Art as Math that Meets Crochet
Gabriele Meyer is a Senior Lecturer Emerita in the Department of Mathematics, at the University of Wisconsin. She creates beautiful art by crocheting mathematical shapes.
In a recent article on a crocheting website, quoting an earlier statement, Gabi explained the connection this way:
“In math, if you want to prove something really beautiful, you have to understand the structure. And the structure means you understand the beauty of an object and with that knowledge you oftentimes can make a very important and deep proof. That’s why beauty matters tremendously in mathematics.”
Today I want to share some of Gabi’s work of crocheting shapes that follow hyperbolic geometry. This includes crocheted algae, flowers, sea anemones, and other organic shapes.
Some of Her Art
The crocheting article relates that Gabi is “happy to give a new spin to a traditional European women’s craft while also connecting it to mathematics.” Besides the sea anemones, flower blossoms, and algae, she draws inspiration from abstract forms in topology. One could say she bridges between what is structurally ideal and what is biologically real, as well as from mathematics to art and culture more generally.
Gabi’s work was featured in the Bridges 2013 conference on mathematical connections in art, music, architecture, and culture. This began a streak of nine consecutive appearances at Bridges conferences, and in 2022 at the Joint Mathematics Meetings. Some more of her hyperbolic art is on her own site.
Her pieces are made by creating hyperbolic crochet around an original shaping line, giving it structure in three dimensions. One principle noted in her talk slides for Bridges 2019 is that neighborhoods of any point in hyperbolic geometry have more stuff than in flat Euclidean geometry or spherical geometry, forcing a local saddle structure having negative curvature. Crochet enables embodying this locality more robustly than weaving or knitting would. The effect of increasing the local stitch count is explained in greater detail in a nice 2016 article by Anna Lambert. That crocheting is friendlier than William Thurston’s paper models for hyperbolic surfaces was discovered in 1997 by Cornell’s Daina Taimina, who has also exhibited at Bridges and elsewhere.
Open Problems
My dear wife, Kathryn Farley and I, have had one of her artworks in our house for years. We knew her first through her husband, Jin-Yi Cai, and Ken has also known both as colleagues in Buffalo. We have also just purchased some more of her art work for our new condo. Gabi also has a separate line of linoleum prints.
John Conway famously kept myriad models of polyhedra and networks in his office for inspiration. The polyhedra illustrate positive curvature. What kind of mathematical creativity is best inspired by surfaces of negative curvature?
Cargo Cult Redo
Richard Feynman during his 1974 commencement address at the California Institute of Technology coined the term cargo cult science. The term was just used over at the blog of Scott Aaronson at Shtetl-Optimized. Read his post and skip the rest here if you will. Or read the rest here and then his post.
Logicians are Everywhere
So where were they between 1720 and 1820?
Helena Rasiowa was a famous logician from Poland. She visited Case Western Reserve University when I was an undergraduate a million years ago—in the 1960s.
I have always loved mathematical logic. I took undergraduate courses with two famous logicians. Richard Vesley taught me my first logic course at Case. I later took an advanced course there, also as an undergraduate, from Rasiowa. Vesley became a Professor in the University at Buffalo mathematics department, where Ken also knew him before he passed away in 2016.
One of my memories from her class is about a statement. One day in class we were stuck on a tricky insight and we as a class were asking lots of questions. Perhaps too many. She finally said:
“You will understand.”
I still recall this like it happened yesterday. She was eventually right. But at the time we were scared that we might have trouble getting it.
Logicians Named Lewis Are Everywhere
Rasiowa’s dissertation was titled Algebraic Treatment of the Functional Calculus of Lewis and Heyting. The names are Clarence Lewis and Arend Heyting. Although Lewis was American, he adopted the British habit of going by his initials as C.I. Lewis. This made him confusable with C.S. Lewis, the writer Clive Lewis in this blog’s style.
An unrelated Lewis is Harry Lewis, who is an American computer scientist known for his research in logic—and for books on theoretical computer science, computing, higher education, and technology. He is the Gordon McKay Professor of Computer Science at Harvard University, and was Dean of Harvard College from 1995 to 2003.
Another logician named Lewis whom I could have known at Princeton was David Lewis. He is best known in logic for rigorizing counterfactual conditionals. An example he gave is, “if kangaroos had no tails, they would topple over.” In complexity theory, many results are like, “if pigs could whistle then horses could fly.” Scott Aaronson wrote about one such result here. Maybe we could have used Lewis to organize the logic of these results.
Lewis is also famous for actuating the condition of the implication “if a cat could get a published paper then …” His pet named Bruce Le Catt was credited for this article until the journal recently corrected it. The Cheshire Cat—
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—brings up another logical Lewis: Lewis Carroll. Well, in this blog’s style he is Charles Dodgson. By whichever name, his work in mathematical logic was substantial.
Our point is, there have been so many logicians in the past century-plus that we can point to several with the same name. But that was not always the case. There is a previous century-plus when we can find hardly any logicians at all. To explain why this surprises us, we need to go back further, to Gottfried Leibniz.
Leibniz
Leibniz was of course one of the great mathematicians of all time. He published nothing on formal logic in his lifetime—he wrote only working drafts. Bertrand Russell claimed that Leibniz had developed logic in his drafts to a level which was reached only two centuries later.
Harry Lewis wrote a book with Lloyd Strickland titled Leibniz on Binary: The Invention of Computer Arithmetic.
Harry’s website says about it:
The definitive edition and translation of 32 of Leibniz’s works on binary arithmetic. He works out all the arithmetic operations, and realizes that base-16 would be a more usable notation, so invents several different notations for what we now call the hexadecimal digits.
Leibniz may have been the first computer scientist and information theorist. Early in life, he documented the binary numeral system (base 2), then revisited that system throughout his career.
Among testimonials on the book’s MIT Press page is this by Donald Knuth:
“This book is a model of how the history of computer science and mathematics should be written. Leibniz pointed out the importance of putting ourselves into the place of others, and here we get to put ourselves into the shoes of Leibniz himself, as we’re treated to dozens of his private notes, carefully translated into idiomatic English and thoroughly explained.”
The publisher’s description, echoed on the book’s Amazon page, chimes in about readability:
The [translated] texts are prefaced by a lengthy and detailed introductory essay, in which Strickland and Lewis trace Leibniz’s development of binary, place it in its historical context, and chart its posthumous influence, most notably on shaping our own computer age.
A Historical Logic Gap?
The “shaping of the computer age” seems to have started no earlier than the work of Charles Babbage on mechanical computation beginning in the 1820s. Even so, Babbage’s Difference Engine dealt only with numerical calculations. It took his later Analytical Engine to involve programming logic as we conceive it.
Ken has had several thoughts along these lines, going back to his graduate student days at Merton College, Oxford University:
- The Merton College Library had one half-height stack of mathematics books. Shelved right alongside modern texts—this was the early 1980s—was an 1854 first edition of George Boole’s book The Laws of Thought. We refer to Boolean logic and Boolean algebra because of this book. These terms came from a book placed with the moderns, not from the centuries-older books growing out of Aristotle and other Classical-era works that Ken could find in the grand Upper Library. This struck Ken as a warp of time.
- Ken says that the watershed for doing computational logic is realizing that NAND and NOR are universal gates. The older name for NAND is the Sheffer stroke, after the American logician Henry Sheffer. But that wasn’t until 1913, when Russell and Alfred Whitehead picked it up. The polymath Charles Peirce had discovered this about NAND and NOR in the 1880s, so NOR is also called the Peirce Arrow. He also conceived electrical implemenation of AND and OR:
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The century-plus between Leibniz and Babbage had Leonhard Euler. It had Carl Gauss. It had all the Bernoullis. It had Joseph-Louis Lagrange, Augustin-Louis Cauchy, Adrien-Marie Legendre, Jean-Baptiste Fourier, and Marie-Sophie Germain. But where are the logicians? As we quoted Russell above, Leibniz’s preliminary work connects only to two centuries later.
So why the gap? That is the puzzle. One further question is how close Leibniz came to perceiving the universality of NAND and its significance. Harry, who also drew the Peirce drawing to our attention, tells us that Leibniz invented XOR and also wrote bitwise AND for binary strings. Another is how far back the ideas of Polish notation go. Gottlob Frege anticipated it, but that was still in the late 1800s. Polish notation and its reverse form have had enduring value in programming and compilation since Jan Łukasiewicz invented the notation in 1924. Being taught by Rasiowa in the 1960s brought me closer to the origins of these logical fundamentals than I might expect, on historical reflection.
Open Problems
Are you puzzled by the gap? Can you explain it?
How might history have changed if the French greats had developed Boolean logic? For some jokes, spelling Louis in their names as Lewis might have helped. There are no US Senators named Lewis, though the introducer at Monday’s Rose Bowl tried to create one. Not jokes: both Leibniz and C.I. Lewis figure in this section of Wikipedia’s bio of David Lewis. Maybe now the rapper Louis Logic will help people named Louis catch up—to the logician Louise Hay, whom Ken knew when she visited Peter Neumann and others at Oxford.
From the Previous ’20s Decade to This One
A woman of ninety said to M. de Fontenelle, then [95]: “Death has forgotten us.” “Hush!” [he replied]
Bruce Arden and Juris Hartmanis both passed away at age 94. Bruce actually left us in December 2021, but I am pointing to a February 2022 obit on Princeton’s website. Juris’s Cornell obit came just a few days afterward, on August 4.
Today in the news I am reading about the passing of Barbara Walters age 93 and Pope Benedict age 95. Not to mention Elizabeth Windsor last September at 96. All were born in the 1920s. Now we are deep into the 2020s.
Read more…
The Gift of Nonconstructivity
Can we quantify “nonconstructive advantage”?
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Péter Frankl has been in the news again this year. The news is substantial partial progress on his famous conjecture that for any finite family of nonempty sets that is closed under union, some element must belong to at least half of the sets. The progress shows that some 
Today, amid twelve days of gift-giving, we wonder why nonconstructive proofs are often easier than constructive ones—and by how much?
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Fusion Breakthrough or Increment?
And could a barrier be lurking?
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Jennifer Granholm is the U.S. Secretary of Energy in President Joe Biden’s cabinet.
Last week, she said at a press conference:
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A Mutation Carol 2
Ghosts of creations past and citations not present
Domenico Amalfitano, Ana Paiva, Alexis Inquel, Luis Pinto, Anna Rita Fasolino, and René Just are the authors of an article in this month’s Communications of the ACM. Their article is on the program testing method called mutation.
Today we discuss how far back citations should go.
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Thanks
Thanks to you all. I must explain why I have not been active in the last six months. I have had several illnesses. I had a broken hip that needed surgery
I also had several COVID related illnesses, and some terrible GI issues. I am trying to get better. Thanks to my dear wife Kathryn Farley I have finally started to get better. I cannot ever thank her enough. Kenneth Regan and Rich DeMillo and many others have also helped me. Thanks to you all.
Quantum Circuits in the New York Times
Can quantum circuits have something to do with wormholes?
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Maria Spiropulu, a physicist at the California Institute of Technology, is featured in an article in last week’s Tuesday Science section of the New York Times about teleporting qubits through what might be described as a wormhole. The article says that physicists such as she
like to compare the teleportation process to two cups of tea. Drop a cube of sugar into one teacup, and it promptly dissolves—then, after a tick of the quantum clock, the cube reappears intact in the other teacup.
Today (yesterday? tomorrow? years ago?—with a quantum clock, does it matter?) we ask what the paper referenced in the NYT story might really be about. This is after trying to work through a lot of hype and pushback.
The first trouble is that we cannot tick a real quantum clock quite yet. But Spiropulu and her group were able to simulate a quantum clock via using a quantum computer. The group comprises: first author Daniel Jafferis, then Alexander Zlokapa, Joseph Lykken, David Kolchmeyer, Samantha Davis, Nikolai Lauk, Hartmut Neven, and then Spiropulu. Their paper was just reported in Nature. The whole buzz makes us recall an exchange that never took place:
Albert Einstein: “Reality is merely an illusion, albeit a persistent one.”
Woody Allen: “[If] everything is an illusion and nothing exists, [then] I definitely overpaid for my carpet.”
The Quantum Experiment
Spiropulu’s group could not build a quantum experiment to directly test this kind of teleportation. At least it seemed like that would be impossible. But they cleverly used the Sycamore chip developed by Google in 2019. It uses a type of quantum computing called superconducting qubits, which send electric currents flowing through superconducting materials to store and process information. Google created the chip to study “quantum supremacy” as we covered three years ago. A wonderful article on Medium by Jonathan Hui runs helpful commentary around circuit diagrams of the processor from the Google team’s original paper. Hui says:
Regardless of other claims, Google’s processor is a significant milestone because it demonstrates a problem with some real value. Whether classical computers will take 100,000 years or 2.5 days for the pseudo-random generator, this kind of speed improvement is rarely or never demonstrated with a general-purpose quantum computer.
What we find significant is that Google’s processor is not limited to the pseudo-randomness application to demonstrate what we’ll call advantage. It has plug-and-play capabilities. Google may not have created Sycamore to study quantum teleportation. But it was possible to exploit its quantum ability to demonstrate teleportation. See their Nature paper for how.
Wormholes and Their Duals
The NYT story and an article by Natalie Wolchover in Quanta have some pithy quotes from people well-known to us:
- John Preskill, about ‘the evolving system of qubits in the Sycamore chip’: “[It] has this really cool alternative description. You can think of the system in a very different language as being gravitational.”
- Leonard Susskind: “The really interesting thing here is the possibility of analyzing purely quantum phenomena using general relativity, and who knows where that’s going to go.”
- Scott Aaronson: “If this experiment has brought a wormhole into actual physical existence, then a strong case could be made that you, too, bring a wormhole into actual physical existence every time you sketch one with pen and paper.” (See also Scott’s post here.)
These quotes allude to a duality, both ends of which connect Einstein and the Israeli-American physicist Nathan Rosen. Their 1935 paper, abbreviated ER, showed how a wormhole could arise in the specific form of an Einstein-Rosen bridge according to the equations of general relativity. They were joined that same year by Boris Podolsky in a paper that described quantum entanglement via the so-called EPR paradox. In 2013, Juan Maldacena wrote a paper with Susskind claiming a correspondence between them. This was expanded by Alexei Kitaev following work by Subir Sachdev and Jinwu Ye. All this draws on Maldacena’s work in the 1990s showing a duality between two cosmic models that have different dimensionalities, in the manner of a hologram. Maldacena and Susskind’s claim is pithily expressed as the title of section 3 of their paper:
Equality is enticing because this could effect the communion of relativity and quantum mechanics. This matter found its way into another New York Times story in October, where Susskind is quoted as saying in 2017:
“It may be too strong to say that gravity and quantum mechanics are exactly the same thing. But those of us who are paying attention may already sense that the two are inseparable, and that neither makes sense without the other.”
That story centered the duality on black holes, but Will Kinney, a colleague of Ken’s at Buffalo, observed that Maldacena and Susskind “conjectured that ANY entangled particle pair is dual to a wormhole solution.” Kinney also quoted Quanta editor Thomas Lin as saying,
“In the context of the article, it should be clear that a quantum system was created that was dual to a holographic wormhole.”
What fascinates us is that the quantum system had the form of direct simulation of quantum circuitry. In fact, Spiropulu’s team used only 9 of 72 available qubits on a Sycamore chip. The NYT article has a big graphic on the role of ordinary quantum circuits, from which we snip this detail:
This leads us to some speculations of our own.
Dualling Quantum Circuits?
Setting aside the question of wormholes or their duals, Spiropulu’s team did observe behavior that could be forecasted only after running an exhaustive classical simulation tied to the Sachdev-Ye-Kitaev model on the 9-qubit scale. This is described in a video by Quanta about their work, starting about 11:00. As Wolchover summarizes:
“[A]n ineffable quantum phenomenon—information teleporting between particles—has a tangible interpretation as a particle receiving a kick of energy and moving at a calculable speed from A to B.”
Maybe there is no actual burst of energy, as Mateus Araújo underscored while criticizing this segment of the video. But is the physical process in the circuit nevertheless subject to instability and noise? We reiterate a question that we posed a year ago after describing how small quantum circuits can embody quantum walks that are chaotic:
When a quantum circuit simulates an unstable physical process with full quantum advantage, what is the boundary between the simulation and the behavior being simulated? Is potential bridging of this boundary a stumbling block to maintaining coherent operation while scaling up quantum hardware?
In the context of the November 2021 post, we noted that when one simulates chaotic physical systems in, say, FORTRAN or C++ or Python on a classical computer, one would not say the physical classical chips are themselves behaving chaotically. In the quantum case, we note Peter Shor’s response to Wolchover’s referencing a 2017 lecture video on simulating physical systems. Shor retorts that a quantum computer simulating fusion doesn’t shine like the sun, nor can quantum circuits both using and simulating superconductivity power the nation’s electric grid. But our instances of chaotic quantum walks, and now possibly wormhole duality, strike us as more closely attached to the relation of physical behavior to computation.
The new question we have is even more speculative, but may work as a counterpoint to the current discussion of the new paper. If ER=EPR is correct, then ordinary entanglements are dual to wormholes. That maximal entanglements are binary-only may correspond to the simplest wormholes going only from a point A to a point B. Entanglements are bread-and-butter to quantum circuits, so if this duality has any force, it should extend to the circuits themselves:
If the team could possibly create a theoretical dual to a wormhole in a quantum circuit, then does the correspondence in the other direction yield duals to quantum circuits that could house wormholes? Might the notion of such a circuit dual, expressed within general relativity, furnish an organizing principle about how the universe processes information to build structures on cosmic scales?
A 2018 paper by Tadashi Takayanagi titled “Holographic Spacetimes as Quantum Circuits of Path-Integrations” seems to open at least one direction of a cosmos-to-quantum-circuits correspondence. A very recent paper by Scott with Jason Pollack, on how Maldacena’s correspondence treats quantum states, draws on earlier work by Takayanagi, but seems to stop short of such duality.
Open Problems
What do you think of the questions raised by the new paper? As we highlighted at the start, it is couched in down-to-earth terms of quantum circuits. Setting aside the rampant controversy on the physics side—there are more links collected by Peter Woit besides those we’ve used—what does this say about realizing the quantum circuit model?
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The Gerrymanders Have It
The real winner of the 2022 midterms in the House
David Wasserman is an elections analyst for the Cook Political Report. He is known for forecasting the results of elections after people have voted. His words “I’ve seen enough” to declare an outcome are taken as seriously as any network election call.
This week, with nine US House races uncalled and control of the chamber still unknown, he is working overtime.
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