1
Math Thematic
106 readers
1 users here now
Sharing of different mathematic elements, stories, archives of all kinds.
founded 1 year ago
MODERATORS
2
3
4
5
6
1
7
8
9
10
11
Pythagoras was many wonderful things. A delirious mystic. A benevolent cult leader. A bean-hating vegetarian. A real person (maybe).
One thing he was not: the guy who gave us the Pythagorean Theorem.
So why does he get his name on it? I cry foul. I cry “no more.” I cry, “Let us band together and vote on a better name for this ancient theorem! Not because it will actually result in a name change, but because it’s a fun debate!”
Who’s with me?!
I submit for your consideration the following names, from Hambrecht and the other clever folks in his thread:
- The Three Squares Theorem. Although we perceive it as a claim about numbers, for most mathematical cultures, this was a claim about shapes. To wit: if you affix squares to the sides of a right triangle, the two smaller areas add up to equal the largest.
- The Babylonian Formula. Give credit where credit is due! As Hambrecht says, this name “has the hint of far-away times and places… Through millenia and continents, this piece of math connects us to strange, alien people, yet so much our equals.” He calls it “fuel for children’s imaginations.” Even more important, as an astute observer points out: it can be abbreviated as “Baby Formula.”
- The Distance Theorem. The theorem’s most ubiquitous use is in finding distances, especially in higher dimensions.
- The Huey Lewis Theorem. Proposed (or, pun-posed) by Susan Burns, because, and I quote: “It’s hyp to be square…or is that b-squared?”
- The Adrakhonic Theorem, because that’s what it’s called in Neal Stephenson’s novel Anathem (which I just added to my reading list).
- Squaring the Triangle. Olaf Doschke’s suggestion, with a ring of the famously impossible “squaring the circle.”
- The Sum of Squares Theorem. Descriptive, clear.
- Garfield’s Theorem. Because if we’re just naming it after a random dude who supplied a proof, why not pick an assassinated U.S. president?
- Theorem 3-4-5. After the most famous Pythagorean triple.
- Euclid, Book I, Proposition 47. “Like chapter and verse in the mathematical bible,” explains George Jelliss.
- The Hypotenuse Theorem. Because it’s all about that longest side.
- The Right Theorem. Because it’s all about that right angle. (Also, because it’s right.)
- The Distance/Area Theorem. Because it’s all about multiple things at once.
- The Benjamin Watson Theorem. Because of this heroic, historic tackle, brought to my attention by Fawn Nguyen in her appearance on My Favorite Theorem.
Now, we could leave it there. We could say, “This has been a fruitful discussion. Let’s call it a day!” We could say, “Obviously a random blog post isn’t going to succeed in renaming the most famous theorem in mathematics, so let’s go home and eat raisins and watch sitcom reruns like the human mediocrities we are.”
But I say no! I say it’s time for a referendum!
What say you, good people of the internet? What is the best name for this fundamental theorem of geometry?
Other ideas are, of course, welcome in the comments below.
12
13
When doing research in computer vision or image processing, it's useful to have a test image or two. Writing programs that reduce noise, alter brightness, or enhance edges is all very well and good, but without test images, we can't know if they work. Early on in vision science, the acquisition of images was hard, and there were a handful of images everyone used. This was partly due to expediency (not everyone had access to a scanner) and partly due to comparability (we want to be able to see the results of each algorithm on the same image or set of images). Today, nearly everyone has a digital camera as part of the device in their pocket, in the 70s and 80s such devices simply didn't exist.
At the very beginning of the discipline that's now become computer vision, sometime in the early 70s - probably early 1973 - a researcher was looking for a test image. Alexander Sawchuk (now a professor at the University of Southern California) scanned one for this researcher: it was the centrefold of a Playboy magazine, and that scan has now entered into our scientific culture in a way the originators could never have imagined. The woman is wearing a floppy hat, and is gazing over her bare shoulder towards the camera. As a 512 pixel squared image, the woman's body is cropped at the shoulders; the suggestion of nudity is there, but the image itself is decidedly safe for work. Her name is Lena Söderberg. The researcher’s name is lost to internet history, as is the name of the person who actually brought the porn to work.
Lena (sometimes spelled Lenna, which is the anglicisation of her name used by Playboy) has been called “The First Lady of the Internet”. Her image has been printed on a chip, shrunk to the size of a human hair, blurred, sharpened, and undoubtedly enhanced. She appeared on the cover of the journal Optical Engineering in 1991, and the model herself was invited to a conference of the Image Science and Technology society in 1997. Her picture is on the wall, tastefully framed, in at least one reputable computer vision lab. For computer vision researchers, her image is everywhere.
In defence of the use of Lena, Hutchinson said “... the image mixes areas of light and dark, fuzzy and sharp, detailed and flat—providing a stiff test for an image processing algorithm”. (I am unsure whether the word stiff in that sentence is a deliberate double-entendre. In a sense, I hope it is: if we're going to have scholarly articles about the use of pornography in our science, let them at least try to be funny.) Interestingly, Hutchinson's article talks about issues with Lena, from the copyright (Playboy were, for a time, not best pleased by the widespread distribution of one of their images) to the equalities issues. Many journal editors, whilst uncomfortable with the idea of such images appearing in their journals, were even less comfortable with the idea of banning such images.
But, as Hutchinson's article was published in 2001, we can surely assume that things have changed – that was over ten years ago. Our discipline is becoming more aware of the issues of minorities: the fact that women in technology can feel isolated is something that everyone working in the field probably knows by now. The usage of Lena must have gone down. The controversial nature of the image (there are discussion threads on the topic stretching back years, on some social media platforms) must make people think twice. Surely people realise that women, as a minority in the field, might be put off by statements like “... the Lena image is a picture of an attractive woman. It is not surprising that the (mostly male) image processing research community gravitated toward an image that they found attractive.”
Right?
Wrong. At ISISPA2013, the IEEE International Symposium on Image and Signal Processing and Analysis, during a single day of the talks I saw her six times. She's included in the standard OpenCV download (at least twice – as a test image in the both the c++ and python directories). On the machine I'm using to type this blog post, despite never having used Lena in a piece of research, and never having deliberately downloaded her, I've just done a search, and I'm amazed. I've got 18 copies (excluding cached thumbnails and the directory of files I've gathered whilst researching this article). It seems that whenever you download a piece of vision code, Lena still comes along for the ride.
As a vision researcher, I'm used to being the odd one out: at the first workshop I ever spoke at I was the only woman in the room. Generally, that doesn't bother me. I'm not entirely sure what I think about Lena though – having decided she would be a good topic for an article, I read up on the history, and to be honest, I still find it a bit bizarre. But despite my avowedly feminist stance, I’m somehow unable to get that annoyed about it.
The fact that there's a historic Playboy image at pretty much every conference I go to, and on the walls of my colleague's labs, and downloaded with every single image processing library I use, well... on the one hand, it's part of that drip-drip-drip of strangeness that comes from working in a male-dominated field, where the topics of conversation and the general attitude can be a little disconcerting. But on the other hand, with changing cultural attitudes, and the effect the internet has had on pornography, the entire centrefold (yes, you can easily find it online if you look) seems very tame indeed by today's standards. And the crop that is used in image processing research is, well... I've developed quite an affection for the picture. It's one of the quirks of computer science. So when I was asked what picture we should use to illustrate this blog post, there was only one choice.
Of course, we really had to use a Lena image on this page (although subject to Gaussian Blur with a 60px by 60px kernel). You can probably still see that it’s just a shoulder and a head in a floppy hat: what’s all the fuss about? Well, we all know what it represents and we all know where it comes from. And sometimes, there’s more to an image than meets the eye.
https://www.software.ac.uk/blog/how-photo-playboy-became-part-scientific-culture
In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).
It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination.
Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) would more accurately reproduce the bokeh effect.
Since the Fourier transform of a Gaussian is another Gaussian, applying a Gaussian blur has the effect of reducing the image's high-frequency components; a Gaussian blur is thus a low-pass filter.
A halftone print rendered smooth through Gaussian blur
14
15
16
Marguerite's Theorem (French: Le Théorème de Marguerite) is 2023 French-Swiss drama film co-written and directed by Anna Novion [fr]. It is about a female mathematics student at ENS whose career is upended when an error is discovered in her work.[4][5] It premiered on 22 May 2023 at the 76th Cannes Film Festival. It was distributed in France on 1 November 2023.
Marguerite is a young and brilliant mathematician, the only girl in her class at the ENS, entirely devoted to her passion. The day an error is discovered in her thesis, she is devastated. In a dizzy spell, she leaves the school, wiping out the past. She then dives into the real world, discovers autonomy, befriends the young Noa, and has sex for the first time. Matured by her experiences, it is in this new momentum that she manages to find a correct proof of her theorem.
https://en.wikipedia.org/wiki/Marguerite%27s_Theorem
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
Letter from Goldbach to Euler dated 7 June 1742 (Latin–German)
The conjecture has been shown to hold for all natural numbers less than 4×10^18^, but remains unproven despite considerable effort.
https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and Faber offered a $1000000 prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March 20, 2002, but the prize went unclaimed and the conjecture remains open.
https://mathworld.wolfram.com/GoldbachConjecture.html
Le Théorème de Marguerite (Marguerite’s Theorem), as explained by Anna Novion
What is the starting point of your feature film?
I remember a period when I was seriously ill. I had to stay home for a long time to get better. I was around 20 and remember being isolated from the world. When I had recovered, I felt that a distance had been created with the carefree attitude of young people of my age. When I start writing a feature film, I always rely on an emotion I’ve felt and the idea is to transpose it. The atmosphere of elite schools and the way students live in a vacuum, devoted to their work, seemed a good way of talking about this isolation.
Why did you choose to focus on the world of mathematics?
I first met with literary scholars from elite institutions. But they didn’t strike me as all that isolated. On the contrary, they were rather open to the world. I found mathematics students far more inspiring. And then there was my encounter with Ariane Mézard, which was decisive. She’s one of the rare French women mathematicians and the two of us really hit it off. The way she presented mathematics to me really provoked something within me. She discussed it in a really artistic way. She spoke to me of everything that animates me in my profession.
Which is?
Passion, necessity, difficulty, tenacity, relentlessness… I realized that there was a real parallel to be drawn between mathematics and artistic creation. What connects mathematics and directing is the risk and the passion that sometimes leads us to work for years without knowing if our work is going to amount to anything. It’s a very personal film that evokes my relationship to creation. I also wanted to talk about what it’s like to be a woman in a masculine milieu. I felt this pressure connected to the fact of being a sort of exception that pushes us to have to prove that we belong. Marguerite, my character, considers herself as a kind of anomaly. She feels this competition all the more so as she is the only woman.
How did you prepare for the film?
I spent four months at the École Normale Supérieure school meeting with mathematicians. I didn’t want them to say that I had just skimmed over the subject. With this ambition: how to make mathematicians at work captivating on the screen when you’re not from this world. The idea was to show to what point doing mathematics means working all the time.
How did you collaborate with Ella Rumpf?
She worked four hours a week over four months with Ariane Mézard. We quickly realized that it was useless for her to have the mathematics that she was going to write explained to her. She thus learned the formulas by heart. We needed it to appear extremely natural on the screen. It’s an American-style role, where we worked a lot on the posture of the character: her way of walking, her way of expressing herself, the speed of her speech and the singularity of her gaze on others.
Visually, light floods the crescendo of the film. Can you explain this aesthetic choice?
The character goes through many stages on the way to her flourishing. There are also several stages in the light and in the framing of shots. Her life is very structured at the beginning, so the shots are rather geometric and monochromatic. And little by little, with the irrational in her life that she’s going to discover, including the world of feelings, light and colour appear.
17
This picture displays the process for the 64 first prime numbers:
{2,3,5,7,11,13,17,19,23,29,,31,37,41,43,47,53,59,61,67,71,,73,79,83,89,97,101,103,107,109,113,,127,131,137,139,149,151,157,163,167,173,,179,181,191,193,197,199,211,223,227,229,,233,239,241,251,257,263,269,271,277,281,,283,293,307,311}
-top row of the picture- with the following colors regarding the numbers:
0 = Dark Yellow, 1 = Cyan, 2 = Light Yellow,
when all other numbers -{3,5,7,11,...}- are Dark Red...
According to the the Gilbreath Conjecture the left-hand side column must be Cyan ('1') except the upper square that is Light Yellow ('2', the first prime number).
http://www.lactamme.polytechnique.fr/Mosaic/images/GILB.22.D/display.html
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had published the same observations.
https://en.wikipedia.org/wiki/Gilbreath's_conjecture
Proth-Gilbreath Conjecture
Beat the Andrew Odlyzko Record G(Pi(1.1x10^14^))=635 (1993)
G(Pi(10^14^))=693 on october 5 2025 at 20:59:18
G(Pi(1.145x10^14^))=744 on 10/27/2025 at 12:10:26
1-Definition:
This conjecture was stated in 1958 by Norman L. Gilbreath but published earlier in 1878 by François Proth. It is related to the prime numbers and to the sequences generated by taking the absolute value of the difference between each prime number and its successor and then repeating this process ad infinitum:
The conjecture states that the first value of each line is 1 (except the first one where it is a 2 -the only even prime number-) and was studied by Andrew Odlyzko in 1993. He did check it for all prime numbers less than 10^13^.
On sunday 10/05/2025 20:45 (Paris time, France) I did succeed to check it up to 10^14^ and on tuesday 10/07/2025 02:25 pm (East Time), Simon Plouffe (Canada) did the same. Moreover he did confirm the maximal value (693) of the G(Pi(x)) function with x ∈ [2,10^14^] that was anticipated on 09/25/2025.
2-The Theory:
Obviously one cannot check the Proth-Gilbreath Conjecture for there is an infinity of prime numbers. Only a demonstration can solve this unless a counter-example is discovered, that is a line not starting with a '1' (except the first one).
Let p~n~ be the prime numbers:
p~1~ = 2
p~2~ = 3
p~3~ = 5
etc...
Let's define the suite d~k~(n):
d~0~(n) = pn for all n such as n > 0
d~k~(n) = |d~k-1~(n) - d~k-1~(n+1)| for all k such as k > 0 and for all n such as n > 0
Then one must check that:
d~k~(1) = 1 for all k such as k > 0
Due to the finite limits of computers, it is impossible to exhaustively check this property. Fortunately Andrew Odlyzko noticed that if for a certain N there exists K such that:
d~K~(1) = 1
d~K~(n) ∈ {0,2} for all n such as 0 < n < N+1
then:
d~k~(1) = 1 for all k such as K-1 < k < N+K
Let's call G(N) the smallest k (if it exists) such that:
d~j~(1) = 1, 0 < j < k+1
d~k~(n) ∈ {0,2} for all n such as 0 < n < N+1
A trivial reasoning shows that G(N) does exist for all N and that the process can be stopped as soon as there are only '0's, '1's and '2's on the current line of rank k. For example:
More at https://www.lactamme.polytechnique.fr/Mosaic/descripteurs/GilbreathConjecture.01.Ang.html
Verification and attempted proofs
Several sources write that, as well as observing the pattern of Gilbreath's conjecture, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. However, Zachary Chase disputes this, writing that although Proth called the observation a "theorem", there is no evidence that he published a proof, or false proof, of it.
[...] but the conjecture remains an open problem. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0s and 2s for the next n numbers. This implies that the next n rows begin with a 1.
Simon Plouffe has announced a computational verification for the primes up to 10^14^.
18
In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon :
The a needle lies across a line, while the b needle does not.
Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is p = 2/π * l/t .
This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question. The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.
https://en.wikipedia.org/wiki/Buffon%27s_needle_problem
Needle : A really weird way to find Pi
Discovery of Calculus in the 17th century opened a new window for estimating π more precisely. Since then, we've seen plenty of methods to do it. Some of them are pretty weird but simple enough to understand. Buffon's Needle Problem is one of them. This problem was first posed by the French Naturalist Georges-Louis Leclerc, Comte de Buffon in 1733. It goes like this:
Suppose you are given a floor with equally spaced parallel lines on it. If we drop a needle on the floor, what is the probability that the needle will lie across a line?
Let's say you've thrown n needles (or a single needle for n times or whatever). m number of needles cross/intersect with a line (or m times if you have one needle). The probability P that a needle crosses the line:
P = m/n
Buffon's Needle Problem. The green needles are the ones crossing the parallel lines
It may seem a bit odd how π is related to this problem. Hang on tight. I'll explain.
To solve this problem, we'll require some basic knowledge of Probability and Integral Calculus. Let's assume that the spacing between two consecutive parallel lines is D, the length of the needle is L. Now, let the distance from the middle point of a needle on the floor to its closest line be x and the acute angle between the line and needle be θ (See figure above). All needles are of equal length.
Since we are considering x to be the smallest distance from the center of the needle to any one of the lines, it can vary only within 0 and D/2. And since θ is an acute angle, it can take any value between 0 and π/2. Using trigonometry, we can find out that the vertical component of length L is L/2 * sin(θ). The needle will cross the line if x is less than L/2 * sin(θ).
Okay. Now let's visualize possible outcomes in a graph. Let the horizontal axis be θ and the vertical axis be x. So a rectangle with sides π/2 and D/2 represents all possible outcomes in this experiment.
Horizontal axis refers to θ and vertical axis refers to x. The rectangle represents all possible outcomes
Now I am going to shade all those points in the rectangle that represent the events where a needle crosses a line according to the two conditions above. Think about it. It's just the area under the curve D/2 * sin(θ) where 0 ≤ θ ≤ π .
The shaded area indicates the probability that a needle will cross a line
The probability we are looking for would be the ratio of the area under the sine curve and the rectangle.
Area of the rectangle:
Area under the sine curve:
Ratio of the areas:
Rearranging, we get
In 1901, Italian mathematician Mario Lazzarini performed Buffon's needle experiment and, tossing a needle 3408 times, obtained π up to six decimal points correctly!
https://tahsin314.github.io/writings/maths/buffons_needle.html
Finally, since Buffon’s needle problem is interested in estimating π – and since I have mostly focused on the error in this method instead – I thought it might be fun to take a look at the the empirical distribution of $\hat{\pi}$ for one of my experimental setups. The following figure shows how this distribution varies as a function of the number of needles thrown for the l = h case. The data are plotted using a dot-plot, while box-plots have been overlaid on top.
19
Here's an interesting quote from the correspondence of Isaac Newton:
This is from the 2nd letter that Newton wrote to Leibniz (via Oldenburg) in 1677. He was responding to some questions from Leibniz about his method of infinite series and came close to revealing his "fluxional method" (i.e., calculus), but then decided to conceal it in the form of an anagram. After describing his methods of tangents and handling maxima and minima, he wrote
The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.
The anagram expresses, in Newton's terminology, the fundamental theorem of the calculus: "Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa", which means "Given an equation involving any number of fluent quantities to find the fluxions, and vice versa." Arranging the characters in his Latin sentence in alphabetical order (and assuming he counted the dipthong "ae" as a separate character, and u's and v's are counted as the same character), the number of occurrences of each character are as follows
This agrees with Newton's anagram
except that I count nine t's instead of eight. Possibly Newton's original Latin spelling used one fewer t's, although I can't see which one of them could plausibly be omitted. It could also be that the anagram has been incorrectly copied, but it agrees with the version in both Westfall's and Christianson's biographies, as well as the transcription of Newton's letter contained in Calinger's Classics of Mathematics. Another possibility is that Newton simply mis-counted. This isn't as implausible as it might seem at first, since there is a well-known psychological phenomenon of overlooking the second letter in short connective words (like the f in "of") when quickly counting the number of occurrences of a certain letter in a string of text. It's very easy, when counting the number of t's in Newton's Latin phrase, to neglect the "t" in the word "et".
Ironically, neither Leibniz nor Newton had published anything on calculus at the time this letter was exchanged, although both are believed to have been in possession of the calculus, so if Newton had just come right out with a complete and explicit statement of his calculus he would have placed Leibniz in a very difficult position, and would have established his own priority beyond doubt (since the letter passed through Oldenburg). Instead, Newton's very protectiveness and secrecy caused him to lose whatever unambiguous claim to priority he might have had (and led to an acrimonious priority dispute that embittered both his and Leibniz's later lives).
On a very deep level it was natural for Newton to express himself in anagrams, because he seems to have regarded "contrived obscurity" (in Domson's words) as an essential feature of God's design for the world, and Newton adopted this mode of operation in his own work. Recall that he said he had made the Principia "designedly abstruse" (to avoid being baited by "little smatterers" in mathematics). Even more pointedly, Newton spent many years attempting to interpret the prophesies in the Bible, which he believed were presented in deliberately disguised form so that their meaning could only be inferred by solving them like puzzles. Interestingly, he had disdain for people who tried to unravel prophesies of future events. In his view, this was misguided and doomed to failure. He believed the encoded prophesies were intended to be understood only after the fact. In his "Observations upon the Prophecies of the Apocalypse of St. John" he wrote
The folly of interpreters has been to foretell times and things by this Prophecy, as if God designed to make them Prophets. By this rashness they have not only exposed themselves, but brought Prophecy also into contempt. The design of God was much otherwise. He gave this [Revelations] and the Prophecies of the Old Testament, not to gratify men's curiosities by enabling them to foreknow things, but that after they were fulfilled they might be interpreted by the event, and [that] his own Providence, not the Interpreters, be then manifested thereby to the world. For the event of things predicted many ages before, will then be a convincing argument that the world is governed by Providence.
We might say that Newton saw the biblical prophesies serving exactly the same function as his anagram on fluxions and fluents, i.e., it is presented in a form that cannot be interpreted to reveal its meaning before the fact, but after the facts have transpired and the solution of the puzzle is found, it can serve as irrefutable evidence of the Providence of the creator.
20
In musical tuning and harmony, the Tonnetz (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.
A modern rendering of the Tonnetz. The A minor triad is in dark blue, and the C major triad is in dark red. Interpreted as a torus, the Tonnetz has 12 nodes (pitches) and 24 triangles (triads).
Euler's Tonnetz
Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a major third above F). Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst, discusses the relationships between keys, presenting them in a network analogous to Euler's Tonnetz, but showing keys rather than notes. The Tonnetz itself was rediscovered in 1858 by Ernst Naumann in his Harmoniesystem in dualer Entwickelung., and was disseminated in an 1866 treatise of Arthur von Oettingen. Oettingen and the influential musicologist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) explored the capacity of the space to chart harmonic modulation between chords and motion between keys. Similar understandings of the Tonnetz appeared in the work of many late-19th century German music theorists.
The appeal of the Tonnetz to 19th-century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C♯-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E major (E-G♯-B) is diagonally across the E vertex, and shares no other vertices. One important point is that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared vertices, the more shared pitches the chord will have. This provides a visualization of the principle of parsimonious voice-leading, in which motions between chords are considered smoother when fewer pitches change. This principle is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships.
https://en.wikipedia.org/wiki/Tonnetz
Harmonic Trajectories in the Tonnetz
Introduction
The Neo-Riemannian Tonnetz is a way of visualizing musical relationships between chords. It was developed by music theorists to help understand how chords can transition smoothly from one to another.
Imagine a grid or network of interconnected points. Each point represents a different chord. The horizontal lines connect chords that are closely related, while the vertical lines connect chords that share similar tonal qualities.
The Tonnetz is based on the idea that chords can be transformed or changed into one another through small movements. These transformations are represented by diagonal lines on the Tonnetz. For example, a chord can be transformed into another chord by changing one note at a time, moving in a specific direction on the grid.
By studying the Tonnetz, musicians and theorists can analyze chord progressions and see how different chords are related to each other. It provides a visual representation of the harmonic possibilities and helps to explain the underlying structure of music.
In simple terms, the Neo-Riemannian Tonnetz is a grid that shows how chords in music are connected and can be transformed smoothly from one to another. It helps musicians and theorists understand how chords fit together and how they can create pleasing transitions in music.
21
By combining theoretical abstraction with practical impact, Stéphane Mallat has left a lasting mark on mathematics and computer science. From the JPEG 2000 image compression standard to the mathematical foundations of artificial intelligence, he has shaped tools that have become essential. He is the 2025 recipient of the CNRS Gold Medal.
“We often imagine mathematics as a collection of abstract concepts that apply ‘from above’ onto reality. But more often than not, it works the other way around: real-world problems push us to invent new mathematical tools. And to shape them, one has to ‘get one’s hands dirty,’ building bridges between abstract theory and concrete questions from the world. That frontier, between the two, is precisely where I feel comfortable.”
The scientific work of the 62-year-old researcher – broad forehead topped with unruly hair, gentle blue eyes, and a warm smile – makes the point. His contributions have profoundly influenced the field of applied mathematics to signal processing. He is best known as the inventor of a key algorithm behind the JPEG 2000 compression format, and for pioneering the mathematical insights that help us understand deep learning models at the heart of modern artificial intelligence.
Stéphane Mallat, holder of the Chair of Data Science at the Collège de France and researcher at the École Normale Supérieure, member of the French Academy of Sciences and of the U.S. National Academy of Engineering, co-signatory of ten patents, and recipient of the CNRS Innovation Medal along with numerous other prestigious distinctions, has now been awarded France’s highest scientific distinction: the CNRS Gold Medal.
From an early age, Stéphane showed a passion for mathematics – “a bubble in which I felt at ease” – yet to him, they seemed too ethereal to imagine as a future career. As a child, he loved “building things, giving shape to ideas, like an engineer,” through woodworking.
"If I returned to mathematics, it was thanks to intuitions sparked by practical applications. That's when I realised the extraordinary power and beauty of abstract concepts, their ability to capture the essence of realities that, on the surface, look completely different.”
After excelling at École Polytechnique, he left for the University of Pennsylvania in the United States. There, in 1988, he completed a PhD in mathematics applied to image processing under the guidance of Ruzena Bajcsy – “a pioneer in the field” – at a time when digital technology was booming.
An image of 1000x1000 pixels contains a million numerical values; each pixel is a number between 0 (black) and 255 (white). How to extract information from such an avalanche of bytes? His PhD supervisor proposed trying to do so by changing image resolution.
Throughout his PhD, and during the subsequent eight years at the prestigious Courant Institute in New York, he focused on uncovering the principles governing the extraction of information from various types of digital data – images, sounds, electrocardiograms – with a central objective: to represent large-scale data as a superposition of a minimal number of elementary structures.
“This is somewhat akin to constructing a house from Lego blocks, using the fewest possible bricks while retaining the ability to define the shape of these elementary components,” he explains. The question of sparse representation, reminiscent of the principle of simplicity underlying Ockham’s razor in philosophy, arises across all fields.
For instance in music, a polyphonic melody consists of a succession of elementary blocks that are the notes, each with its own pitch and duration. “With an image, being sparse means focusing on significant variations, such as a contour or abrupt change in colour. In mathematics, the goal is to capture the essence of the problem – the pursuit of sparsity – while freeing oneself from the context of specific applications, in order to discover general solutions that can later have a wide range of applications." From the very beginning of his career, Stéphane Mallat set out in search of these fundamental structures capable of representing any type of data sparsely. Serendipity would eventually guide him toward these elementary building blocks.
One summer, while on the beach, a friend mentioned the work of the mathematician Yves Meyer on “wavelets.” In mathematics, a wavelet is a curve that oscillates over a small domain and then vanishes. Intrigued, Mallat obtained Meyer’s paper, which showed, among other things, that any complex curve can be represented as a superposition of very particular wavelets. The mathematical problem raised by Yves Meyer was to determine whether it was possible to construct other types of wavelets capable of producing sparser decompositions^[1]^.
“I found a solution to this mathematical question based on the image processing problem posed by Ruzena Bajcsy,” explains Stéphane Mallat. “In image processing, wavelets can be interpreted as details that progressively increase the resolution of an image. Following this approach, I introduced the theory of multiresolution analyses, which provides a framework for constructing all mathematical wavelets. In this way, the intuition derived from image processing led me to the solution of the mathematical problem, but it was the mathematical abstraction that enabled me to understand how to compute the ‘wavelet transform.’” This is a fast algorithm, known as the Mallat's algorithm, capable of rewriting any digital data – such as an image composed of millions of pixels – as a superposition of a much smaller number of wavelets, each representing a local variation within the image.
“While Bajcsy was my mentor on the applications side, Meyer was undoubtedly the one on abstraction, and I move back and forth from one to the other.”
Mallat’s powerful algorithm, which is capable of rapidly compressing images without any no loss of information, was central to the many applications that emerged around the turn of the millennium, including the JPEG 2000 image compression standard. Under Mallat’s leadership, the mathematical language of wavelets generated a global standard used not only in software, but also in numerous medical, meteorological, and astronomical databases.
Already celebrated and recognised worldwide as a scientist whose work commands attention, the bridge-builder continues his rapid ascent. He aims to further advance the sparsity in data representation of which he is the architect. “In writing, using a limited vocabulary, one can certainly express complex ideas, but this comes with the risk of resorting to long circumlocutions and, ultimately, producing approximations. To create shorter, more impactful sentences, one must enrich the vocabulary. This is why I introduced the concept of a ‘mathematical dictionary,’ comprising a large number of elementary building blocks, more specialised than wavelets.”
Back in France, where he served as the Director of the Mathematics Department at Polytechnique beginning in 1998, he applied these results by building dictionaries of bandlets, to more effectively represent images and the geometry of contours. This work ultimately prompted him to make a significant change in his professional life.
In 2001, he founded the start-up Let It Wave with three of his former doctoral students. “Almost overnight, I went from being an academic to a CEO, and I discovered an entirely new world: marketing, negotiating funding rounds, concern that the venture would abruptly stop for lack of subsidies… It was exhilarating, and in some ways similar to research: entrepreneurs also need to be excited like children about an idea they believe will revolutionise the world, even if it might collapse in a fortnight. They have a vision and are never jaded, which, in my view is an essential quality. But by moving into this world, I realised how much I missed research and teaching.” Too much concreteness, not enough abstraction.
So after profitably selling Let It Wave, he returned to Polytechnique in 2007, where he introduced entrepreneurship classes for students, a way of passing the torch of the builder. “Yet as a researcher, I went through a dry spell. I had no desire to repeat what I had done before, all the ideas I had in mind seemed already explored. I was in doubt, wondering whether at 45 years of age I was too old to take up research again, to invent new mathematics or algorithms.” And then the horizon brightened. In 2008, he discovered Yann LeCun’s results on deep neural networks. “I knew enough about image processing applications to realise that these computer programs inspired by the human brain, did not merely represent incremental progress, but they constituted a genuine paradigm shift.”
Mallat plunged headfirst into the world of artificial intelligence, with a clear objective: to develop mathematical models to understand the remarkable performance of neural networks. These networks learn to answer a question by analysing data, for example identifying the animal in an image. During their training, they are provided with millions of examples, each paired with the correct answer – the name of the animal corresponding to each image. Much like a student practicing exercises, the network learns by adjusting its internal parameters to make fewer mistakes. “But how does it manage to provide so many correct answers for new images it has never seen? It’s a mystery, because these problems are highly complex. What type of information has it learned to extract from the data? I observed that these neural networks initially compute a ‘wavelet transform’. This reminded me of the results of neurophysiologists, who have also identified ‘wavelet transforms’ in the primary areas of our visual cortex, as well as in the cochlea in the ear.”
Building on his expertise and interdisciplinary vision, Mallat showed that a neural network builds hierarchical representations. It separates the largest structures, for example the coarse outline of a face in an image, and represents finer components relative to the broader ones. For instance, the eyes relative to the face, and the pupil relative to the eye. “The wavelet transform is a first step in constructing this hierarchy,” he explains.
By elucidating this mechanism, he laid the mathematical foundations for deep learning models, which underpin many AI systems today. “But the deeper one goes into the network layers, the more sophisticated the structures the network detects. Certain neurons activate for very specific features, such as a melody or a face. It is as if these deep layers represented data with very rich and highly specialised ‘mathematical dictionaries,’ whose properties remain poorly understood by scientists.” All these results, he emphasizes, were achieved collectively: “In science, one almost never moves forward alone. Throughout my career, I have worked extensively with my doctoral students and numerous collaborators. They have supported me in formulating the right questions, sharing both successes and setbacks. Each, in their own way, has brought fundamental contributions.”
Do these artificial intelligences, which Mallat is still studying today, pose a threat to our societies? “They bring remarkable advances, for example in medicine, but like any technology, they also carry risks – for privacy, and because of their potential military use,” he points out. "It is therefore crucial to control and regulate them, but that is not solely the responsibilityof governments. Each of us is confronted with this revolution, and will need to adapt to take advantage of the best it offers, while avoiding the pitfalls. This requires understanding AI, and not mythologizing it. It is with this goal in mind that I created MathAData^[2]^, a high school teaching programme for mathematics directly linked to solving practical AI problems. We can see that middle and high school students are much more motivated to learn maths when they understand it lies at heart of major issues and the tools of their everyday lives.”
How do you, Stéphane Mallat, spend your time when not navigating oceans of data, or building bridges between great ideas and reality? “I love to dance. Tango, rock-and roll… sometimes on the banks of the Seine. When I dance, I’m in another world, that of the music and my partner. I disconnect.” After all, don’t builders sometimes need to take a breather?
Footnotes
-
Or, in mathematical language: “build new orthogonal wavelet bases”.
-
mathadata.fr/en
22
painter
Johnson, Crockett
Description
Some of Crockett Johnson's paintings reflect relatively recent research. Mathematicians had long been interested in the distribution of prime numbers. At a meeting in the early 1960s, physicist
Stanislaw Ulam
Stanisław Marcin Ulam (13 April 1909 – 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved a number of theorems and proposed several conjectures.
https://en.wikipedia.org/wiki/Stanis%C5%82aw_Ulam
of the Los Alamos Scientific Laboratory in New Mexico passed the time by jotting down numbers in grid. One was at the center, the digits from 2 to 9 around it to form a square, the digits from 10 to 25 around this, and the spiral continued outward.
Circling the prime numbers, Ulam was surprised to discover that they tended to lie on lines. He and several colleagues programmed the MANIAC computer to compute and plot a much larger number spiral, and published the result in the American Mathematical Monthly in 1964. News of the event also created sufficient stir for Scientific American to feature their image on its March 1964 cover. Martin Gardner wrote a related column in that issue entitled “The Remarkable Lore of the Prime Numbers.”
The painting is #77 in the series. It is unsigned and undated, and has a wooden frame painted white.
date made
ca 1965
Object Name
painting
Physical Description
masonite (substrate material) wood (frame material)
Measurements
overall: 82 cm x 85 cm x 1.3 cm; 32 5/16 in x 33 7/16 in x 1/2 in
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
Ulam spiral of size 201×201. Black dots represent prime numbers. Diagonal, vertical, and horizontal lines with a high density of prime numbers are clearly visible.
For comparison, a spiral with random odd numbers colored black (at the same density of primes in a 200x200 spiral).
Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x^2^ − x + 41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be.
The Ulam spiral is constructed by writing the positive integers in a spiral arrangement on a square lattice:
and then marking the prime numbers:
In the figure, primes appear to concentrate along certain diagonal lines. In the 201×201 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern to that point. Horizontal and vertical lines with a high density of primes, while less prominent, are also evident. Most often, the number spiral is started with the number 1 at the center, but it is possible to start with any number, and the same concentration of primes along diagonal, horizontal, and vertical lines is observed. Starting with 41 at the center gives a diagonal containing an unbroken string of 40 primes (starting from 1523 southwest of the origin, decreasing to 41 at the origin, and increasing to 1601 northeast of the origin), the longest example of its kind.
Explanation
Diagonal, horizontal, and vertical lines in the number spiral correspond to polynomials of the form
f(n) = 4n^2^ + bn + c
where b and c are integer constants. When b is even, the lines are diagonal, and either all numbers are odd, or all are even, depending on the value of c. It is therefore no surprise that all primes other than 2 lie in alternate diagonals of the Ulam spiral. Some polynomials, such as 4n^2^ + 8n + 3, while producing only odd values, factorize over the integers (4n^2^ + 8n + 3) = (2n + 1)(2n + 3) and are therefore never prime except possibly when one of the factors equals 1. Such examples correspond to diagonals that are devoid of primes or nearly so.
To gain insight into why some of the remaining odd diagonals may have a higher concentration of primes than others, consider 4n^2^ + 6 n + 1 and 4n^2^ + 6 n + 5. Compute remainders upon division by 3 as n takes successive values 0, 1, 2, .... For the first of these polynomials, the sequence of remainders is 1, 2, 2, 1, 2, 2, ..., while for the second, it is 2, 0, 0, 2, 0, 0, .... This implies that in the sequence of values taken by the second polynomial, two out of every three are divisible by 3, and hence certainly not prime, while in the sequence of values taken by the first polynomial, none are divisible by 3. Thus it seems plausible that the first polynomial will produce values with a higher density of primes than will the second. At the very least, this observation gives little reason to believe that the corresponding diagonals will be equally dense with primes. One should, of course, consider divisibility by primes other than 3. Examining divisibility by 5 as well, remainders upon division by 15 repeat with pattern 1, 11, 14, 10, 14, 11, 1, 14, 5, 4, 11, 11, 4, 5, 14 for the first polynomial, and with pattern 5, 0, 3, 14, 3, 0, 5, 3, 9, 8, 0, 0, 8, 9, 3 for the second, implying that only three out of 15 values in the second sequence are potentially prime (being divisible by neither 3 nor 5), while 12 out of 15 values in the first sequence are potentially prime (since only three are divisible by 5 and none are divisible by 3).
While rigorously-proved results about primes in quadratic sequences are scarce, considerations like those above give rise to a plausible conjecture on the asymptotic density of primes in such sequences, which is described in the next section.
Variants
Klauber triangle with prime numbers generated by Euler's polynomial x^2^ − x + 41 highlighted
Sacks spiral
Ulam spiral of size 150×150 showing both prime and composite numbers
Hexagonal number spiral with prime numbers in green and more highly composite numbers in darker shades of blue
Number spiral with 7503 primes visible on regular triangle
Ulam spiral with 10 million primes
23
Sangaku or san gaku (Japanese: 算額, lit. 'calculation tablet') are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.
A sangaku dedicated to Konnoh Hachimangu (Shibuya, Tokyo) in 1859.
A sangaku dedicated at Emmanji Temple in Nara
The sangaku were painted in color on wooden tablets (ema) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas, as challenges to the congregants, or as displays of the solutions to questions. Many of these tablets were lost during the period of modernization that followed the Edo period, but around nine hundred are known to remain.
Fujita Kagen (1765–1821), a Japanese mathematician of prominence, published the first collection of sangaku problems, his Shimpeki Sampo (Mathematical problems Suspended from the Temple) in 1790, and in 1806 a sequel, the Zoku Shimpeki Sampo.
During this period Japan applied strict regulations to commerce and foreign relations for western countries so the tablets were created using Japanese mathematics, developed in parallel to western mathematics. For example, the connection between an integral and its derivative (the fundamental theorem of calculus) was unknown, so sangaku problems on areas and volumes were solved by expansions in infinite series and term-by-term calculation.
https://en.wikipedia.org/wiki/Sangaku
Of the world's countless customs and traditions, perhaps none is as elegant, nor as beautiful, as the tradition of sangaku, Japanese temple geometry. From 1639 to 1854, Japan lived in strict, self-imposed isolation from the West. Access to all forms of occidental culture was suppressed, and the influx of Western scientific ideas was effectively curtailed. During this period of seclusion, a kind of native mathematics flourished.
Devotees of math, evidently samurai, merchants and farmers, would solve a wide variety of geometry problems, inscribe their efforts in delicately colored wooden tablets and hang the works under the roofs of religious buildings. These sangaku, a word that literally means mathematical tablet, may have been acts of homage--a thanks to a guiding spirit--or they may have been brazen challenges to other worshipers: Solve this one if you can! For the most part, sangaku deal with ordinary Euclidean geometry. But the problems are strikingly different from those found in a typical high school geometry course. Circles and ellipses play a far more prominent role than in Western problems: circles within ellipses, ellipses within circles. Some of the exercises are quite simple and could be solved by first-year students. Others are nearly impossible, and modern geometers invariably tackle them with advanced methods, including calculus and affine transformations
https://www.cut-the-knot.org/pythagoras/Sangaku.shtml
The tablet was called a SANGAKU which means a mathematics tablet in Japanese. Many skilled geometers dedicated a SANGAKU in order to thank the god for the discovery of a theorem. The proof of the proposed theorem was rarely given. This was interpreted as a challenge to other geometers, "See if you can prove this."
More at http://www.wasan.jp/index.html
24
The icosian game is a mathematical game invented in 1856 by Irish mathematician William Rowan Hamilton. It involves finding a Hamiltonian cycle on a dodecahedron, a polygon using edges of the dodecahedron that passes through all its vertices. Hamilton's invention of the game came from his studies of symmetry, and from his invention of the icosian calculus, a mathematical system describing the symmetries of the dodecahedron.
Hamilton sold his work to a game manufacturing company, and it was marketed both in the UK and Europe, but it was too easy to become commercially successful. Only a small number of copies of it are known to survive in museums. Although Hamilton was not the first to study Hamiltonian cycles, his work on this game became the origin of the name of Hamiltonian cycles. Several works of recreational mathematics studied his game. Other puzzles based on Hamiltonian cycles are sold as smartphone apps, and mathematicians continue to study combinatorial games based on Hamiltonian cycles.
Game play
A Hamiltonian cycle on a dodecahedron
Planar view of the same cycle
The game's object is to find a three-dimensional polygon made from the edges of a regular dodecahedron, passing exactly once through each vertex of the dodecahedron. A polygon visiting all vertices in this way is now called a Hamiltonian cycle.) In a two-player version of the game, one player starts by choosing five consecutive vertices along the polygon, and the other player must complete the polygon.
Édouard Lucas describes the shape of any possible solution, in a way that can be remembered by game players. A completed polygon must cut the twelve faces of the dodecahedron into two strips of six pentagons. As this strip passes through each of its four middle pentagons, in turn, it connects through two edges of each pentagon that are not adjacent, making either a shallow left turn or a shallow right turn through the pentagon. In this way, the strip makes two left turns and then two right turns, or vice versa.
One version of the game took the form of a flat wooden board inscribed with a planar graph with the same combinatorial structure as the dodecahedron (a Schlegel diagram), with holes for numbered pegs to be placed at its vertices. The polygon found by game players was indicated by the consecutive numbering of the pegs. Another version was shaped as a "partially flattened dodecahedron", a roughly hemispherical dome with the pentagons of a dodecahedron spread on its curved surface and a handle attached to its flat base. The vertices had fixed pegs. A separate string, with a loop at one end, was wound through these pegs to indicate the polygon.
The game was too easy to play to achieve much popularity, although Hamilton tried to counter this impression by giving an example of an academic colleague who failed to solve it. David Darling suggests that Hamilton may have made it much more difficult for himself than for others, by using his theoretical methods to solve it instead of trial and error.
https://en.wikipedia.org/wiki/Icosian_game
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish mathematician, physicist, and astronomer who made numerous major contributions to algebra, classical mechanics, and optics. His theoretical works and mathematical equations are considered fundamental to modern theoretical physics, particularly his reformulation of Lagrangian mechanics. His research included the analysis of geometrical optics, Fourier analysis, and quaternions, the last of which made him one of the founders of modern linear algebra.
https://en.wikipedia.org/wiki/William_Rowan_Hamilton
A graph having a Hamiltonian cycle, i.e., on which the Icosian game may be played, is said to be a Hamiltonian graph. While the skeletons of all the Platonic solids and Archimedean solids (i.e., the Platonic graphs and Archimedean graphs, respectively) are Hamiltonian, the same is not necessarily true for the skeletons of the Archimedean duals, as shown by Coxeter (1946) and Rosenthal (1946) for the rhombic dodecahedron (Gardner 1984, p. 98).
Wolfram (2022) analyzed the icosian game as a multicomputational process, including through the use of multiway and branchial graphs. In particular, the multiway graph for the icosian game begins as illustrated above.
https://mathworld.wolfram.com/IcosianGame.html
The Original Icosian Game
In 1857 Sir William Rowan Hamilton invented the Icosian game. In a world based on the dodecahedral graph, a traveler must visit 20 cities, without revisiting any of them. Today, when the trip makes a loop through all the vertices of the graph, it is called a Hamiltonian tour (or cycle). When the first and last vertices in a trip are not connected, it is called a Hamiltonian path (or trail). The first image shown is a tour; the second is a path.
Hamiltonian cycles gained popularity in 1880, when P. G. Tait made the conjecture: “Every cubic polyhedron has a Hamiltonian cycle through all its vertices”. Cubic means that three edges meet at every vertex. Without the cubic requirement, there are smaller polyhedra that are not Hamiltonian. The simplest counterexample is the rhombic dodecahedron. Every edge connects one of six valence-four vertices to one of eight valence-three vertices. The six valence-four vertices would need to occupy every other vertex in the length-14 tour. Six items cannot fill seven slots, so this is impossible.
Any noncubic graph can be made cubic by placing a small disk over the exceptions.
The word “polyhedral” implies that the graph must be 3-connected. If a line is drawn to disconnect the map, it must pass through at least three borders. Central Europe is not 3-connected, since a line through Spain will disconnect Portugal. France, the Vatican, and various islands also make the shape of Europe nonpolyhedral.
Tait’s method turns a Hamiltonian cycle on a cubic polyhedral graph into a four-coloring, by the following method.
Alternately color the edges of the Hamiltonian cycle blue and purple. Color the other edges red.
Throw out thin edges, and color the resulting polygon blue.
Throw out dashed edges, and color the resulting polygon(s) red.
Overlay the two colorings to get a four-coloring.
For 66 years, Tait’s conjecture held. In 1946, W. G. Tutte found the first counterexample, now known as Tutte’s graph. Since then, some smaller cubic polyhedral non-Hamiltonian graphs have been found, with the smallest such graph being the Barnette-Bosák-Lederberg graph, found in 1965. Seven years earlier, Lederberg had won the Nobel Prize in Medicine.
https://www.mathematica-journal.com/2010/02/05/the-icosian-game-revisited/#Dalgety
Tutte's fragment
The key to this counter-example is what is now known as Tutte's fragment [...].
If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex (and either one of the lower ones). It cannot go in one lower vertex and out the other.
The counterexample
The fragment can then be used to construct the non-Hamiltonian Tutte graph, by putting together three such fragments as shown in the picture.
The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle.
The resulting Tutte graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. In total it has 25 faces, 69 edges and 46 vertices. It can be realized geometrically from a tetrahedron (the faces of which correspond to the four large faces in the drawing, three of which are between pairs of fragments and the fourth of which forms the exterior) by multiply truncating three of its vertices.
25
The number π is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors.
Because it relates to a circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas having little to do with geometry, such as number theory and statistics, and in modern mathematical analysis can be defined without any reference to geometry. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.
Definition
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.
π is commonly defined as the ratio of a circle's circumference C to its diameter d:
π = C/d
The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio C/d.
In modern mathematics, this definition is not fully satisfactory for several reasons. Firsly, it lacks a rigorous definition of the length of a curved line. Such a definition requires at least the concept of a limit, or, more generally, the concepts of derivatives and integrals. Also, diameters, circles and circumferences can be defined in Non-Euclidean geometries, but, in such a geometry, the ratio C / d need not to be a constant, and need not to equal to π. Also, there are many occurrences of π in many branches of mathematics that are completely independent from geometry, and in modern mathematics, the trend is to built geometry from algebra and analysis rather than independently from the other branches of mathematics.
https://en.wikipedia.org/wiki/Pi
Archimedes’ Method of Approximating Pi
Since the true value of pi could not be measured directly, Archimedes developed a geometric technique using polygons to establish upper and lower bounds for its value. His method relied on inscribing and circumscribing regular polygons around a circle and calculating their perimeters. By progressively increasing the number of sides, he was able to narrow the range within which pi must lie. This approach was a precursor to the concept of limits, which later became a fundamental idea in calculus.
The Inscribed and Circumscribed Polygon Method
In his work Measurement of a Circle Archimedes considered a circle with diameter d and radius r. He inscribed a regular hexagon inside the circle and circumscribed another hexagon outside it. By calculating the perimeters of these polygons, he obtained lower and upper estimates for the circumference of the circle. Since the ratio of the circumference to the diameter is pi (C / d = pi), these perimeters provided bounds for pi.
He then systematically increased the number of sides of the polygons, doubling them from 6-sided to 12-sided, 24-sided, 48-sided, and finally 96-sided polygons. As the number of sides increased, the perimeters of the inscribed and circumscribed polygons became closer to the true circumference of the circle, refining the estimate of pi.
Using this method, Archimedes established the following inequality:
223/71 < pi < 22/7
This meant that pi was approximately 3.1408 < pi < 3.1429, a remarkably accurate estimate for the time.
Mathematical Process Behind Archimedes’ Approximation
To derive these values, Archimedes used the Pythagorean theorem and properties of similar triangles to calculate the side lengths of the polygons. By repeatedly applying trigonometric relationships (though without the formal notation used today), he determined the perimeters of each successive polygon. His method can be broken down as follows:
- For an inscribed n-sided polygon:
The perimeter Pi provides a lower bound for the circle’s circumference.
Formula: Pi = n * s~i~, where s~i~ is the side length.
- For a circumscribed n-sided polygon:
The perimeter P~c~ gives an upper bound for the circumference.
Formula: P~c~ = n * s~c~, where sc is the side length.
- Refining the estimate:
Archimedes doubled the number of sides, recalculating the new perimeters iteratively.
The values of P~i~ and P~c~ converged toward the true circumference of the circle, P~i~ < C < P~c~.
By the time he reached a 96-sided polygon, his estimates were precise to two decimal places. This level of accuracy was unprecedented and remained the best approximation of pi for nearly 1,000 years.
Circle circumscribed and inscribed by a square where n=4.
The Limitations of Archimedes’ Approach
Archimedes' method had several inherent limitations. First, the computational intensity of his approach increased significantly as the number of sides in his polygons grew. Without the tools of modern algebra or trigonometry, he had to rely solely on geometric reasoning, making the process increasingly complex. Additionally, his method could only provide an approximation of pi rather than an exact value. Since pi is an irrational number that cannot be expressed as a finite fraction, Archimedes' approach was necessarily limited in its precision. Another challenge was the laborious nature of manual computation. Each successive step required extensive geometric derivations, making further refinements impractical beyond a certain point. Despite these limitations, Archimedes' work demonstrated a systematic method for refining numerical approximations and laid the foundation for future mathematical advancements.
Implications of Archimedes’ Work on Pi
Archimedes' method of approximating pi was groundbreaking, not only for its accuracy but also for its influence on the development of mathematical techniques. His approach established a systematic way of refining numerical approximations, which later became essential in calculus and numerical analysis. His work remained the most accurate estimate of pi for over a millennium and laid the foundation for future mathematicians to further refine the calculation of pi.
Archimedes’ method set the stage for many mathematicians across different cultures to refine and improve the approximation of pi. In the 3rd century CE, the Chinese mathematician Liu Hui built upon Archimedes' technique and extended it to a 3072-sided polygon, achieving a more precise approximation of pi at 3.14159. Two centuries later, Zu Chongzhi improved on this result, determining that pi was approximately 355/113 (3.1415929), an extraordinarily precise fraction that remained the most accurate estimate for over a thousand years.
In the Islamic Golden Age, mathematicians such as Al-Khwarizmi and Al-Kashi expanded on these ideas using decimal notation and further refinements of the polygonal method. The Renaissance period saw renewed interest in Archimedes' approach, with European scholars like Ludolph van Ceulen extending the method to polygons with millions of sides. This allowed for calculations of pi accurate to more than 30 decimal places. Despite these advancements, Archimedes’ geometric method remained the dominant approach for approximating pi until the development of calculus in the 17th century.
Ludolph van Ceulen (8 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim known for the Ludolphine number, his calculation of the mathematical constant pi to 35 digits.
Ludolph van Ceulen spent a major part of his life calculating the numerical value of the mathematical constant π, using essentially the same methods as those employed by Archimedes some seventeen hundred years earlier. He published a 20-decimal value in his 1596 book Van den Circkel ("On the Circle"), which was published before he moved to Leiden, and he later expanded this to 35 decimals.
Van Ceulen's 20 digits is more than enough precision for any conceivable practical purpose. Even if a circle was perfect down to the atomic scale, the thermal vibrations of the molecules of ink would make most of those digits physically meaningless. Future attempts to calculate π to ever greater precision have been driven primarily by curiosity about the number itself.
https://en.wikipedia.org/wiki/Ludolph_van_Ceulen
The above image is the title page of Vanden Circkel, a book about the circle and π by Ludolph Van Ceulen (1540–1610). Published in 1596 in Dutch, it contains the longest decimal approximation of π at the time—20 decimal places. In fact, below the portrait of Van Ceulen, the engraving on the title page has a circle with diameter of 10^20^. Across the top semicircle is “314159265358979323846 te cort” (too short), and “314159265358979323847 te lanck” (too long) is along the bottom semicircle. Later, Van Ceulen would determine π to 35 decimal places. A modified Latin version of the work was published in 1619, images of which can also be found on Convergence here and here.
Part of what little is known of Van Ceulen’s life before 1578 comes from the Preface of Vanden Circkel. Starting in 1566, he earned a living as a mathematics teacher, and in 1580 he opened his first fencing school. A few years later Archimedes’ method of approximating π was translated from the Greek for him, and Van Ceulen proceeded to use the technique to improve on approximations of π, publishing Vanden Circkel in 1596. Below are images from folio 1 and folio 7.
Chapter 21 is devoted to analyzing a work of Joseph Justice Scaliger (1540–1609) called Cyclometrica Elementa (Elements of Circle Measurement), which had several incorrect results, including a “proof” that the area of a circle is equal to 6/5 of the area of an inscribed regular hexagon, which results in π=(9/5)√3 or approximately 3.117691454. Van Ceulen doesn’t mention Scaliger by name, but rather calls him a “highly learned man”. Below is Folio 63a.
https://old.maa.org/press/periodicals/convergence/mathematical-treasure-van-ceulen-s-vanden-circkel
Van Ceulen is famed for his calculation of π to 35 places which he did using polygons with 2^62^ sides. Having published 20 places of π in his book of 1596, the more accurate results were only published after his death. In 1615 his widow Adriana Simondochter published a posthumous work by Van Ceulen entitled De arithmetische en geometrische fondamenten. This contained his computation of 33 decimal places for π. The complete 35 decimal place approximation was only published in 1621 in Snell's Cyclometricus. Having spent most of his life computing this approximation, it is fitting that the 35 places of π were engraved on Van Ceulen's tombstone. In fact Van Ceulen had purchased a grave in the Pieterskerk on 11 November 1602 but, after Van Ceulen's death on 31 December 1610, his widow Adriana exchanged this grave for another, still in the Pieterskerk, and it was in this second grave that Van Ceulen was buried on 2 January 1611. The tombstone gave both Van Ceulen's lower bound of 3.14159265358979323846264338327950288 and his upper bound of 3.14159265358979323846264338327950289. However, the original tombstone disappeared around 1800 to be replaced by a replica two hundred years later. The original text on the tombstone was known since it had been recorded in a guidebook of 1712 and after that reprinted in many articles. Vajta writes :
On July 5, 2000 a very special ceremony took place in the St Pieterskerk (St Peter's Church) at Leiden, the Netherlands. A replica of the original tombstone of Ludolph Van Ceulen was placed into the Church since the original disappeared. ... It was therefore a tribute to the memory of Ludolph Van Ceulen, when on Wednesday 5 July, 2000 prince Willem-Alexander (heir to the throne), unveiled the memorial tombstone in the St Peter's Church, in Leiden.
In Germany π was called the "Ludolphine number" for a long time.
https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/
view more: next ›