• Sabbatical trip to Europe – Part 3 (Otto Rössler)

    Niklas Manz

    After the conference in Switzerland, I stopped in Tübingen, Germany to visit Otto Rössler. Nearly everyone who learned about nonlinear systems knows the nowadays named Rössler attractor and his work in chaos theory in the 1970s.
    For the last four years we were in contact for my science history project and this year, I finally visited him. We had a great day and I received MANY documents from conferences 40-50 years ago. He saved ‘everything’ in countless binders, sorted by years, and kept them in ceiling-heigh bookshelfs at his home.

    Otto Rössler and Niklas Manz on 20 July 2024 at Schwärzlocher Hof (farm and restaurant), which started as an abby at around 1100.
    Otto and Reimara Rössler on 20 July 2024 at Schwärzlocher Hof – with the local peacock mom.

    In the 1970s, Otto Rössler designed a famous 3D flow that mimics the folding and bending of taffy in a taffy machine, which is now named the Rössler attractor. Described by the 3 coupled differential equations

    \begin{aligned}\dot x &= -y – z, \\ \dot y &= x + a y,\\ \dot z &= b – c z + x z,\end{aligned}

    with just one nonlinear term, the x z in the third equation. The Rössler velocity field results from the interaction of two crossed vortices, one near the origin and the other far away pointing at the “fold” in the Rössler band. The parameter space \{a,b,c\} is large, but the system undergoes a period-doubling route to chaos in \{0<a<2, b=2, c=4\}, as in the animation below, which culminates in a period-3 window.

    Post-transient {x ,y, z} solutions to the Rössler system for 0.1 < a < 0.411, b = 2, c = 4, exhibiting a period-doubling route to chaos culminating in a 3-cycle window. (You may need to “click” on the figure to see the animation.)

    The animation was created by John Lindner.

  • Sabbatical trip to Europe – Part 2 (Switzerland)

    Niklas Manz

    The second stop of my Europe trip was Switzerland. In Zürich, I visited places where Boris Belousov (the discoverer of the Belousov-Zhabotinsky reaction I am using in my lab) lived and studied during his time in exile (1910-1915). But the theme of that part of my science history project keeps validating “Searching for a ghost”.

    None of the buildings mentioned in publications still exist. And searching for original document is two archived did not provide any new information. Therefore, two impressions from Zürich: the beautiful Zürich train station and the street sign of the street Belousov supposedly lived in.

    Zürich train station
    Schmelzbergsstrasse at the ETH campus

    From Zürich, I traveled to Les Diablerets in the southwest of Switzerland to attend the Gordon Research Conference on Oscillations and Dynamic Instabilities in Chemical Systems. This is a beautiful location at 1200 m to interact with scientists from all over the world and to spend the free afternoons. One afternoon, I ‘ran’ up the switchback road to the Col de la Croix and took several hiking trails back to the village – to be back in time for a poster session.

    Les Diablerets, Switzerland (1200 m)
    View from Col de la Croix (1750 m)

    After the conference, I walked towards the Lake Geneva to take the ‘train’ at the last possible ‘train station’ before the track winds into the next valley. It was a beautiful hike to relax the brain.

    Cows in the Swiss Alps
    Hiking to the train station
    Passing a small village
    Last train station

  • Sabbatical trip to Europe – Part 1 (Lviv, Ukraine)

    Niklas Manz

    Since about 2018, I was interested in the work of Julian Hirniak, who published an article on periodic chemical systems in 1908 (and a follow-up in 1911), before Alfred Lotka’s famous theoretical 1910 paper and William Bray’s experimental work in 1921.
    The article had been published in a journal of the Shevchenko Society in Lviv (Austria-Hungarian empire at that time) in, as I read everywhere, in Ruthenian language. All those years, I could not get a scan of the article, or the journal, not even from the Shevchenko Society archive in New York City.
    In January, I contacted a physicist at the Ukrainian National Academy of Science, who recently published about the Shevchenko Society and asked him about Julian Hirniak and the journal. He immediately responded and shortly after, he sent me pictures of the article in question. He translated that article into English, I translated Hirniak’s German 1911 article into English and, together with another co-author, we submitted a manuscript in June.
    When I mentioned that I will be in Europe this summer, he invited me to visit him in Lviv, Ukraine. It became the first trip during my nearly 6-week time in Europe.

    Impression of Lviv, located close to the Polish border in Western Ukraine: surreal!
    Despite the war in the Eastern/Southern part of the country, life is going on. Electricity outages are compensated by power generators outside every shop/restaurant. On Saturday evening, we saw Mozart’s Don Giovanni in Lviv’s Opera Theater.

    With Yurij Holovatch on an overlook in Lviv, Ukraine
    Lviv protected church
    Lviv protected basement shelter
    Lviv Coffee shop with power generator
    Lviv Opera Theatre
    Lviv Opera Theater

    We also visited the archive of the Shevchenko society – on a Saturday morning. The director Kostiantyn Kurylyshyn (Head of the Department at the Vasyl Stefanyk National Scientific Library of Ukraine in Lviv) invited us because he is, as he said, nevertheless there and could show us around. Finally, I visited the place, build in 1912, where ‘every’ Ukrainian publication can be found. This is similar to the Library of Congress in Washington DC. And I could finally see the original 1908 publication – and take my own picture.

    Archive of the Stefanyk National Science Library in Lviv
    Archive of the Stefanyk National Science Library in Lviv
    Cover page of the 1908 journal volume
    With Kostyantyn Kurylyshyn in the Stefanyk National Science Library in Lviv. The brown books between our heads are 8 volumes of his planned 11 volumes about the oldest Ukrainian newspaper running from 1880 to 1939. He started that project more than 20 years ago.

  • The Longest Flight

    John F. Lindner

    As a kid pouring over the Guinness Book of World Records, I was astonished by the record longest flight; instead of lasting hours – as I would have guessed – it lasted more than two months! Today, nearly 65 years later, that amazing achievement remains one of aviation’s most enduring records.

    For over 64 days in 1958-1959, Robert Timm and John Cook flew a modified Cessna 172 above and around Las Vegas. Modifications included an extra fuel tank, a mattress, a small steel sink, and a camping toilet. The duo took turns piloting, and they refueled and resupplied every 12 hours by flying low and slow above a speeding truck.

    Robert Timm (right) and John Cook (left) flying their modified Cessna 172 near Las Vegas, Nevada, 1958-1959. (Howard W. Cannon Aviation Museum)
    Twice a day Timm and Cook refueled and resupplied from a fast truck. (Howard W. Cannon Aviation Museum)

  • 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.

    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 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

    \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. 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 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!

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