• Copy, Moon Joy

    jlindner

    Carrying the torch from Apollo, through shuttle and station, to a hoped-for new era of space exploration, the Artemis 2 lunar flyby exceeded expectations

    All last week, I monitored the NASA mission coverage livestream. As the crew approached the Moon (Luna), it waned from gibbous to half to crescent in just a few hours, as its apparent size grew to a basketball’s held at arm’s length. Earth appeared to set and rise. And then, partially lit by earthshine, haloed by the solar corona, floating in the void of space spangled with stars, amidst a parade of planets bathed in zodiacal light, the Moon eclipsed the Sun (Sol) and commander Reid Weisman declared, “We have not evolved to see such a sight.” Up until then, the crew photographs had done justice to their experiences, but no longer.

    Responding to the astronauts waxing poetic and ecstatic, one capsule communicator, channelling Project Hail Mary‘s fictional Rocky, replied, “Amaze, amaze, amaze,” while another cap com replied, “Copy, moon joy.”

    The crew travelled farther from Earth than any other humans, over a quarter million miles, about 1.3 light-seconds, and pilot Victor Glover advised, “Let’s actually savor the com delay we have.”

    NASA Associate Administrator Amit Kshatriya remarked, “The [crew’s] expressions of love and devotion to family … is a great example of why we go and do these missions. If you can’t take love to the stars, then what are we doing? … why do we even go? That’s why we send humans instead of robots, [for] that firsthand witness. They’ll go through a whole range of emotions, like we who are watching them, and that’s the whole point: that we can share that experience.”

    Earth out the window
    Earth outside the window of the Artemis 2 Orion crew capsule “Integrity”.

    Earthiest
    Earth appears to set as the Artemis 2 crew moves behind the Moon. The lunar surface is comparatively dark and Earth is so bright it was difficult to look at.

    Eclipse
    The Sun eclipsed by the Moon (right) as seen by Artemis 2 (left) from a camera at the end of one of its solar arrays. The Moon is partially lit by Earthshine (upper left, out of frame) with Venus (upper left, nearly eclipsed by the spacecraft) and Saturn and Mars (lower right) amidst stars of the constellation Pisces.

    Command and service modules separate
    Broadcast live, ESA’s service module (left) separates from NASA’s command module (right) with the crew shortly before Artemis 2’s atmospheric reentry, again recorded from a solar array end.

  • The Dream Is Alive

    jlindner

    As a child of the Apollo program and a lifelong dreamer of spaceflight, I am thrilled to follow the Artemis 2 mission, carrying the first humans around the Moon (Luna) in over half a century, with the intent to pick up where we left off, establish a permanent lunar presence, and proceed to Mars and beyond.

    This evening, the Artemis 2 crew of Reid Wiseman, Victor Glover, Kristina Koch, and Jeremy Hansen approaches the Moon’s gravitational sphere of influence, where lunar gravity exceeds terrestrial gravity. Kristina recently remarked, “Our strong hope is that this mission is the start of an era where everyone — every person on Earth — can look at the Moon and think of it as … a destination.” The dream is alive.

    In Greek mythology, Artemis is the sister of Apollo; in reality, Artemis is safer (better computers), cheaper (as a fraction of US budget), and bigger (crew of 4 rather than 3) than Apollo. More importantly, with international and commercial help, I am hopeful that Artemis will evolve to a sustainable program so the Moon really does enter the human sphere as a destination, dramatically and irreversibly expanding the range of human experience.

    Earth by moonlight from Artemis 2
    This Reid Wiseman Artemis 2 photo shows Earth illuminated by moonlight, except for a thin crescent illuminated by sunlight, with Venus in zodiacal light at 4 o’clock joined by aurora at 1 o’clock and 7 o’clock, shortly after translunar injection (TLI), 2026 April 2.

    Artemis 2 crew in Orion en route to the Moon
    Artemis 2 crew — Reid, Jeremy, Kristina, Victor — in their Orion spacecraft “Integrity” en route to the Moon, 2026 April 4.

  • A Method of Reaching Extreme Altitudes

    jlindner

    100 years ago, physicist Robert Goddard designed and built the first liquid-fueled rocket. Powered by gasoline and liquid oxygen and launched from his Aunt Effie’s farm in Auburn, Massachusetts on 1926 March 16, the first flight lasted 2.5 seconds and reached an altitude of 12.5 meters.

    7 years earlier, in 1919, Goddard published the seminal treatise A Method of Reaching Extreme Altitudes, whose final section is “Calculation of minimum mass required to raise one pound to an ‘infinite’ altitude”, including to the Moon. Goddard eschewed publicity, but his ideas were nonetheless widely ridiculed.

    In 1920, an unsigned New York Times editorial denied that a rocket could work in a vacuum and suggested that Goddard “seems to lack the knowledge ladled out daily in high schools.” In 1929, a local newspaper mocked one of Goddard’s experiments with the headline “Moon rocket misses target by 298,799 ½ miles”.

    Goddard remarked, “It is difficult to say what is impossible, for the dream of yesterday is the hope of today and the reality of tomorrow.”

    Robert Goddard did not live to see the Space Age he helped create, but his wife Esther Goddard, who championed his work after his death, did live to see the 1969 July 16 launch of Apollo 11, which used a liquid-fueled rocket based on principles pioneered by him to reach the Moon. The crew included Buzz Aldrin, the son of one of his students.

    The day after Apollo 11 launched, the New York Times corrected its 1920 error and acknowledged that rockets can fly in a vacuum (by expelling mass in one direction and recoiling in the opposite direction).

    Goddard standing next to the first liquid-fueled rocket
    Robert Goddard and the first liquid-fueled rocket on 1926 March 8. (The combustion chamber is above the propellant tanks, but he reversed the order in later versions.) Photo by Esther Goddard.

  • Guided Flame

    jlindner

    Yuhe Ren, Niklas Manz, and I recently published an article Guided flame: reaction-diffusion of fire pulses in narrow channels in the journal Open Transport. Tim Siegenthaler helped machine the channels. This work had been gestating for a long time, but has recently became a hot topic. Fortunately, Yuhe was able to acquire all our data in the last year, spanning his Junior I.S., Wooster summer REU (thanks to the Koontz Endowed Fund), and Senior I.S.

    We studied fire propagation in annular channels whose rectangular cross-sections are a few millimeters wide and high and whose circumferences are hundreds of millimeters long. If a channel is partially filled with a volatile flammable hydrocarbon fluid, locally igniting the vapor above the fluid can start a fire pulse that rapidly propagates around the annulus at hundreds of millimeters per second leaving behind an unexcitable region of depleted vapor, a refractory tail. Further evaporation of the volatile fluid restores the vapor and the corresponding excitable condition, allowing the returning pulse to propagate, provided the channel’s circumference is sufficiently long.

    Experimentally, we explored this quasi-one-dimensional reaction-diffusion system, discovering simple trends connecting refractory tail length and pulse propagation speed to channel length, height, and width. Computationally, we introduced phenomenological computer simulations that simply reproduce the guided flame and elucidate the underlying physics.

    Guided flame experiment.
    Yuhe’s overhead video of a blue hydrocarbon flame propagating at almost one meter per second counterclockwise in a narrow channel in a fume hood. (You may need to click or tap to see the motion.)

  • Moon Trees

    jlindner

    As command module pilot for the 1971 Apollo 14 mission, Stuart Roosa was one of 24 people to travel around the Moon* in the heroic first age of lunar exploration. He was also a former U.S. Forest Service smokejumper, and he carried into lunar orbit about 500 seeds to test the effects of spaceflight on the resulting trees. Upon returning to Earth, almost all the seeds germinated successfully, and many of the seedlings were distributed widely for the 1976 U.S. Bicentennial. After 50 years, no differences have been noted between Moon trees and Earth trees.

    NASA repeated this experiment for the 2022 Artemis I test flight. While we await the next 4 people to travel around the Moon, during Artemis II later this year, I recently visited Asheville Botanical Garden to see its Apollo Moon Tree. It was a clear and unseasonably warm February day, and I found the sycamore barren of leaves but apparently healthy, only distinguishable by the plaque at its base.

    *Luna is arguably a better name for Earth’s natural satellite.

    Moon Tree and I
    At the Asheville Botanical Garden with a sycamore tree planted from a seed that travelled around the moon with astronaut Stuart Roosa during Apollo 14 in 1971.

  • The Mathematics of Wonder

    jlindner

    Since childhood I have been fascinated by M. C. Escher‘s extraordinary graphics. Escher once wrote, “I never feel quite at home among my artist colleagues; what they are striving for, first and foremost is “beauty” … I guess the thing I mainly strive after is wonder … .”

    In 1945 Escher produced a lithograph called “Balcony“, of buildings overlooking a Malta harbor, whose center is enlarged four times compared to its edges to emphasize a single balcony. Working without computers, Escher accomplished the inflation by first manually constructing a blowup of a square grid.

    Later he imagined “a cyclic expansion or bulge, without beginning or end”. The result was his amazing 1956 lithograph “Print Gallery“, one of his favorite prints, and one of two that I have on the walls beside me as I write this. A young man in a print gallery gazes up at a print of a Malta harbor town, and as we follow his gaze rightward the scene enlarges until we see a woman looking out a window above the entrance to the print gallery containing the man — who is simultaneously inside and outside it!

    Like the circular bulge of “Balcony”, Escher accomplished the cyclic bulge of “Print Gallery” by first transforming a grid of squares, intuitively ensuring that right angles were mapped to right angles. The transformed squares became very small near the print’s center, and Escher left that part blank.

    In 2003, mathematicians B. de Smit and H. W. Lenstra Jr. published a refinement and extension of “Print Gallery” that filled the center with an infinite regress of rotated and scaled down copies of itself. Using complex variables

    z = x + i y = r e^{i \theta} \in \mathbb{C},

    and noting that moving 256 units in the original square grid corresponded to moving 22.6 units and rotating 158° in the cyclic bulge, they defined the conformal mapping

    h(z) = z^\alpha = \exp(\alpha \log z)

    by

    h^{-1}(256)\approx 22.6\, e^{-i\, 158^\circ},

    which has the principle branch solution

    \alpha \approx 1.33\,e^{-i\,41.4^\circ}.

    Working with artists and a computer programmer, they then created the final figure below, which would likely have pleased Escher.

    Complex conformal map
    Conformal map underlying the cyclic bulge (and inward spiral) of Escher’s “Print Gallery” depends on the complex constant \alpha in the exponent.

    Refined Print Gallery
    Escher’s “Print Gallery” as refined and extended by de Smit and Lenstra. The observer is both inside and outside the print!

  • Chemical Wires

    jlindner

    With Mahala Wanner and Gus Thomas, Niklas Manz and I recently published an article Chemical wires: reaction-diffusion waves as analogues of electron drift in the journal Transport Phenomena. Mahala began the work during our summer 2022 REU, and Gus continued it for his 2025 Senior IS.

    We used chemical reaction-diffusion waves in narrow channels to model electron drift in wires. By varying the initial conditions of an excitable Belousov–Zhabotinsky (BZ) medium, we achieved careful, quantitative control of BZ wave speeds in the range of electron drift speeds in conductors, a few millimeters per minute. We compared the speeds of the easily observable BZ waves and their computer simulations with theoretical electron drift speeds to explore the effects of wire radius, electric current, and material composition. Such BZ waves are compelling visual analogues of electron drift.

    The slow effective speed of typical electrical currents, despite the large quantum and thermal speeds of electrons in common wires, contributes to misconceptions about the nature of information transfer via current; individual electron trajectories do not transmit electrical signals at high speeds, but perturbations in the accompanying electromagnetic fields do. To build better intuition, below is an animation of a 10 mm / min electron drift, represented by the filling bar, with a penny for scale.

    Electron drift simulation to scale
    Electron drift simulation to scale.
    (You may need to click or tap to see the animation.)

  • e is Transcendental

    jlindner

    Introduction

    The Euler-Napier-Bernoulli constant e =2.71828\ldots is not just irrational, it is transcendental, as first proved by Charles Hermite in 1873. Inspired by the work of Mathologer (Burkard Polster with Marty Ross), here I offer an elementary proof of e‘s transcendence. As warmup, I first present a well-known proof of its irrationality while hinting at the proof of its transcendence.

    e is irrational

    The infinite series expansion of the exponential function

    e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^m}{m!} + \cdots

    implies the Euler-Napier-Bernoulli constant

    e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{m!} + \cdots .

    By way of contradiction, assume e can be written as a root of the linear polynomial

    be-a = 0

    for positive integers a,b \in \mathbb{Z}^+ or, equivalently, that e is the rational number

    e = \frac{a}{b}.

    Use b as a cutoff to separate e‘s infinite series expansion into a \color{blue}\text{large body} large body and a \color{red}\text{small tail}

    \frac{a}{b} = e = \color{blue} 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{b!} \color{red}+ \frac{\delta}{b!}\color{black},

    where

    \begin{aligned}0 < \delta &= \frac{b!}{(b+1)!} + \frac{b!}{(b+2)!} + \frac{b!}{(b+3)!} + \cdots \\&= \frac{1}{b+1} + \frac{1}{(b+1)(b+2)} + \frac{1}{(b+1)(b+2)(b+3)} + \cdots \\&< \frac{1}{b+1} + \frac{1}{(b+1)^2} + \frac{1}{(b+1)^3} + \cdots \\&= \frac{1/(b+1)}{1-1/(b+1)} = \frac{1}{b} \le 1\end{aligned}

    using the sum formula for an infinite geometric series. But

    \frac{a}{b} = e = \frac{b!\left(1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{b!}\right) + \delta}{b!} \color{red} = \frac{N^\prime + \delta}{N}

    and so

    a\color{red}N\color{black}-b\color{red}N^\prime\color{black}-b\color{red}\delta\color{black}=0

    and

    a(b-1)!-b!\left(1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{b!}\right)-\delta = 0,

    which is impossible, as all terms are integers accept 0 < \delta <1. Hence e is not the root of a linear polynomial with integer coefficients and, equivalently, is an irrational number.

    e is transcendental

    First consider the polynomial

    f(x) = \color{red}x^{p-1}\color{black}(x-1)^p(x-2)^p \cdots (x-n)^p,

    where n > 1 and p is a large prime. As |x-m|\le n on the interval x\in [0,n], it is bounded by

    |f| \le \color{red}n^{p-1}\color{black}n^p n^p \cdots n^p = n^{np+p-1},

    and its expansion begins

    \begin{aligned}f(x) &= (-1)^p(-2)^p\cdots (-n)^p \color{red}x^{p-1}\color{black} + \mathcal{O}(x^{p})\\&= (-1)^{np}(n!)^p \color{red}x^{p-1}\color{black} + \mathcal{O}(x^{p}).\end{aligned}

    Use the factorial integral

    \int_0^\infty \hspace{-0.8em}\text{d}x \,e^{-x} x^m = m!

    to expand powers of e like

    e^m = \frac{ \int_0^\infty \hspace{-0.4em}\text{d}x \,e^{m-x} f(x)}{ \int_0^\infty \hspace{-0.4em}\text{d}x \,e^{-x} f(x)} \color{red}=\ \frac{N_m + \delta_m}{N}\color{black},

    where

    \begin{aligned}N &=\frac{1}{(p-1)!} \int_0^\infty \hspace{-0.8em}\text{d}x \,e^{-x} f(x)\\&= (-1)^{np}(n!)^p + M p,\end{aligned}

    where M \in \mathbb{Z} and so N \in \mathbb{Z}, and if p > n then N\neq 0, as p divides the second term but not the first,

    \begin{aligned}N_m &=\frac{1}{(p-1)!} \int_m^\infty \hspace{-0.8em}\text{d}x \,e^{m-x} f(x)\\&= \frac{1}{(p-1)!} \int_0^\infty \hspace{-0.8em}\text{d}y \,e^{-y} f(y+m) = M^\prime p,\end{aligned}

    where M^\prime \in \mathbb{Z}, as f(y+m) has a factor of y^p enabling the integral to absorb the denominator, and

    \begin{aligned}\delta_m &=\frac{1}{(p-1)!}\int_0^m \hspace{-0.8em}\text{d}x \,e^{m-x}f(x)\\&\le \frac{1}{(p-1)!} \color{red}e^m\int_0^\infty \hspace{-0.8em}\text{d}x \,e^{-x}\color{blue} n^{np+p-1}\color{black}\\&=\frac{1}{(p-1)!} \color{red}e^m\color{blue}\frac{n^{(n+1)p}}{n}\color{black}\\&\le \frac{1}{(p-1)!} \color{red}e^n\color{blue} n^{(n+1)p}\color{black} = \frac{c\,d^p}{(p-1)!} \rightarrow 0,\end{aligned}

    as p\rightarrow \infty, where c = e^n and d = n^{n+1}>1.

    Now, by way of contradiction, assume e can be written as a root of the polynomial

    a_n e^n+a_{n-1}e^{n-1}+\cdots+a_0 = 0

    of lowest degree with a_m\in\mathbb{Z}, for m=1,\ldots,n, and a_0\neq 0. Replace powers of e to get

    a_n \left(\frac{N_n+\delta_n}{N}\right)+a_{n-1}\left(\frac{N_{n-1}+\delta_{n-1}}{N}\right)+\cdots+a_0 = 0

    and so

    a_0 \color{red}N\color{black}+(a_1 \color{red}N_1\color{black} + \cdots + a_n \color{red}N_n\color{black}) + (a_1 \color{red}\delta_1\color{black} + \cdots + a_n\color{red}\delta_n\color{black}) = 0,

    which is impossible, as the N and N_m terms are integers but the \delta_m terms are arbitrarily small. (For large enough p >|a_0| and p >n, p does not divide a_0 N but does divide N_m, and the integer parts cannot vanish.) Hence e is not the root of a polynomial with integer coefficients and, therefore, is a transcendental number.

  • My First Patent

    jlindner

    In last month’s blog post, I described my second patent, which raises the question, What was my first patent?

    In 1998, my colleagues and I were issued United States Patent No. US 5 789 961  “Nose- and coupling-tuned signal processor with arrays of nonlinear elements”. The work began during my 1994-1995 sabbatical with the Applied Chaos Lab at Georgia Tech and continued for several years, including the two publications listed below.

    Already well-known was the phenomenon of stochastic resonance, where the ability of a bistable system to detect a weak periodic signal could be optimized by non-zero noise, which might seem counter-intuitive. In simulations I performed on a NeXT computer, we explored the effects of coupling such stochastic resonators into arrays and found that signal detection was further enhanced by intermediate coupling, even if it were local and linear.

    The March 1996 Physics Today cover was an illustration, created in my Wooster office, summarizing our results: with coupling increases rightward and noise increases upward, each square represents the spatiotemporal evolution of the array at that coupling and noise with space increasing rightward and time increasing upward and doppler colors indicating movement between the two stable states (coded red and blue). Note the regular alternating bands at the optimal coupling and noise, \{10^2,10^5\}.

    This Physics Today cover prompted vehement letters-to-the-editor, not about the figure itself, but about the vertical split in the cover, which opened to an advertisement! I believe this is the only such cover Physics Today has ever published.

    Henon-Heiles flows
    Physics Today cover I created in my Wooster office based on our array enhanced stochastic resonance research, for which we were awarded a US patent. The remarkable cover features a split that opens to an advertisement!

    Array Enhanced Stochastic Resonance and Spatiotemporal Synchronization, J. F. Lindner, B. Meadows, W. Ditto, M. Inchiosa, A. Bulsara, Physical Review Letters, volume 75, pages 3-6 (3 July 1995) doi.org/10.1103/PhysRevLett.75.3

    Scaling Laws for Spatiotemporal Synchronization and Array Enhanced Stochastic Resonance, J. F. Lindner, B. Meadows, W. Ditto, M. Inchiosa, A. Bulsara, Physical Review E, volume 53, pages 2081-2086 (March 1996) doi.org/10.1103/PhysRevE.53.2081

  • My Second Patent

    jlindner

    Today, about six years after beginning the relevant research, my colleagues and I were issued United States Patent No. US 12 450 468 B2, “Physics augmented neural networks configured for operating in environments that mix order and chaos”. The work began during my 2019-2020 sabbatical at the Nonlinear Artificial Intelligence Lab at North Carolina State University and continued for two years, across the four publications listed below, accompanied by years of legal work by NCSU.

    Neural networks are powerful computational tools but they can be confounded by the mix of order and chaos in natural and artificial phenomena. Our solution to this “chaos blindness” exploits an elegant and deep structure to everyday motion discovered by William Rowan Hamilton, who remarkably re-imagined such motion as an incompressible energy-conserving flow in an abstract, higher-dimensional space of positions and momenta. In this phase space, motions are unique trajectories confined to constant-energy surfaces, and regular motions are further confined to donut-like hypertorii. This structure constrains our Hamiltonian neural networks to better forecast systems that mix order and chaos, such as the Hénon-Heiles motion of stars in galaxies.

    Henon-Heiles flows
    Two sample Hénon-Heiles flows for different initial conditions forecast by conventional neural network (left), Hamiltonian neural network (center), and Hénon-Heiles differential equations (right), for small (bottom), medium (middle), and large (top) bounded energies. Hues code momentum magnitudes, from red to violet; orbit tangents code momentum directions. Orbits fade into the past.

    Physics enhanced neural networks learn order and chaos, A. Choudhary, J. F. Lindner, E. G. Holliday, S. T. Miller, S. Sinha, W. L. Ditto, Physical Review E, volume 101, pages 062207(1-8) (June 2020) doi.org/10.1103/PhysRevE.101.062207

    The scaling of physics-informed machine learning with data and dimensions, S. T. Miller, J. F. Lindner, A. Choudhary, S. Sinha, W. L. Ditto, Chaos, Solitons & Fractals: X, volume 5, pages 100046(1-7) (2020) doi.org/10.1016/j.csfx.2020.100046

    Forecasting Hamiltonian dynamics without canonical coordinates, A. Choudhary, J. F. Lindner, E. G. Holliday, S. T. Miller, S. Sinha, W. L. Ditto, Nonlinear Dynamics, volume 103 number 2, pages 1553-1562 (2021) doi.org/10.1007/s11071-020-06185-2

    Negotiating the separatrix with machine learning, S. T. Miller, J. F. Lindner, A. Choudhary, S. Sinha, W. L. Ditto, Nonlinear Theory and Its Applications, volume 12, number 2, pages 1-9 (April 2021) doi.org/10.1587/nolta.12.134

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