#euler — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #euler, aggregated by home.social.
-
#Euler and the Seven #Bridges of #Königsberg : Medium
#Mathematicians make a #Breakthrough on #Gauss’s #Riddle, unsolved for 200 years : Sci Am
A newly found #World #Blurs the boundary between #Planet and #Moon : Misc
Latest #KnowledgeLinks
-
#Euler and the Seven #Bridges of #Königsberg : Medium
#Mathematicians make a #Breakthrough on #Gauss’s #Riddle, unsolved for 200 years : Sci Am
A newly found #World #Blurs the boundary between #Planet and #Moon : Misc
Latest #KnowledgeLinks
-
#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/ -
#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/ -
From a Mathematical Curiosity to the Foundation of Internet Security 🧮🧐🔐✨
In 1636, Pierre de Fermat made a remarkable observation: if you take an integer a, raise it to a prime power p, and then subtract a, the result is always divisible by p. In other words, for a prime p,
aᵖ ≡ a (mod p).
For example, when p = 5:
2⁵ − 2 = 30, which is divisible by 5, and
3⁵ − 3 = 240, which is also divisible by 5.Fermat found this property elegant and tried to spark interest in this curious corner of mathematics. Not everyone was impressed. John Wallis reportedly dismissed such results, remarking, “Big deal; I could find other relationships just as interesting without much effort, and none of them are important.”
About a century later, Leonhard Euler was encouraged by Christian Goldbach to study Fermat’s ideas. Though initially unenthusiastic, Euler soon uncovered deeper structure within them. His work laid much of the foundation for what we now call modern number theory.
#Mathematics #Math #NumberTheory #Fermat #Euler #EulerTheorem #FermatsLittleTheorem #ModularArithmetic #RSA #Cryptography #CyberSecurity #HistoryOfMathematics #MathHistory #PrimeNumbers #EulerTotient #AbstractMath #PureMathematics #STEM #ScienceCommunication #MathEducation #EducationalContent #DidYouKnow #MathematicalBeauty #InternetSecurity #Encryption #Curiosity #Innovation #History #Learning #Knowledge
-
From a Mathematical Curiosity to the Foundation of Internet Security 🧮🧐🔐✨
In 1636, Pierre de Fermat made a remarkable observation: if you take an integer a, raise it to a prime power p, and then subtract a, the result is always divisible by p. In other words, for a prime p,
aᵖ ≡ a (mod p).
For example, when p = 5:
2⁵ − 2 = 30, which is divisible by 5, and
3⁵ − 3 = 240, which is also divisible by 5.Fermat found this property elegant and tried to spark interest in this curious corner of mathematics. Not everyone was impressed. John Wallis reportedly dismissed such results, remarking, “Big deal; I could find other relationships just as interesting without much effort, and none of them are important.”
About a century later, Leonhard Euler was encouraged by Christian Goldbach to study Fermat’s ideas. Though initially unenthusiastic, Euler soon uncovered deeper structure within them. His work laid much of the foundation for what we now call modern number theory.
#Mathematics #Math #NumberTheory #Fermat #Euler #EulerTheorem #FermatsLittleTheorem #ModularArithmetic #RSA #Cryptography #CyberSecurity #HistoryOfMathematics #MathHistory #PrimeNumbers #EulerTotient #AbstractMath #PureMathematics #STEM #ScienceCommunication #MathEducation #EducationalContent #DidYouKnow #MathematicalBeauty #InternetSecurity #Encryption #Curiosity #Innovation #History #Learning #Knowledge
-
Today we share our Euler's formula tee:
https://redbubble.com/i/camiseta/La-ecuaci%C3%B3n-m%C3%A1s-elegante-de-tudi/78240774.NL9AC
#euler #eulersformula #eulersequation #eulersidentity #mathematics #maths #geek #nerd #nerdy #formula #equation #identity #formuladeeuler #science #ciencia #trigonometry #geekygifts #gifts #giftideas #Tshirt #hoodie
https://bsky.app/profile/tuditees.bsky.social/post/3mpgmokgqc22t
-
Today we share our Euler's formula tee:
https://redbubble.com/i/camiseta/La-ecuaci%C3%B3n-m%C3%A1s-elegante-de-tudi/78240774.NL9AC
#euler #eulersformula #eulersequation #eulersidentity #mathematics #maths #geek #nerd #nerdy #formula #equation #identity #formuladeeuler #science #ciencia #trigonometry #geekygifts #gifts #giftideas #Tshirt #hoodie
https://bsky.app/profile/tuditees.bsky.social/post/3mpgmokgqc22t
-
E, já que estamos falando tanto em #Euler hoje, nada melhor do que corrigir trabalhos tomando #cappuccino na minha xícara do Euler rs.
-
E, já que estamos falando tanto em #Euler hoje, nada melhor do que corrigir trabalhos tomando #cappuccino na minha xícara do Euler rs.
-
Results ranging from visualizable theorems of solid geometry to abstract propositions of analysis were called beautiful by Leonhard Euler (1707–83). For instance, he thought beautiful the following result:
If an elliptical cylinder is cut by any plane at an angle θ, then the ratio of the product of the principal axes of the section and of the product of the principal axes of the base is 1:cos θ (see attached image).
Aesthetic concerns seem to have been part of what drew Euler to number theory. Christian Goldbach (1690–1764) persuaded him to take an interest in the subject and to make a serious study of Fermat's work. His attention was drawn by the theorem:Every natural number can be expressed as a sum of four squares.
With presumably deliberate understatement, Euler described it as a ‘not inelegant theorem’. The result remained unproven in Euler's time, and the first proof was given by Joseph-Louis Lagrange (1736–1813), becoming known as ‘Lagrange’s four-square theorem’.
Thus, for Euler, *unproven* conjectures could have aesthetic value. And so he judged another well-known then-unproven result of Fermat:
‘In Fermat there is another very beautiful theorem for which he claims to have found a proof. […] the formula $a^n + b^n = c^n$ is impossible whenever $n > 2$’1/2
[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#Euler #Fermat #Goldbach #Lagrange #FermatsLastTheorem #MathematicalBeauty
-
Results ranging from visualizable theorems of solid geometry to abstract propositions of analysis were called beautiful by Leonhard Euler (1707–83). For instance, he thought beautiful the following result:
If an elliptical cylinder is cut by any plane at an angle θ, then the ratio of the product of the principal axes of the section and of the product of the principal axes of the base is 1:cos θ (see attached image).
Aesthetic concerns seem to have been part of what drew Euler to number theory. Christian Goldbach (1690–1764) persuaded him to take an interest in the subject and to make a serious study of Fermat's work. His attention was drawn by the theorem:Every natural number can be expressed as a sum of four squares.
With presumably deliberate understatement, Euler described it as a ‘not inelegant theorem’. The result remained unproven in Euler's time, and the first proof was given by Joseph-Louis Lagrange (1736–1813), becoming known as ‘Lagrange’s four-square theorem’.
Thus, for Euler, *unproven* conjectures could have aesthetic value. And so he judged another well-known then-unproven result of Fermat:
‘In Fermat there is another very beautiful theorem for which he claims to have found a proof. […] the formula $a^n + b^n = c^n$ is impossible whenever $n > 2$’1/2
[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#Euler #Fermat #Goldbach #Lagrange #FermatsLastTheorem #MathematicalBeauty
-
Zum #Euler-Tag ein kleines #Calliopemini-Projekt gebaut (im Simulator von MakeCode nutzbar):
https://makecode.calliope.cc/_YvdYr29wmdWh<Hav_e_fun/>!
// CC @calliope
-
Zum #Euler-Tag ein kleines #Calliopemini-Projekt gebaut (im Simulator von MakeCode nutzbar):
https://makecode.calliope.cc/_YvdYr29wmdWh<Hav_e_fun/>!
// CC @calliope
-
#Euler-Tag:
* Datum (Europa): 27. Januar (27.1.)
* Datum (USA/International): 7. Februar (2/7)Zur Feier des Tages differenzieren und integrieren wir e^x.
-
#Euler-Tag:
* Datum (Europa): 27. Januar (27.1.)
* Datum (USA/International): 7. Februar (2/7)Zur Feier des Tages differenzieren und integrieren wir e^x.
-
2026-02-05 #SpookyKittiesSciences: there is also preliminary evidence that #Euler likes Tatarian Honeysuckle. She very deliberately licked the corner I put the honeysuckle in.
-
2026-02-05 #SpookyKittiesSciences: there is also preliminary evidence that #Euler likes Tatarian Honeysuckle. She very deliberately licked the corner I put the honeysuckle in.
-
Learned from my dad today that apparently #Euler the catnip addict has been meowing up a storm. He was outside in the afternoon when she went down the hill, meowing her little heart out. #SpookyKittiesFerals
-
Learned from my dad today that apparently #Euler the catnip addict has been meowing up a storm. He was outside in the afternoon when she went down the hill, meowing her little heart out. #SpookyKittiesFerals
-
It appears that the #CatnipWhirlpool froze just enough that her #Euler-ness was willing to use it to get up onto the shelf. #SpookyKittiesFerals
-
It appears that the #CatnipWhirlpool froze just enough that her #Euler-ness was willing to use it to get up onto the shelf. #SpookyKittiesFerals
-
Not 100% sure on the identity of the meow, but I am like 90% sure it is #Euler. I can rule out Dorian, as he went SW around the house, which would require an entire trip around the house and down the neighbors driveway in 10 seconds. X22 is really the only other possibility, but I have never heard his what I am sure is a very sweet voice. The Inspectors status as an equally long feral with Dorian makes me doubt it was him.
-
Not 100% sure on the identity of the meow, but I am like 90% sure it is #Euler. I can rule out Dorian, as he went SW around the house, which would require an entire trip around the house and down the neighbors driveway in 10 seconds. X22 is really the only other possibility, but I have never heard his what I am sure is a very sweet voice. The Inspectors status as an equally long feral with Dorian makes me doubt it was him.
-
Further progressing with #Euler the catnip addict today. Got a full long pet around her tail, including the tail!
It did add to the body of evidence that she has not been inside for a while. That collar and her behavior all but confirms that she had humans at one point, but her fur has a roughness that only comes from being out 24/7. I definitely made the right decision in naming her as a member of my #SpookyKitties family. #SpookyKittiesFerals
-
Further progressing with #Euler the catnip addict today. Got a full long pet around her tail, including the tail!
It did add to the body of evidence that she has not been inside for a while. That collar and her behavior all but confirms that she had humans at one point, but her fur has a roughness that only comes from being out 24/7. I definitely made the right decision in naming her as a member of my #SpookyKitties family. #SpookyKittiesFerals
-
2026-02-01 #SpookyKittiesSciences: in a surprise development, #Euler the catnip addict has revealed she likes silvervine. This is after 4 whole days of rolling everywhere but on the vine. And, like an X22+Dorian earlier in the morning, a very confused Inspector Clouseau(for those not in the know, he appears to no react to nip/vine) was watching her. Unlike X22, Euler did not roll over and almost land in the water bowl. #Cat #Cats #TuxedoCat #GrayCat
-
2026-02-01 #SpookyKittiesSciences: in a surprise development, #Euler the catnip addict has revealed she likes silvervine. This is after 4 whole days of rolling everywhere but on the vine. And, like an X22+Dorian earlier in the morning, a very confused Inspector Clouseau(for those not in the know, he appears to no react to nip/vine) was watching her. Unlike X22, Euler did not roll over and almost land in the water bowl. #Cat #Cats #TuxedoCat #GrayCat
-
It’s incredible that these two identities can explain so much.
Link: https://arxiv.org/abs/2505.03754
#math #calculus #euler #unification #generalization #integrals #integration
-
Weierstrass Substitution as a Special Case of the USM Framework:
https://geometriadominicana.blogspot.com/2025/12/weierstrass-as-special-case-of-usm.html#math #calculus #Weierstrass #USM #framework #euler #unification #integral #integration
-
Никлаус Вирт и язык Pascal — легенды 80-х
В 1970 году молодой швейцарский учёный и программист Никлаус Вирт (Niklaus Wirth) выпустил первую версию Pascal. Прошло более полувека, автор умер в 89 лет, а вот Паскаль остаётся актуальным и популярным языком программирования.
https://habr.com/ru/companies/ruvds/articles/968706/
#Никлаус_Вирт #Pascal #Turbo_Pascal #Delphi #Niklaus_Wirth #Free_Pascal #Lazarus_IDE #Algol #Euler #ruvds_статьи
-
When a calculus student ends up solving this integral using partial fraction decomposition (PFD) or integration by parts (IBP), what they have actually done is successfully applied a classical algorithm. It’s like solving the Rubik’s cube by following an algorithm you found on the Internet: the heavy lifting of inventing PFD and IBP was done centuries ago by Bernoulli, Leibniz, and Gregory.
That said, the USM is, as far as I know, the only published general scheme that, for this kind of radical–rational integrals, systematically reduces the integrand to polynomial-type functions (Laurent polynomials, to be more precise). There’s no need to set up complicated PFDs or wrestle with sec³, which can be tricky with IBP. That’s what I call solving a problem, if you’ll allow me a bit of self-promotion.
You can find a first draft (I’ll soon upload a second version with benchmarks and some improvements) of the USM method on arXiv: https://arxiv.org/abs/2505.03754
#calculus #math #symmetrymatters #halfangleapproach #euler #integration
-
“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…
Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?…
[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]
… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…
“What Is a Manifold?” from @quantamagazine.bsky.social.
Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”
* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)
###
As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.
#BernhardRiemann #calculus #cryptography #culture #Euler #geometry #history #JohnWallis #LeonhardEuler #manifold #manifolds #maps #Mathematics #Physics #Reimann #RoyalSociety #Science #Technology
-
Gave a lecture today about Euler's theorem.
Then somehow it led to:
"There's a face, next muffin!"
Wondering what kind of vibe that sent out... -
🎩💡 Math sorcery alert! #RSA swapped Euler's math magic for Carmichael's card trick without even telling anyone! Who knew #encryption had a sense of humor? 🤷♂️🔒
https://www.johndcook.com/blog/2025/10/06/a-quiet-change-to-rsa/ #MathMagic #Humor #Carmichael #Euler #HackerNews #ngated -
🎩💡 Math sorcery alert! #RSA swapped Euler's math magic for Carmichael's card trick without even telling anyone! Who knew #encryption had a sense of humor? 🤷♂️🔒
https://www.johndcook.com/blog/2025/10/06/a-quiet-change-to-rsa/ #MathMagic #Humor #Carmichael #Euler #HackerNews #ngated -
Played with the BEST theorem a little yesterday:
-
Played with the BEST theorem a little yesterday:
-
2/ Wusstet Ihr, dass #Euler die Planung der Trockenlegung des #Oderbruchs gemacht hat. Steht nicht in #Wikipedia, sollte es aber.
Ansonsten das Oderbruch ist wunderbar. Da kann man auch #Urlaub machen. Auf dem Deich Rad fahren usw.
-
2/ Wusstet Ihr, dass #Euler die Planung der Trockenlegung des #Oderbruchs gemacht hat. Steht nicht in #Wikipedia, sollte es aber.
Ansonsten das Oderbruch ist wunderbar. Da kann man auch #Urlaub machen. Auf dem Deich Rad fahren usw.
-
One day, one decomposition
A001838: Numbers k such that phi(k+2) = phi(k) + 23D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A001838.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A001838.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A001838.html#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #phi #Euler #totient #function #graph #threejs #webGL
-
One day, one decomposition
A001838: Numbers k such that phi(k+2) = phi(k) + 23D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A001838.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A001838.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A001838.html#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #phi #Euler #totient #function #graph #threejs #webGL
-
One day, one decomposition
A001494: Numbers k such that phi(k) = phi(k+2)3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A001494.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A001494.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A001494.html#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #Euler #totient #function #phi #graph #threejs #webGL
-
"Euler calculated without any apparent effort, as men breathe, as eagles sustain themselves in the air." – François Arago (1786-1853)
#quote #mathematics #maths #math #Euler -
"Euler calculait sans aucun effort apparent, comme les hommes respirent, comme les aigles se soutiennent dans les airs." – François Arago (1786-1853)
#citation #mathématiques #maths #math #Euler
