Distance to Luna

    Measure the duration of a lunar eclipse.

    Step 3:  Distance to Sol

    Measure the angle  between Sol and Luna at first quarter moon, when Luna appears to be ahead of Earth in its orbit, like a signpost to a car on a road. This is the most difficult step, as the angle is nearly but not quite ninety degrees, but the result is the astronomical unit. (Alternately, measure the ratio of time Luna is crescent to gibbous.)

    Step 4:  Diameter of Luna

    Measure the time for Luna to enter Earth’s shadow during a lunar eclipse. Consistent with Step 3, again assume Earth is far from Sol, so Earth’s shadow is nearly uniform.

    Step 5:  Diameter of Sol

    Note that Sol and Luna have about the same apparent angular size (both subtending about half a degree). This is most spectacularly evident during a solar eclipse, where Luna just barely covers Sol — if you were ever fortunate enough to experience a solar eclipse.

  • Wooster’s Time Crystals

    John F. Lindner

    Saturday, March 8, 2008. A heavy snow, one of the heaviest I remember, shuts down the city of Wooster. Streets are undriveable, so I walk to Taylor Hall, getting snow in my boots.

    Taylor is deserted, as the College has begun Spring Break, but like yesterday, Kelly and I work all afternoon and evening in the Physics Shop, me with wet socks. We’re leaving tomorrow, weather permitting, for the New Orleans American Physical Society meeting, where Kelly will present her senior thesis research, but with just hours until our departure, and just days until her I.S. is due, the apparatus is still not working.

    Todd brings us sandwiches, which we eat for dinner in the empty Reading Room. Then, just after dinner, back in the shop, the apparatus works for the first time, just as in our computer simulations. But simulation is one thing, reality is another! With no time to celebrate, we video record the dynamics, and move upstairs to the intro physics lab. On one computer, Kelly steps through the video frame-by-frame recording the motion, while at another, I work on the APS poster, which we output with the Taylor large-format printer shortly before midnight.

    Next morning is sunny and white, a winter wonderland. The roads are plowed but still snowy. We cautiously drive to the airport and fly to New Orleans via Houston. Kelly’s poster presentation goes well, but I have a fever, sore throat, and stuffy nose. I blame the wet socks.

    Wooster Physics at the March 2008 APS meeting in New Orleans. Kelly is on the right, and I am next to her. Todd is on the left.

    Kelly’s work spawned three publications in refereed journals, two in the Physical Review and one in Chaos, involving many other undergraduates, as we gradually refined the apparatus over the next decade. The final version is on a shelf in my living room beside me as I write this 16 years later.

    We designed and constructed a mechanical array of bistable pendulums coupled one-way in a topological ring, so each element affected the next one, but not vice versa, simultaneously violating Newton’s third law of action and reaction, momentum conservation, and energy conservation. Nevertheless, we achieved this by powering the device with a constantly flowing fluid, first water and later air. Each element directed the fluid flow on the next element to rotate it from one stable state to the other.

    In this way, solitary waves or solitons of see-sawing elements propagated in one direction along the array, each soliton undoing what the previous one had done. Periodic boundaries enabled solitons to annihilate pairwise in arrays with an even number of elements, but solitons propagated indefinitely in arrays with an odd number of elements, where the oddness frustrated pairing and forbade a “ground state” of alternating elements. The frequency of motion depended continuously on the fluid speed and discretely on the number of elements.

    We called them one-way arrays, although today we’d probably call them time crystals, as they are not only periodic in space, but periodic in time, for as long as the constant fluid flow continues.

    3D printed final design after a decade of development. Wind blows down, solitons move right. Deflector of one element selectively shields the wing of the next, causing wind torque to rotate the next’s tail to the opposite (red) stopping rod. Periodic boundary element, shown in motion, is split into two (cyan) pieces connected coaxially via (gray) rod and (cyan) gears.

    REFERENCES (* indicates undergraduate coauthor)

    “Experimental observation of soliton propagation and annihilation in a hydromechanical array of one-way coupled oscillators”, J. F. Lindner, K. M. Patton*, P. M. Odenthal*, J. C. Gallagher*, B. J. Breen, Physical Review E, volume 78, pages 066604(1-5) (2008)

    “Electronic and mechanical realizations of one-way coupling in one and two dimensions”, B. J. Breen, A. B. Doud*, J. R. Grimm*, A. H. Tanasse*, S. J. Tanasse*, J F. Lindner, K. J. Maxted*, Physical Review E, volume 83, pages 037601(1-4) (2011)

    “A wind-powered one-way bistable medium with parity effects”, T. Rosenberger*, G. Schattgen*, M. King-Smith*, P. Shrestha*, K. J. Maxted*, J. F. Lindner, Chaos: An Interdisciplinary Journal of Nonlinear Science, volume 27, pages 023114(1-5) (2017)

  • Venus’s Supercritical Ocean

    John F. Lindner

    The pressure and temperature near the surface of Venus are so high that its carbon dioxide atmosphere is a global ocean of a remarkable state of matter, a supercritical fluid, which fills any container like a gas but is as dense as a liquid.

    I created a carbon dioxide pressure versus temperature phase diagram using Mathematica and its curated computable data. Phase transitions separate single-phase regions. Moving along the boiling-condensing curve from the triple point, the liquid and gas densities converge at the critical point, beyond which carbon dioxide can transition between liquid and gas without boiling or condensing! I added points representing the near-surface atmospheres of Earth and Venus, with the latter being in the supercritical region above both the critical temperature and pressure.

    Only the Soviet Union‘s Venera spacecraft have landed on Venus’s alien surface, and only between 1975 and 1982. Their cameras provided us our first and so far only glimpses of Venus from beneath its supercritical “ocean”.



    Carbon dioxide pressure versus temperature phase diagram, created in Mathematica. Carbon dioxide is a gas near Earth’s surface (blue dot) but is a supercritical fluid near Venus’s surface (white dot).

    At the bottom of a supercritical “ocean”, the surface of Venus (top) reconstructed by Don Mitchell based on Venera 14 panoramas (bottom) processed by Ted Stryk.

  • Percy & Ginny

    John F. Lindner

    A chill went through the spaceflight community last week as NASA reported that it had lost contact with the Ingenuity Mars helicopter. Delivered to Mars underneath the Perseverance rover and intended as a 5-flight 30-sol tech demo, it had vastly exceeded expectations, including the first successful powered flight off Earth, a true Wright brothers moment, and more than two hours cumulative flying time in 72 flights over 1000 sols.

    Although Perseverance (nicknamed Percy) was able to reestablish communication with Ingenuity (nicknamed Ginny), NASA reports today that one of Ginny’s rotors is broken, and the first Mars helicopter will not fly again. Like the Apollo 11 lunar lander Eagle, Ginny carries a swatch from the original Wright flyer, and just last month the Smithsonian’s National Air and Space Museum added one of Ginny’s prototypes to its collection.

    Flying Ingenuity helicopter photographs grounded Perseverance rover (left) and vice versa (right) -- on Mars! (NASA photos.)
    Flying Ingenuity helicopter photographs grounded Perseverance rover (left) and vice versa (right)
    — on Mars! (NASA photos.)

    Ginny and Percy selfie. (NASA photo.)
    Ginny and Percy selfie. (NASA photo.)

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