Math caricature: Math Cartoons — Glasbergen Cartoon Service

Posted on 26 ноября, 2022 by alexxlab

Learning math? Think like a cartoonist. – BetterExplained

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.

A cartoonist might draw this:

Wow! The cartoonist recognizes:

The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.
The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.

So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.

Numbers Began With Cartoons

Think about the first numbers, the tally system:

I, II, III, IIII …

Those are… drawings! Cartoons! Caricatures of an idea!

They capture the essence of “existing” or “having something” without the specifics of what it represents.

Og the Cavemen Accountant might have tried drawing individual stick figures, buffalos, trees, and so on. Eventually he might realize a shortcut: draw a line and call it a buffalo. This captures the essence of “something is there” and our imaginations do the rest.

Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8” instead of “three plus five is equal to eight”. Much better, right?

So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence.

Technically Correct: The Worst Kind Of Correct

We agree that multiplication makes things bigger, right?

Ok. Pick your favorite number. Now, multiply it by a random number. What happens?

If that random number is negative, your number goes negative
If that random number is between 0 and 1, your number is destroyed or gets smaller
If that random number is greater than 1, your number will get larger

Hrm. It seems multiplication is more likely to reduce a number. Maybe we should teach kids “Multiplication generally reduces the original number.” It’ll save them from making mistakes later.

No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior of multiplication is misleading, can you imagine what happens when we study the formal definitions of more advanced math?

There’s a fear that without every detail up front, people get the wrong impression. I’d argue people get the wrong impression because you provide every detail up front.

As George Box wrote, “All models are wrong, but some are useful.”

A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)

Takeaways

I wrap my head around math concepts by reducing them to their simplified essence:

Imaginary numbers let us rotate numbers. Don’t start by defining i as the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.
The number e is a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.
The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape.
Euler’s Formula makes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.

Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yet accurate truth of the idea.

Happy math.

PS. Here’s my cheatsheet full of “cartoonified” descriptions of math ideas.

Math Cartoon Prize 2022 | Scholar Source

Math Cartoon Prize 2022

Holding Data
: 2022-09-12

Deadline:

2022-08-31

Location:

Online

Duration:

6 Months

Country of Origin:

Germany

Description

March 14, 2022: On the occasion of the International Day of Mathematics 2022, the German Mathematical Society / DMV (Deutsche MathematikerVereinigung) publishes the call for its cartoon prizes 2022 in mathematics. The DMV, with its partners toonpool.com and imaginary.org, are looking for the funniest and most sophisticated cartoons on the subject of mathematics.

All artists are invited to upload their math cartoons on www.toonpool.com until August 31, 2022. Up to 100 of the best entries will be collected for a virtual exhibition to be displayed globally and in local presentations throughout the year 2022.

All submitted cartoons will be evaluated by a jury (representatives of mathematics, media and arts) to honour three outstanding artworks with the «DMV Math Cartoon Prize 2022». The award ceremony will take place in September 2022 during the DMV annual meeting in Berlin.

Starting date for submitting your cartoons is March 14, 2022. The deadline for your submissions is August 31, 2022 (24:00 h CET).

Timing

14.03.2022: Start of the DMV Math Cartoon Competition 2022
02.05.2022: A first virtual exhibition is presenting cartoons submitted up to that point (will be successively expanded). You will be informed of your participation in the exhibition.
31.08.2022: Deadline — the submission period ends at 24:00 h CET
05.09.2022: Jury meeting – in order to generate the final prize decisions (followed by personal information to the winners)
07.09.2022: The winners of the DMV Math Cartoon Prize 2022 will be announced in a press release and press conference organized by the DMV, toonpool.com and imaginary.org
07.09.2022: Start of the final virtual exhibition highlighting the winner’s artworks
12.09.2022: Award Ceremony

Contacts

DMV / Thomas Vogt: [email protected]

toonpool.com / Bernd Pohlenz: [email protected]

Benefits

The cartoonist in the first place will receive a prize of 1000 euros for the best humorous drawing on mathematics, 500 euros for second place, and 250 euros for third place. In addition, up to five honourable mentions can be awarded. The prize money is donated by the DMV.

Requirements

Important: For participating in the competition, each cartoon must be uploaded on toonpool.com (artworks sent by e-mail or by other forms of submission can not be considered)
Also required: While uploading it, each cartoon entering the contest has to be tagged with math3022
If possible: Please upload high-resolution cartoons for the competition
The number of entries is limited to 5 cartoons for each participant
Any legal recourse to the competition is excluded.

Apply
Website

German Mathematical Society

Deutsche Mathematiker-Vereinigung (DMV) is the German Mathematical Society. Founded in 1890, the DMV promotes and supports mathematics in its full breadth, Mathematics as a Science in the full range from theory to applications, mathematics in Schools, Universities, and Industry, as well as its presentation in Media and Society. It currently represents 4000 mathematicians and has representatives at almost every mathematics department in Germany. It publishes its magazine DMV-Mitteilungen, the web site www.mathematik.de, two research journals Jahresbericht der DMV and Documenta Mathematica, as well as frequent memoranda and press releases.

In honour of its founding president, Georg Cantor, the society awards the Cantor Medal. The DMV publishes two scientific journals, the Jahresbericht der DMV and Documenta Mathematica. It also publishes a quarterly magazine for its membership, the Mitteilungen der DMV.[2] The annual meeting of the DMV is called the Jahrestagung; the DMV traditionally meets every four years together with the Austrian Mathematical Society (ÖMG) and every four years together with the Gesellschaft für Didaktik der Mathematik (GDM). It sometimes organises its meetings jointly with other societies (e.g., in 2014 with the Polish Mathematical Society, PTM, or in 2016 with the Gesellschaft für Angewandte Mathematik und Mechanik, GAMM). Twice annually, it organises the Gauß Lecture, a public audience lecture by well-known mathematicians.

Founded:

September 18, 1890

Founder:

Georg Cantor

Email:

[email protected]

Country:

Germany

City:

Berlin

Website:

https://www.mathematik.de/en

SUSU scientist draws mathematical cartoons

Igor Klebanov looks at the world through a “mathematical microscope”. A SUSU scientist creates cartoons in the form of equations. Creativity allows you to find unexpected solutions and get beautiful results.

Igor Klebanov was born in 1969 in Chelyabinsk. Graduated with honors from the Faculty of Physics of ChSPI (1991) and postgraduate studies at the Department of Theoretical Physics of ChelGU (1994). Associate Professor and Senior Researcher at the System Programming Department of the SUSU Higher School of Electronics and Computer Science, Candidate of Physical and Mathematical Sciences. Hobbies — philosophy, literature, theater, sculpture. Married, has a son.

In the photo: Igor Klebanov

Under the sign of symmetry

My interlocutor is an enthusiastic scientist. For 26 years of scientific work and teaching mathematics in higher education, the subjects of his scientific research were statistical physics, physics of liquid crystals, mathematical modeling in the humanities. Today, the center of his professional attention is the “mathematical microscope”. If scientifically — group analysis of mathematical models.

“Symmetry is at the heart of research using the “mathematical microscope”, – says Igor Iosifovich. – She permeates the entire universe. From the point of view of mathematics, symmetry is some transformation that leaves an object unchanged. Let’s take the human body as an example. If we ignore the internal structure, it is clear that there is a mirror symmetry. Or take an ordinary equilateral triangle, there are three axes of symmetry (angle bisectors).

You can also rotate 0.120 and 240 degrees from the center and it will align with itself. We are doing more complex things. It turns out that if we know the symmetry of the system of equations that model the physical process, then we practically have a microscope in our hands. It allows you to see the solution, which without this technique is simply invisible, well, or can be the object of a brilliant conjecture. It happens that mathematicians who do not know symmetry analysis, due to their intuition, guessed that such solutions are possible. And symmetry analysis makes it possible to obtain these solutions clearly algorithmically.

It all started in the 70s of the XIX century in Norway, where the outstanding European mathematician Marius Sofus Lie, in fact, invented the «mathematical microscope» — a device for symmetry analysis. Contemporaries, as often happens, did not appreciate his discoveries. It was considered that this is just a new language for retelling old things. Only two of his closest friends realized that the discovery had far-reaching consequences. And so it happened.

Sophus Lee passed away at 1899 and for the next 50 years, group analysis did not cause much attention. But everything changed in the middle of the twentieth century, when science began to deal with gas dynamics in connection with military purposes. Specialists in the field of gas dynamics need to solve very complex equations, which, even with a modern computer, requires a lot of effort. And so two outstanding minds — the American mathematician G. Birkhoff and the Soviet scientist Lev Ovsyannikov — realized that just the group analysis of mathematical models can play a big role here. Thus, he began to take root in applied mathematics. The metaphor «mathematical microscope» was introduced by the outstanding modern mathematician Nail Ibragimov.

Microscope for the Universe

– And what did you manage to see with the “mathematical microscope”?

– Now I am doing what the school of Academician Ovsyannikov did in Novosibirsk, but with a correction. They studied the equations of classical gas dynamics, and I added gravity to it. That is, I study the equations that model the gas, taking into account the forces of its own gravitational field. The field of application is astrophysics. The theory of the formation and evolution of stars, the formation of the large-scale structure of the Universe.

These equations have not yet been considered under the «mathematical microscope». In the last three years I have been actively engaged in this, I have published several works. I was able to discover the symmetries of these equations, as well as find some new solutions that may be of interest to the same astrophysicists. By the way, I started as an astrophysicist, I understand the essence of the ongoing processes. But today I am an applied mathematician and mathematical physicist. I no longer keep my finger on the pulse of modern astrophysics.

— Are these «new solutions» a scientific discovery?

— The word «discovery» sounds too pretentious. This is a result, serious, important and beautiful. I don’t like to throw words around. A discovery is something fundamental, something that dramatically moves an entire field forward. I got the result, which over time, perhaps, will lead to some discoveries.

— Are you modest like a true scientist?

– You know, I can’t call myself a modest person. Rather, I am an honest person. A scientist still has to be honest.

— You started talking about the origin of the universe. What theory do you follow?

— Canonical. The Big Bang theory has some difficult moments that make some scientists doubt. But if you do not climb into the wilds, in general, this is a fairly adequate theory. You see, in order not to adhere to some generally accepted version, you need to have some good reasons, to conduct research on this sector of the front. And I just have a different interest — the front of applied mathematics.

Predictions of scientists

– In general, in the field of astrophysics there is a lot that is incomprehensible even for scientists. If you start thinking about the infinity of the universe.

— You know, the Nobel laureate, respected by me Lev Landau , has two good statements about this. «Astrophysicists are always wrong, but they never doubt.» «Modern science allows us to understand what we can no longer imagine.» The first statement is obviously just a joke about astrophysicists, and the second is very deep.

Indeed, there are things that we cannot imagine, but we can only understand. To imagine means to create a visual model that is close to us. And who are we? We are macroscopic 3D beings living on Earth. We think in terms of the space available to our senses. And since the Universe goes beyond our senses, we cannot fully visualize it. But we can understand, that is, make some predictions.

– It turns out that scientists also do fortune-telling on coffee grounds? At the same time scolding the same astrologers, for example.

— This is a completely different base. Scientists make predictions based on models or hypotheses that can be tested. In general, science takes place where verification is possible, that is, verification. When I build a mathematical model, I draw a conclusion based on calculations. If the experiment confirms it, this is a good model, if it does not confirm it, then the model must be changed. True, sometimes it happens that an experiment does not confirm a theory, and then it turns out that the experimenters did not take into account something.

Astrologers have a different methodology — esoteric, philosophical. By the way, I’m not one of those people who scold astrologers. It is clear that there are charlatans who write their forecasts in tabloids for everyone at once. But if you look into history, medieval astrology had nothing to do with modern charlatanism. It was a serious science based on other messages than modern natural science. Similarly, alchemy was based on very deep messages, primarily related to the transformation of man.

– Why then is all this now anathematized?

— I think this happened in modern times, when positive science based on experiment entered into a struggle with the traditional sciences of the Middle Ages. She won because she was stronger in terms of predictions, but the baby was thrown out with water. What did alchemists really do? If it’s rude and vulgar, they were looking for a philosopher’s stone to turn everything in the world into gold. But we are forgetting one important thing. There was a condition: before you find the philosopher’s stone, you must change yourself. Only the one who transforms himself, who can be «born a second time» (there is such a concept in philosophy), will be able to find the philosopher’s stone. It was just thrown out, leaving only some strange scheme.

Brothers in mind

— But back to your research in the field of astrophysics. In the «mathematical microscope» can one see, for example, brothers in mind? When it comes to the Cosmos, the question invariably arises whether we are alone in the Universe…

– The «mathematical microscope» is not looking for brothers in mind (laughs).

– But, obviously, astrophysicists do it?

— Yes. But it’s not in the mainstream. Today, as far as I know, none of the serious astrophysicists devote their lives to catching intelligent signals from the Cosmos.

– It turns out that there are no brothers in mind, because no one is looking for them?

— They are looking for. But, firstly, who said that they should be far beyond the Earth? After all, there is a theory according to which they can be in this room, but in parallel worlds. Not the fact that they have the same communication skills. We’re sending a signal, but can they decode it? Maybe they have a completely different language system. All these games of finding brothers in mind are based on anthropomorphism. That is, we believe that they are similar to us. But remember Solaris. What did the other mind look like? An ocean that reads thoughts and reacts to them. So the search for brothers in mind somewhere in the Alpha Centauri region or even further is an outdated idea of the 60s of the last century. By the way, at that time these searches indirectly led to an interesting discovery. Strictly periodic signals were found, initially taken for the call signs of aliens. But then it turned out that this is a rapidly rotating neutron star. This is how radio pulsars were discovered.

Unfashionable science

– And yet, what is the applied value of your research?

– Huge! In any field of science, the main thing is to have a good mathematical model, that is, equations that can clarify a lot. We find new solutions. We examined the system of gas dynamics under a microscope, taking into account self-gravity — we got interesting results. There is a law of the expansion of the universe, the so-called Hubble law. We proved its group-theoretic nature and saw that this is one of the possible symmetric solutions. But it could be different, there are alternative scenarios. And then astrophysicists should consider more subtle effects and establish: what if there really are significant amendments to the Hubble law?

— But isn’t having an alternative a refutation of Hubble’s law?

— Now, if the alternative is experimentally proven… But today we cannot claim that we have refuted Hubble’s law. Unfortunately, the «mathematical microscope» is a tool that few master. This is a very useful but unfashionable area of mathematics because it requires a lot of analytical work. Now abstract sections are fashionable, which are still far from life: algebraic topology, algebraic geometry, theory of categories and functors. If we gather all the serious specialists in group analysis living on the globe, they are unlikely to fill the SUSU assembly hall. Recall the story of Perelman, who proved the Poincaré conjecture. So 10 people on the globe could really understand his work. And it took them two years to figure it out. In general, fashion in science is stupidity. A scientist should do what interests him and what brings results.

— You are lucky! In your case, this is exactly what happens.

— My work in the field of mathematical physics and mathematical modeling is consonant with the work of a cartoonist. How is a cartoon drawn? You need to see some characteristic feature of a person, draw it out, and take everything else into the background. That is, if a person is thin and tall, I can draw him in the form of spaghetti. So, an applied mathematician and mathematical physicist, in fact, draws caricatures of the object he is researching. He chooses some of the most characteristic features, exaggerates them, and takes the rest into the background, or rather, simply neglects them. For example, I believe that there are factors A, B and C, while the rest do not interest me. And I write mathematical equations as if there were nothing else but these A, B and C. That is, I actually draw a caricature of the object and thus learn something. It turns out that any mathematical model is a caricature. By drawing cartoons, we learn about the world.

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