iiinova — where your digital self keeps evolving

Leo C. Stein @duetosymmetry.com · 25/09/2026

Confidently answering the wrong question… The Netherlands is literally named after being low-lying. Its average elevation is 30m above sea level. (The correct answer is Bhutan at 3.28km, just edging out Nepal at 3.27km)

Screenshot of googling "country with the highest average elevation", and the AI responds with The Netherlands because their men are 184cm tall on average.

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Leo C. Stein @duetosymmetry.com · 26/08/2026

Perfect application of this meme

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Leo C. Stein @duetosymmetry.com · 19/08/2026

The gravity + astrophysics group at UMiss has a totally overhauled web page! Many thanks to João Dinis Álvares for his work overhauling the site. www.phy.olemiss.edu/GR/ ⚛️🧪🔭

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Leo C. Stein @duetosymmetry.com · 11/07/2026

Two more PhDs defended! duetosymmetry.com/news/Two-mor...

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Leo C. Stein @duetosymmetry.com · 28/05/2026

Always relevant m.xkcd.com/1987/

The flowchart from hell of python dependency management

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Leo C. Stein @duetosymmetry.com · 15/05/2026

I just noticed that @inspirehep.net lets you add your bsky profile link to your author page!

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Leo C. Stein @duetosymmetry.com · 30/04/2026

Did you have the soundtrack on repeat the whole time?

Cover art from the 1991 album "We can't dance" by Genesis

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Leo C. Stein @duetosymmetry.com · 10/04/2026

I'm in this picture and I don't like it (ok, I actually think it's very very funny) from @zachweinersmith.bsky.social , see smbc-comics.com/comic/sphere... 🧮

One panel of a comic, with an algebraist (labeled with A on their chest) saying 'Where's your precious "visual intuition" now, Jenkins!? HAHAHAHA!' to a geometer (with G on their chest) who is crying

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Leo C. Stein @duetosymmetry.com · 01/04/2026

Very important paper: No hair but plenty of feathers: are birds black holes? Andrew Laeuger and Taylor Knapp arxiv.org/abs/2603.29064 ⚛️🧪🔭

Fig. 1 from the paper: A cardinal (bird) with a chirp waveform emanating from its mouth; a magnifying glass held over the head of another cardinal reveals a binary black hole in its neck.

Figure caption: The central tenet of this paper: careful investigation of the calls of certain birds may reveal that these creatures actually conceal compact binaries whose radiated gravitational waves match the “chirps” we observe in nature.

Title/authors/abstract of the paper.

Abstract:
The imitative verb “chirp” is thought to originate from 16th-century Middle English. Meanwhile, this same word has been used to describe the gravitational waves (GWs) emitted from the merger of compact objects, such as black holes and neutron stars, since at least the 1990s. Motivated purely by this linguistic overlap, we study whether the chirps of birds can be modeled by compact binary waveforms. In particular, we consider a test case of the Northern cardinal (Cardinalis cardinalis), finding that its time-reversed chirp can be approximately modeled by that of a high mass ratio, precessing black hole binary, with a number of indications towards extreme matter effects or beyond the Standard Model physics. Importantly, this waveform correspondence is not so straightforward for all bird species, as some chirp morphologies are far more akin to glitches seen in GW observatories. With these comparisons made, we propose an alternative solution to the longstanding philosophical conundrum: rather than the chicken or the egg, perhaps it was the Big Bang which truly came first.

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Leo C. Stein @duetosymmetry.com · 27/03/2026

Everyone tell Aniket congrats! arxiv.org/abs/2603.24661 Fixing the center-of-mass frame of numerical relativity waveforms using the post-Newtonian center-of-mass charge Khairnar, Stein, et al. 🧪⚛️🔭🧮

Fig. 2 from the paper. Caption:
FIG. 2. The center-of-mass charge vector G (in blue) obtained from the asymptotic data of a quasicircular nonprecessing system SXS:BBH:2115. Before frame-fixing (left panel), G starts closer to the origin and drifts away in a boosted outspiral. A fit to the boosted post-Newtonian prediction of Eq. (26) (orange, dashed) agrees well with numerical data. Notice the difference in scales after fixing the PNBMS frame (right panel). The small box near the origin in the left plot correspond to the range of CoM charge after fixing the frame. This plot includes times in the 15%–85% region between the metadata’s reference_time and the time of the maximum norm of h across the entire 2-sphere.

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Leo C. Stein @duetosymmetry.com · 11/03/2026

In my head, the title of this paper is "Chaos! At the light ring." arxiv.org/abs/2603.08790 Chaos and fractals of the black hole photon ring Roman Berens, Peter Galison, Trevor Gravely, Alexandru Lupsasca, Leo C. Stein 🧪⚛️🧮🔭

A basin exit plot, showing chaotic structure

$Title/author list/abstract of the paper. The abstract reads: The photon ring of a Kerr black hole decomposes into a self-similar hierarchy of subrings. Here, we show that this self-similar structure persists in phase space. Moreover, near the photon shell of bound photon orbits, dynamics are controlled by a Lyapunov exponent γ, whose role we highlight by computing the first-return map for light rays close to an unstably bound orbit. Despite an exponential e^γ sensitivity to initial conditions, nearly bound rays do not exhibit chaotic behavior. However, as the background spacetime is deformed away from the Kerr geometry, chaos sets in, with its first onset most visible near strongly resonant bound orbits in the photon shell. We display two animations: one illustrating the emergence of chaos near the photon shell, which results in a fractal phase-space structure, and another exhibiting how this chaotic, fractal, self-similar structure is encoded in the first-return map.$

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Leo C. Stein @duetosymmetry.com · 25/02/2026

Classical dynamics Quantum mechanics Electrodynamics Statistical mechanics

Black and white tarot-like drawing of a warrior fighting a skeleton.
What are the four core character classes?

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Leo C. Stein @duetosymmetry.com · 23/02/2026

Oh maybe I should have linked this one instead: arxiv.org/abs/1804.03659

Figure 1 from the above-linked paper, with three panels. Caption:

Figure 1. The normal cone (a) of the evolution equations for the constraint violations in the cotangent space, and the corresponding ray cone (b) in the tangent space. The geometric mean cones (shown in green) are not part of the normal or the ray cone. The causal cone (c) is the convex hull of the causal cones of of the two individual systems, which encloses the ray cones (b).

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Leo C. Stein @duetosymmetry.com · 13/02/2026

Found this in Nouvelles recherches sur les systèmes dynamiques, Birkhoff, 1935

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Leo C. Stein @duetosymmetry.com · 13/02/2026

I found the following figure in Birkhoff and Smith 1928 (www.numdam.org/item/JMPA_19...), but the paper doesn't seem to be making the point about the tangle yet

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Leo C. Stein @duetosymmetry.com · 13/02/2026

So, there are an infinite number of weaves, getting more and more tangled. From Lichtenberg + Lieberman, Fig. 3.4:

Figure 3.4 from Lichtenberg + Lieberman, showing a zoom into a heteroclinic tangle at a hyperbolic fixpoint

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Leo C. Stein @duetosymmetry.com · 13/02/2026

Apparently this is the first* illustration of the "homoclinic tangle" in the dynamics literature, from Melnikov 1962 (www.mathnet.ru/php/archive....). (For reasons that escape me, people write homoclinic even when they mean heteroclinic) ⚛️🧪 * According to the textbook by Lichtenberg and Lieberman

A diagram depicting four hyperbolic fixpoints, each with stable and unstable manifolds asymptoting to/away from them. Because of the area preservation property, there are infinitely intricate folds in these manifolds (only a finite number of which have been illustrated).

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Leo C. Stein @duetosymmetry.com · 12/02/2026

Fun fact: cos(2π/17) can be expressed in terms of radicals. 🧮

$Typeset image of the following LaTeX code: \cos\frac{2\pi}{17} =& -\frac{1}{16} + \frac{\sqrt{17}}{16}+\frac{1}{16} \sqrt{34-2 \sqrt{17}} \\ &{}+\frac{1}{8} \sqrt{17+3 \sqrt{17} -\sqrt{34-2 \sqrt{17}} -2 \sqrt{2 \sqrt{17}+34}}$

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Leo C. Stein @duetosymmetry.com · 10/02/2026

Here's what the Hamiltonian vector field of this system looks like. It vaguely looks like the vector field of an elliptic point on the right, and the vector field of a hyperbolic point on the left!

A vector field in the (q,p) plane, with a fixpoint at (0,0). To the right, it looks like "circulating" motion, qualitatively like that of a harmonic oscillator. To the left, it looks like "hyperbolic" motion, qualitatively like that of an anti-oscillator.

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Leo C. Stein @duetosymmetry.com · 10/02/2026

And here's a smooth vector field of index +2: As the blue dot goes around once, the orange dot goes around twice. 8/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

This "hyperbolic" vector field has an index of -1: note that the while the blue dot goes around counterclockwise, the orange dot goes around clockwise. 7/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

Here are more vector fields with index +1: a repelling center or an attracting center. 6/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

Notice that multiplying the whole vector field by a constant doesn't change the index. Here I flipped the sign of the vector field, but the orange dot still goes around once in the positive direction: 5/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

If we had a constant vector field, it would go around 0 times. Here's a vector field with index +1: 4/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

The *index* of the vector field at the point answers the question: As the blue dot goes around its circle once counterclockwise, how many times does the orange dot go around its circle? Positive number mean counter, while negative mean clockwise. The index can only be an integer. 3/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

We don't care about the magnitude of the vector field, only its direction. So, we can associate it to a point on a unit circle (orange dot on the right). 2/9

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Leo C. Stein @duetosymmetry.com · 10/02/2026

I realized that I hadn't posted my visualizations of the *index* of a vector field on bsky! Pick a point where v=0, and take a tiny circle around that point. Walk around the circle (blue dot on the left) counterclockwise, noting where the vector field points (orange arrow). 1/9 🧮

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Leo C. Stein @duetosymmetry.com · 06/02/2026

Remembered a stupid physics meme I made years ago en.wikipedia.org/wiki/Born_ap... 🧪⚛️🧮

A poster for the movie THE BOURNE IDENTITY, but with the text IDENTITY replaced with APPROXIMATION, and Matt Damon's (Jason Bourne's) face replaced with that of Max Born

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Leo C. Stein @duetosymmetry.com · 01/02/2026

This is who runs this account

Art collector / critic Leo Stein (1872-1947), elder brother of Gertrude Stein

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Leo C. Stein @duetosymmetry.com · 20/01/2026

What's so confusing about this?

Fig. 4.2 from MTW, with "egg-crate" diagrams visualizing differential forms

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Leo C. Stein @duetosymmetry.com · 12/01/2026

I may have misjudged what is an appropriate size for a biscuit (banana for scale)

A cutting board with a cooling rack, upon which sit 6 cheddar jalapeño biscuits, each about 80% the length of a banana

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Leo C. Stein @duetosymmetry.com · 01/01/2026

(Text of footnote in alt text)

"The decay processes discussed in the present paper should not be confused with the oscillatory processes (νₑ↔ν_μ) discussed earlier by B. Pontecorvo, Zh, Eksp. Teor. Fiz. 53, 1717 (1967) [Sov. Phys. JETP 26, 984 (1968)]; V. Gribov and B. Pontecorvo, Phys. Lett. 28B, 493 (1969). The oscillatory processes typically lead to only a factor-of-2 reduction in the terrestrially detected flux of electron neutrinos when properly averaged over the spectrum of energies of solar neutrinos

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Leo C. Stein @duetosymmetry.com · 28/12/2025

If you see this, post a random fact about yourself: I discovered the Kerr separatrix polynomial arxiv.org/abs/1912.07609

A 3d surface plot of the Kerr separatrix

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Leo C. Stein @duetosymmetry.com · 26/12/2025

Throwback to some silly xmas/Newtonmas drawings I made a few years ago (Isaac Newton was born on Dec 25 under the Julian calendar in use in England at the time of his birth)