Category Archives: Mathematics

Arithmetic and its biological roots

Randolph Grace’s (Professor of Psychology, University of Canterbury, England) August 14, 2023 essay for The Conversation delves into an interesting question,

Why have humans invented the same arithmetic, over and over again? Could arithmetic be a universal truth waiting to be discovered?

The point is made (from Grace’s August 14, 2023 essay), Note: A link has been removed,

Humans have been making symbols for numbers for more than 5,500 years. More than 100 distinct notation systems are known to have been used by different civilisations, including Babylonian, Egyptian, Etruscan, Mayan and Khmer.

The remarkable fact is that despite the great diversity of symbols and cultures, all are based on addition and multiplication. For example, in our familiar Hindu-Arabic numerals: 1,434 = (1 x 1000) + (4 x 100) + (3 x 10) + (4 x 1).

Why have humans invented the same arithmetic, over and over again? Could arithmetic be a universal truth waiting to be discovered?

Grace describes a biological phenomenon to support his proposal (from Grace’s August 14, 2023 essay), Note: Links have been removed,

Bees provide a clue

We proposed a new approach based on the assumption that arithmetic has a biological origin.

Many non-human species, including insects, show an ability for spatial navigation which seems to require the equivalent of algebraic computation. For example, bees can take a meandering journey to find nectar but then return by the most direct route, as if they can calculate the direction and distance home.

A graph that shows a bee's zig-zag flight and the direct route home.
Bees can integrate their zig-zag flight path to calculate the straightest route back to the hive. Nicola J. Morton, CC BY-SA

How their miniature brain (about 960,000 neurons) achieves this is unknown. These calculations might be the non-symbolic precursors of addition and multiplication, honed by natural selection as the optimal solution for navigation.

Arithmetic may be based on biology and special in some way because of evolution’s fine-tuning.

He goes on to describe how he and his colleagues tested their hypothesis (read the essay) and concludes with this (from Grace’s August 14, 2023 essay), Note: A link has been removed,

Although this structure [how our perception is structured] is shared with other animals, only humans have invented mathematics. It is humanity’s most intimate creation, a realisation in symbols of the fundamental nature and creativity of the mind.

In this sense, mathematics is both invented (uniquely human) and discovered (biologically-based). The seemingly miraculous success of mathematics in the physical sciences hints that our mind and the world are not separate, but part of a common unity.

The arc of mathematics and science points toward non-dualism, a philosophical concept that describes how the mind and the universe as a whole are connected, and that any sense of separation is an illusion. This is consistent with many spiritual traditions (Taoism, Buddhism) and Indigenous knowledge systems such as mātauranga Māori.

Here’s a link to (or PDF for Grace’s paper) and a citation for the paper,

The Psychological Scaffolding of Arithmetic by Matt Grice, Simon Kemp, Nicola J. Morton, Randolph C. Grace. Psychological Review DOI: https://doi.org/10.1037/rev0000431 Advance online publication June 26, 2023

This paper is open access.

6th annual Girls and STEAM (science, technology, engineering, arts, and mathematics) Summit at Science World in Vancouver (Canada)

Thanks to Rebecca Bollwitt and the October 24, 2023 posting on her Miss 604 blog for the news about the 2023 (or 6th annual) Girls and STEAM (science, technology, engineering, arts, and math) Summit. From Alexis Miles’s October 24, 2023 post,

The 6th annual Girls and STEAM (science, technology, engineering, arts and design, and math), presented by STEMCELL Technologies, is taking place at Science World November 4th [2023].

Girls and STEAM at Science World
Date: Saturday, November 4, 2023
Time: 7:45am to 4:00pm
Location: Science World (1455 Quebec Street, Vancouver)
Admission: Registration is open online for girls aged 12 to 14.

300 young girls, aged 12-14, will take over the Science World dome in a day of hands-on activities, enriching workshops, inspiring mentorship sessions and a keynote presentation.

This year’s keynote presentation features Andini Makosinski, Filipina-Polish Canadian inventor best known for her invention of the Hollow Flashlight that runs off the heat of the human hand, and theeDrink, a coffee mug that harvests the excess heat of a hot drink and converts it into electricity to charge a phone. The inspiration for Andini’s flashlight came from her friend in the Philippines, who had failed a grade in school because she had no light or electricity to study with at night.

A September 25, 2023 STEMCELL Technologies news release announces the company’s participation and support for the event,

STEMCELL Technologies, Canada’s largest biotechnology company, is pleased to announce it will be the presenting partner of the Girls and STEAM Summit at Science World in Vancouver.

The Summit, which takes place on November 4, 2023, is a full-day event with workshops, hands-on activities, a keynote presentation, and sessions with experienced mentors who work in STEAM (science, technology, engineering, art and design, and math).

“Science is about so much more than what happens in the laboratory. It provides a lens that can instill a deep-seated curiosity in young minds and enrich every aspect of our lives,” said Sharon Louis, Senior Vice President of Research and Development, STEMCELL. “Scientific education – in the classroom and out in the world – can lead to life-changing experiences and limitless opportunities for young women and girls. STEMCELL is proud to support the Girls and STEAM program to make science more accessible, and help ignite the passion of the next generation of scientists and leaders.”

About STEMCELL Technologies

STEMCELL Technologies supports life sciences research with more than 2,500 specialized reagents, tools, and services. STEMCELL offers high-quality cell culture media, cell separation technologies, instruments, accessory products, educational resources, and contract assay services that are used by scientists performing stem cell, immunology, cancer, regenerative medicine, and cellular therapy research globally.

[downloaded from https://miss604.com/2023/10/girls-and-steam-at-science-world.html]

You can register here.

Researchers at University of Montéal decode how molecules “talk” to each

An August 15, 2023 news item on ScienceDaily breaks news from the University of Montréal,

Two molecular languages at the origin of life have been successfully recreated and mathematically validated, thanks to pioneering work by Canadian scientists at Université de Montréal.

Fascinating, non? An August 15, 2023 Université de Montréal news release (also on EurekAlert), which originated the news item, explaining how this leads to nanotechnology-enabled applications, Note: A link has been removed,

Published this week in the Journal of American Chemical Society, the breakthrough opens new doors for the development of nanotechnologies with applications ranging from biosensing, drug delivery and molecular imaging.

Living organisms are made up of billions of nanomachines and nanostructures that communicate to create higher-order entities able to do many essential things, such as moving, thinking, surviving and reproducing.

“The key to life’s emergence relies on the development of molecular languages – also called signalling mechanisms – which ensure that all molecules in living organisms are working together to achieve specific tasks,” said the study’s principal investigator, UdeM bioengineering professor Alexis Vallée-Bélisle.

In yeasts, for example, upon detecting and binding a mating pheromone, billions of molecules will communicate and coordinate their activities to initiate union, said Vallée-Bélisle, holder of a Canada Research Chair in Bioengineering and Bionanotechnology.

“As we enter the era of nanotechnology, many scientists believe that the key to designing and programming more complex and useful artificial nanosystems relies on our ability to understand and better employ molecular languages developed by living organisms,” he said.

Two types of languages

One well-known molecular language is allostery. The mechanism of this language is “lock-and-key”: a molecule binds and modifies the structure of another molecule, directing it to trigger or inhibit an activity.

Another, lesser-known molecular language is multivalency, also known as the chelate effect. It works like a puzzle: as one molecule binds to another, it facilitates (or not) the binding of a third molecule by simply increasing its binding interface.

Although these two languages are observed in all molecular systems of all living organisms, it is only recently that scientists have started to understand their rules and principles – and so use these languages to design and program novel artificial nanotechnologies.

“Given the complexity of natural nanosystems, before now nobody was able to compare the basic rules, advantage or limitations of these two languages on the same system,” said Vallée-Bélisle.

To do so, his doctoral student Dominic Lauzon, first author of the study, had the idea of creating a DNA-based molecular system that could function using both languages. “DNA is like Lego bricks for nanoengineers,” said Lauzon. “It’s a remarkable molecule that offers simple, programmable and easy-to-use chemistry.”

Simple mathematical equations to detect antibodies

The researchers found that simple mathematical equations could well describe both languages, which unravelled the parameters and design rules to program the communication between molecules within a nanosystem.

For example, while the multivalent language enabled control of both the sensitivity and cooperativity of the activation or deactivation of the molecules, the corresponding allosteric translation only enabled control of the sensitivity of the response.

With this new understanding at hand, the researchers used the language of multivalency to design and engineer a programmable antibody sensor that allows the detection of antibodies over different ranges of concentration.

“As shown with the recent pandemic, our ability to precisely monitor the concentration of antibodies in the general population is a powerful tool to determine the people’s individual and collective immunity,” said Vallée-Bélisle.

In addition to expanding the synthetic toolbox to create the next generation of nanotechnology, the scientist’s discovery also shines a light on why some natural nanosystems may have selected one language over another to communicate chemical information.

Caption; The illustration depicts two chemical languages at the basis of molecular communication. The same white molecule, represented as a lock, is activated either via allostery (top) or multivalency (bottom). The allosteric activator (cyan) induces a conformational change of the lock while the multivalent activator provides the missing part of the lock, both enabling the activation by the key (pink). Credit: Monney Medical Media / Caitlin Monney

Here’s a link to and a citation for the paper,

Programing Chemical Communication: Allostery vs Multivalent Mechanism by Dominic Lauzon and Alexis Vallée-Bélisle. J. Am. Chem. Soc. 2023, XXXX, XXX, XXX-XXX DOI: https://doi.org/10.1021/jacs.3c04045 Online Publication Date: August 15, 2023 © 2023 American Chemical Society

This paper is behind a paywall.

Simon Fraser University’s (SFU; Canada) Café Scientifique Fall 2023 events: first event is Sept. 26, 2023

From a September 7, 2023 SFU Café Scientifique announcement of their Fall 2023 event schedule (received via email),

We hope you had a great summer and are all excited for a brand new fall line-up:

SFU Café Scientifique lectures and discussions on Zoom 

Tuesdays from 5:00-6:30pm, Zoom invites are sent to those who register.

Email cafe_scientifique@sfu.ca for inquires.

Sept 26, 2023 Vance Williams, Chemistry

Title: (Un)Natural Beauty: Art, Science and Technology

Description: While art is often described in opposition to science and technology, in reality, these disciplines are mutually supporting and reinforcing explorations of the natural and constructed world. In this presentation, I will examine the intersection of art and science and the often blurry distinction between the scientist and the artist.

[Register here for September 26, 2023 event]

October 24, 2023 Ailene MacPherson, Mathematics

Title: Who, What, Where, When, and Why: the power of genomics in public health

Description: Within days of first being identified the full genome sequence of SARS Cov-2 was published online. Here we discuss the extraordinary power and limitations of genomics for understanding disease spread and for designing effective public health interventions.

[Register here for October 24, 2023 event]

November 28, 2023 Dustin King, Molecular Biology and Biochemistry

Title: Decoding how life senses and responds to carbon dioxide gas.

Description: Dustin King’s Indigenous background is central to his work and relationship with the biochemical research he conducts. He brings Indigenous ways of knowing and a two-eye seeing approach to critical questions about humanity’s impact upon the natural world. 

Join Dr. King on a microscopic journey into intricate cellular systems, which make use of CO2 in incredible ways. The presence of CO2 on Earth has given rise to a diverse evolutionary tree, with plants and animals developing ingenious methods for harnessing and using CO2 in their unique habitats. We travel from the depths of the ocean floor to the air we breathe, to understand the implications of increasing CO2 levels in nature and in daily human life.

[Register here for November 28, 2023 event]

I wouldn’t have thought art/science or, as it sometimes called, sciart was a particularly obscure concept these days but it’s a good reminder that much depends on the community from which you draw your audience.

Punctuation: a universal complement to the mathematical perfection of language

Before getting to the research into mathematics and punctuation, I’m setting the scene with snippets from a February 13, 2023 online article by Dan Falk for Aperio magazine, which seems to function both as a magazine and an advertisement for postdoctoral work in Israel funded by the Azrieli Foundation,

Four centuries ago, Galileo famously described the physical world as a realm that was rooted in mathematics. The universe, he wrote, “cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”

Since Galileo’s time, scientists and philosophers have continued to ponder the question of why mathematics is so shockingly effective at describing physical phenomena. No one would deny that this is a deep question, but for philosopher Balthasar Grabmayr, an Azrieli International Postdoctoral Fellow at the University of Haifa, even deeper questions lie beneath it. Why does mathematics work at all? Does mathematics have limits? And if it does, what can we say about those limits?

Grabmayr found his way to this field from a very different passion: music. Growing up in Vienna, he attended a music conservatory and was set on becoming a classical musician. Eventually, he began to think about what made music work, and then began to think about musical structure. “I started to realize that, actually, what I’m interested in — what I found so attractive in music — is basically mathematics,” he recalls. “Mathematics is the science of structure. I was completely captured by that.”

One of Grabmayr’s main areas of research involves Gödel coding, a technique that, roughly put, allows mathematics to study itself. Gödel coding lets you convert statements about a system of rules or axioms into statements within the original system.

Gödel coding is named for the Austrian logician Kurt Gödel, who in the 1930s developed his famous “incompleteness theorems,” which point to the inherent limitations of mathematics. Although expressed as an equation, Gödel’s proof was based on the idea that a sentence such as “This statement is unprovable” is both true and unprovable. As Rebecca Goldstein’s biography of Gödel declares, he “demonstrated that in every formal system of arithmetic there are true statements that nevertheless cannot be proved. The result was an upheaval that spread far beyond mathematics, challenging conceptions of the nature of the mind.”

Grabmayr’s work builds on the program that Gödel began nearly a century ago. “What I’m really interested in is what the limitations of mathematics are,” he says. “What are the limits of what we can prove? What are the limits of what we can express in formal languages? And what are the limits of what we can calculate using computers?” (That last remark shows that Gödel coding is of interest well beyond the philosophy of mathematics. “We’re surrounded by it,” says Grabmayr. “I mean, without Gödel coding there wouldn’t be any computers.”)

Another potential application is in cognitive science and the study of the mind. Psychologists and other scientists have long debated to what extent the mind is, or is not, like a computer. When we “think,” are we manipulating symbols the way a computer does? The jury is still out on that question, but Grabmayr believes his work can at least point toward some answers. “Cognitive science is based on the premise that we can use computational models to capture certain phenomena of the brain,” he says. “Artificial intelligence, also, is very much concerned with trying to formally capture our reasoning, our thinking processes.”

Albert Visser, a philosopher and logician at Utrecht University in the Netherlands and one of Grabmayr’s PhD supervisors, sees a number of potential payoffs for this research. “Balthasar’s work has some overspill to computer science and linguistics, since it involves a systematic reflection both on coding and on the nature of syntax,” he says. “The discussion of ideas from computer science and linguistics in Balthasar’s work is also beneficial in the other direction. [emphases mine]

Now for the research into punctuation in European languages. From an April 19, 2023 Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences press release (also on EurekAlert but published April 20, 2023),

A moment’s hesitation… Yes, a full stop here – but shouldn’t there be a comma there? Or would a hyphen be better? Punctuation can be a nuisance; it is often simply neglected. Wrong! The most recent statistical analyses paint a different picture: punctuation seems to “grow out” of the foundations shared by all the (examined) languages, and its features are far from trivial.

To many, punctuation appears as a necessary evil, to be happily ignored whenever possible. Recent analyses of literature written in the world’s current major languages require us to alter this opinion. In fact, the same statistical features of punctuation usage patterns have been observed in several hundred works written in seven, mainly Western, languages. Punctuation, all ten representatives of which can be found in the introduction to this text, turns out to be a universal and indispensable complement to the mathematical perfection of every language studied. Such a remarkable conclusion about the role of mere commas, exclamation marks or full stops comes from an article by scientists from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, published in the journal Chaos, Solitons & Fractals.

“The present analyses are an extension of our earlier results on the multifractal features of sentence length variation in works of world literature. After all, what is sentence length? It is nothing more than the distance to the next specific punctuation mark –  the full stop. So now we have taken all punctuation marks under a statistical magnifying glass, and we have also looked at what happens to punctuation during translation,” says Prof. Stanislaw Drozdz (IFJ PAN, Cracow University of Technology).

Two sets of texts were studied. The main analyses concerning punctuation within each language were carried out on 240 highly popular literary works written in seven major Western languages: English (44), German (34), French (32), Italian (32), Spanish (32), Polish (34) and Russian (32). This particular selection of languages was based on a criterion: the researchers assumed that no fewer than 50 million people should speak the language in question, and that the works written in it should have been awarded no fewer than five Nobel Prizes for Literature. In addition, for the statistical validity of the research results, each book had to contain at least 1,500 word sequences separated by punctuation marks. A separate collection was prepared to observe the stability of punctuation in translation. It contained 14 works, each of which was available in each of the languages studied (two of the 98 language versions, however, were omitted due to their unavailability). In total, authors in both collections included such writers as Conrad, Dickens, Doyle, Hemingway, Kipling, Orwell, Salinger, Woolf, Grass, Kafka, Mann, Nietzsche, Goethe, La Fayette, Dumas, Hugo, Proust, Verne, Eco, Cervantes, Sienkiewicz or Reymont.

The attention of the Cracow researchers was primarily drawn to the statistical distribution of the distance between consecutive punctuation marks. It soon became evident that in all the languages studied, it was best described by one of the precisely defined variants of the Weibull distribution. A curve of this type has a characteristic shape: it grows rapidly at first and then, after reaching a maximum value, descends somewhat more slowly to a certain critical value, below which it reaches zero with small and constantly decreasing dynamics. The Weibull distribution is usually used to describe survival phenomena (e.g. population as a function of age), but also various physical processes, such as increasing fatigue of materials.

“The concordance of the distribution of word sequence lengths between punctuation marks with the functional form of the Weibull distribution was better the more types of punctuation marks we included in the analyses; for all marks the concordance turned out to be almost complete. At the same time, some differences in the distributions are apparent between the different languages, but these merely amount to the selection of slightly different values for the distribution parameters, specific to the language in question. Punctuation thus seems to be an integral part of all the languages studied,” notes Prof. Drozdz, only to add after a moment with some amusement: “…and since the Weibull distribution is concerned with phenomena such as survival, it can be said with not too much tongue-in-cheek that punctuation has in its nature a literally embedded struggle for survival.”

The next stage of the analyses consisted of determining the hazard function. In the case of punctuation, it describes how the conditional probability of success – i.e. the probability of the next punctuation mark – changes if no such mark has yet appeared in the analysed sequence. The results here are clear: the language characterised by the lowest propensity to use punctuation is English, with Spanish not far behind; Slavic languages proved to be the most punctuation-dependent. The hazard function curves for punctuation marks in the six languages studied appeared to follow a similar pattern, they differed mainly in vertical shift.

German proved to be the exception. Its hazard function is the only one that intersects most of the curves constructed for the other languages. German punctuation thus seems to combine the punctuation features of many languages, making it a kind of Esperanto punctuation. The above observation dovetails with the next analysis, which was to see whether the punctuation features of original literary works can be seen in their translations. As expected, the language most faithfully transforming punctuation from the original language to the target language turned out to be German.

In spoken communication, pauses can be justified by human physiology, such as the need to catch one’s breath or to take a moment to structure what is to be said next in one’s mind. And in written communication?

“Creating a sentence by adding one word after another while ensuring that the message is clear and unambiguous is a bit like tightening the string of a bow: it is easy at first, but becomes more demanding with each passing moment. If there are no ordering elements in the text (and this is the role of punctuation), the difficulty of interpretation increases as the string of words lengthens. A bow that is too tight can break, and a sentence that is too long can become unintelligible. Therefore, the author is faced with the necessity of ‘freeing the arrow’, i.e. closing a passage of text with some sort of punctuation mark. This observation applies to all the languages analysed, so we are dealing with what could be called a linguistic law,” states Dr Tomasz Stanisz (IFJ PAN), first author of the article in question.

Finally, it is worth noting that the invention of punctuation is relatively recent – punctuation marks did not occur at all in old texts. The emergence of optimal punctuation patterns in modern written languages can therefore be interpreted as the result of their evolutionary advancement. However, the excessive need for punctuation is not necessarily a sign of such sophistication. English and Spanish, contemporarily the most universal languages, appear, in the light of the above studies, to be less strict about the frequency of punctuation use. It is likely that these languages are so formalised in terms of sentence construction that there is less room for ambiguity that would need to be resolved with punctuation marks.

The Henryk Niewodniczański Institute of Nuclear Physics (IFJ PAN) is currently one of the largest research institutes of the Polish Academy of Sciences. A wide range of research carried out at IFJ PAN covers basic and applied studies, from particle physics and astrophysics, through hadron physics, high-, medium-, and low-energy nuclear physics, condensed matter physics (including materials engineering), to various applications of nuclear physics in interdisciplinary research, covering medical physics, dosimetry, radiation and environmental biology, environmental protection, and other related disciplines. The average yearly publication output of IFJ PAN includes over 600 scientific papers in high-impact international journals. Each year the Institute hosts about 20 international and national scientific conferences. One of the most important facilities of the Institute is the Cyclotron Centre Bronowice (CCB), which is an infrastructure unique in Central Europe, serving as a clinical and research centre in the field of medical and nuclear physics. In addition, IFJ PAN runs four accredited research and measurement laboratories. IFJ PAN is a member of the Marian Smoluchowski Kraków Research Consortium: “Matter-Energy-Future”, which in the years 2012-2017 enjoyed the status of the Leading National Research Centre (KNOW) in physics. In 2017, the European Commission granted the Institute the HR Excellence in Research award. As a result of the categorization of the Ministry of Education and Science, the Institute has been classified into the A+ category (the highest scientific category in Poland) in the field of physical sciences.

Here’s a link to and a citation for the paper,

Universal versus system-specific features of punctuation usage patterns in major Western languages by Tomasz Stanisz, Stanisław Drożdż, and Jarosław Kwapień. Chaos, Solitons & Fractals Volume 168, March 2023, 113183 DOI: https://doi.org/10.1016/j.chaos.2023.113183

This paper is behind a paywall but the publishers do offer a preview of sorts.

There is also an earlier, less polished, open access version on the free peer review website arXiv,

Universal versus system-specific features of punctuation usage patterns in~major Western~languages by Tomasz Stanisz, Stanislaw Drozdz, Jaroslaw Kwapie. arXiv:2212.11182 [cs.CL] (or arXiv:2212.11182v1 [cs.CL] for this version) DOI: https://doi.org/10.48550/arXiv.2212.11182 Postede Wed, 21 Dec 2022 16:52:10 UTC (1,073 KB)

Ada Lovelace’s skills (embroidery, languages, and more) led to her pioneering computer work in the 19th century

This is a cleaned up version of the Ada Lovelace story,

A pioneer in the field of computing, she has a remarkable life story as noted in this October 13, 2014 posting, and explored further in this October 13, 2015 posting (Ada Lovelace “… manipulative, aggressive, a drug addict …” and a genius but was she likable?) published to honour the 200th anniversary of her birth.

In a December 8, 2022 essay for The Conversation, Corinna Schlombs focuses on skills other than mathematics that influenced her thinking about computers (Note: Links have been removed),

Growing up in a privileged aristocratic family, Lovelace was educated by home tutors, as was common for girls like her. She received lessons in French and Italian, music and in suitable handicrafts such as embroidery. Less common for a girl in her time, she also studied math. Lovelace continued to work with math tutors into her adult life, and she eventually corresponded with mathematician and logician Augustus De Morgan at London University about symbolic logic.

Lovelace drew on all of these lessons when she wrote her computer program – in reality, it was a set of instructions for a mechanical calculator that had been built only in parts.

The computer in question was the Analytical Engine designed by mathematician, philosopher and inventor Charles Babbage. Lovelace had met Babbage when she was introduced to London society. The two related to each other over their shared love for mathematics and fascination for mechanical calculation. By the early 1840s, Babbage had won and lost government funding for a mathematical calculator, fallen out with the skilled craftsman building the precision parts for his machine, and was close to giving up on his project. At this point, Lovelace stepped in as an advocate.

To make Babbage’s calculator known to a British audience, Lovelace proposed to translate into English an article that described the Analytical Engine. The article was written in French by the Italian mathematician Luigi Menabrea and published in a Swiss journal. Scholars believe that Babbage encouraged her to add notes of her own.

In her notes, which ended up twice as long as the original article, Lovelace drew on different areas of her education. Lovelace began by describing how to code instructions onto cards with punched holes, like those used for the Jacquard weaving loom, a device patented in 1804 that used punch cards to automate weaving patterns in fabric.

Having learned embroidery herself, Lovelace was familiar with the repetitive patterns used for handicrafts. Similarly repetitive steps were needed for mathematical calculations. To avoid duplicating cards for repetitive steps, Lovelace used loops, nested loops and conditional testing in her program instructions.

Finally, Lovelace recognized that the numbers manipulated by the Analytical Engine could be seen as other types of symbols, such as musical notes. An accomplished singer and pianist, Lovelace was familiar with musical notation symbols representing aspects of musical performance such as pitch and duration, and she had manipulated logical symbols in her correspondence with De Morgan. It was not a large step for her to realize that the Analytical Engine could process symbols — not just crunch numbers — and even compose music.

… Lovelace applied knowledge from what we today think of as disparate fields in the sciences, arts and the humanities. A well-rounded thinker, she created solutions that were well ahead of her time.

If you have time, do check out Schlombs’ essay (h/t December 9, 2022 news item on phys.org).

For more about Jacquard looms and computing, there’s Sarah Laskow’s September 16, 2014 article for The Atlantic, which includes some interesting details (Note: Links have been removed),

…, one of the very first machines that could run something like what we now call a “program” was used to make fabric. This machine—a loom—could process so much information that the fabric it produced could display pictures detailed enough that they might be mistaken for engravings.

Like, for instance, the image above [as of March 3, 2023, the image is not there]: a woven piece of fabric that depicts Joseph-Marie Jacquard, the inventor of the weaving technology that made its creation possible. As James Essinger recounts in Jacquard’s Web, in the early 1840s Charles Babbage kept a copy at home and would ask guests to guess how it was made. They were usually wrong.

.. At its simplest, weaving means taking a series of parallel strings (the warp) lifting a selection of them up, and running another string (the weft) between the two layers, creating a crosshatch. …

The Jacquard loom, though, could process information about which of those strings should be lifted up and in what order. That information was stored in punch cards—often 2,000 or more strung together. The holes in the punch cards would let through only a selection of the rods that lifted the warp strings. In other words, the machine could replace the role of a person manually selecting which strings would appear on top. Once the punch cards were created, Jacquard looms could quickly make pictures with subtle curves and details that earlier would have take months to complete. …

… As Ada Lovelace wrote him: “We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.”

For anyone who’s very curious about Jacquard looms, there’s a June 25, 2019 Objects and Stories article (Programming patterns: the story of the Jacquard loom) on the UK’s Science and Industry Museum (in Manchester) website.

Illustrating math at the University of Saskatchewan (Canada)

Art and math intersect in Dr. Steven Rayan’s work on quantum materials at the University of Saskatchewan (USask).

An illustration by Elliot Kienzle (undergraduate research assistant, quanTA Centre, USask) of a hyperbolic crystal in action

A May 2, 2022 USask news release (also received via email) describes Rayan’s work in more detail,

Art and mathematics may go hand-in-hand when building new and better materials for use in quantum computing and other quantum applications, according to University of Saskatchewan (USask) mathematician Dr. Steven Rayan (PhD).

Quantum materials are what futuristic dreams are made of. Such materials are able to efficiently conduct and insulate electric currents – the everyday equivalent of never having a lightbulb flicker. Quantum materials may be the fabric of tomorrow’s supercomputers, ones that can quickly and accurately analyze and solve problems to a degree far beyond what was previously thought possible.

“Before the 1700s, people were amazed that metals could be melted down and reshaped to suit their needs, be it the need for building materials or for tools. There was no thought that, perhaps, metals were capable of something much more — such as conducting electricity,” said Rayan, an associate professor of mathematics and statistics in the USask College of Arts and Science who also serves as the director of the USask Centre for Quantum Topology and its Applications (quanTA).

“Today, we’re at a similar juncture. We may be impressed with what materials are capable of right now, but tomorrow’s materials will redefine our expectations. We are standing at a doorway and on the other side of it is a whole new world of materials capable of things that we previously could not imagine.”

Many conducting materials exhibit a crystal-like structure that consists of tiny cells repeating over and over. Previous research published in Science Advances had highlighted Rayan and University of Alberta physicist Dr. Joseph Maciejko’s (PhD) success in defining a new type of quantum material that does not follow a typical crystal structure but instead consists of “hyperbolic” crystals that are warped and curved. 

“This is an immense paradigm shift in the understanding of what it means to be a ‘material’,” said Rayan.

It is expected that hyperbolic materials will exhibit the perfect conductivity of current quantum materials, but at slightly higher temperatures. Today’s quantum materials often need to be supercooled to extremely low temperatures to reach their full potential. Maintaining such temperatures is an obstacle to implementing widespread quantum computing, which has the potential to impact information security, drug design, vaccine development, and other crucial tasks. Hyperbolic materials may be part of the solution to this problem.

Hyperbolic materials may also be the key to new types of sensors and medical imaging devices, such as magnetic resonance imaging (MRI) machines that take advantage of quantum effects in order to be more lightweight for use in rural or remote environments.

USask recently named Quantum Innovation as one of its three new signature areas of research [Note: Link removed] to respond to emerging questions and needs in the pursuit of new knowledge.

“All of this comes at the right time, as new technologies like quantum computers, quantum sensors, and next-generation fuel cells are putting new demands on materials and exposing the limits of existing components,” said Rayan.

This year has seen two new articles by Rayan together with co-authors extending previous research of hyperbolic materials. The first is written with Maciejko and appears in the prestigious journal Proceedings of the National Academy of Sciences (PNAS). The second has been written with University of Maryland undergraduate student Elliot Kienzle, who served as a USask quanTA research assistant under Rayan’s supervision in summer of 2021.

In these two articles, the power of mathematics used to study quantum and hyperbolic crystals is significantly extended through the use of tools from geometry. These tools have not typically been applied to the study of materials. The results will make it much easier for scientists experimenting with hyperbolic materials to make accurate predictions about how they will behave as electrical conductors.

Reflecting on the initial breakthrough of considering hyperbolic geometry rather than ordinary geometry, Rayan said, “What is interesting is that these warped crystals have appeared in mathematics for over 100 years as well as in art – for instance, in the beautiful woodcuts of M.C. Escher – and it is very satisfying to see these ideas practically applied in science.”

The work also intersects with art in another way. The article with Kienzle, which was released in pre-publication form on February 1, 2022 [sic], was accompanied by exclusive hand drawings provided by Kienzle. With concepts in mathematics and physics often being difficult to visualize, the artwork helps the work to come to life and invites everyone to learn about the function and power of quantum materials. 

The artwork, which is unusual for mathematics or physics papers, has garnered a lot of positive attention on social media.

“Elliot is tremendously talented not only as an emerging researcher in mathematics and physics, but also as an artist,” said Rayan. “His illustrations have added a new dimension to our work, and I hope that this is the start of a new trend in these types of papers where the quality and creativity of illustrations are as important as the correctness of equations.”

Here are links to and citations for both of Rayan’s most recent papers,

Hyperbolic band theory through Higgs bundles by Elliot Kienzle and Steven Rayan. arXiv:2201.12689 (or arXiv:2201.12689v1 [math-ph] for this version) DOI: https://doi.org/10.48550/arXiv.2201.1268 Submitted on 30 Jan 2022

This paper is open access and open for peer review.

Automorphic Bloch theorems for hyperbolic lattices by Joseph Maciejko and Steven Rayan. PNAS February 25, 2022 | 119 (9) e2116869119 DOI: https://doi.org/10.1073/pnas.2116869119

This peer-reviewed paper is behind a paywall.

Entropic bonding for nanoparticle crystals

A January 19, 2022 University of Michigan news release (also on EurekAlert) is written in a Q&A (question and answer style) not usually seen on news releases, Note: Links have been removed),

Turns out entropy binds nanoparticles a lot like electrons bind chemical crystals

ANN ARBOR—Entropy, a physical property often explained as “disorder,” is revealed as a creator of order with a new bonding theory developed at the University of Michigan and published in the Proceedings of the National Academy of Sciences [PNAS]. 

Engineers dream of using nanoparticles to build designer materials, and the new theory can help guide efforts to make nanoparticles assemble into useful structures. The theory explains earlier results exploring the formation of crystal structures by space-restricted nanoparticles, enabling entropy to be quantified and harnessed in future efforts. 

And curiously, the set of equations that govern nanoparticle interactions due to entropy mirror those that describe chemical bonding. Sharon Glotzer, the Anthony C. Lembke Department Chair of Chemical Engineering, and Thi Vo, a postdoctoral researcher in chemical engineering, answered some questions about their new theory.

What is entropic bonding?

Glotzer: Entropic bonding is a way of explaining how nanoparticles interact to form crystal structures. It’s analogous to the chemical bonds formed by atoms. But unlike atoms, there aren’t electron interactions holding these nanoparticles together. Instead, the attraction arises because of entropy. 

Oftentimes, entropy is associated with disorder, but it’s really about options. When nanoparticles are crowded together and options are limited, it turns out that the most likely arrangement of nanoparticles can be a particular crystal structure. That structure gives the system the most options, and thus the highest entropy. Large entropic forces arise when the particles become close to one another. 

By doing the most extensive studies of particle shapes and the crystals they form, my group found that as you change the shape, you change the directionality of those entropic forces that guide the formation of these crystal structures. That directionality simulates a bond, and since it’s driven by entropy, we call it entropic bonding.

Why is this important?

Glotzer: Entropy’s contribution to creating order is often overlooked when designing nanoparticles for self-assembly, but that’s a mistake. If entropy is helping your system organize itself, you may not need to engineer explicit attraction between particles—for example, using DNA or other sticky molecules—with as strong an interaction as you thought. With our new theory, we can calculate the strength of those entropic bonds.

While we’ve known that entropic interactions can be directional like bonds, our breakthrough is that we can describe those bonds with a theory that line-for-line matches the theory that you would write down for electron interactions in actual chemical bonds. That’s profound. I’m amazed that it’s even possible to do that. Mathematically speaking, it puts chemical bonds and entropic bonds on the same footing. This is both fundamentally important for our understanding of matter and practically important for making new materials.

Electrons are the key to those chemical equations though. How did you do this when no particles mediate the interactions between your nanoparticles?

Glotzer: Entropy is related to the free space in the system, but for years I didn’t know how to count that space. Thi’s big insight was that we could count that space using fictitious point particles. And that gave us the mathematical analogue of the electrons.

Vo: The pseudoparticles move around the system and fill in the spaces that are hard for another nanoparticle to fill—we call this the excluded volume around each nanoparticle. As the nanoparticles become more ordered, the excluded volume around them becomes smaller, and the concentration of pseudoparticles in those regions increases. The entropic bonds are where that concentration is highest. 

In crowded conditions, the entropy lost by increasing the order is outweighed by the entropy gained by shrinking the excluded volume. As a result, the configuration with the highest entropy will be the one where pseudoparticles occupy the least space.

The research is funded by the Simons Foundation, Office of Naval Research, and the Office of the Undersecretary of Defense for Research and Engineering. It relied on the computing resources of the National Science Foundation’s Extreme Science and Engineering Discovery Environment. Glotzer is also the John Werner Cahn Distinguished University Professor of Engineering, the Stuart W. Churchill Collegiate Professor of Chemical Engineering, and a professor of material science and engineering, macromolecular science and engineering, and physics at U-M.

Here’s a link to and a citation for the paper,

A theory of entropic bonding by Thi Vo and Sharon C. Glotzer. PNAS January 25, 2022 119 (4) e2116414119 DOI: https://doi.org/10.1073/pnas.2116414119

This paper is behind a paywall.