not open access :(, hopefully it will become open access at some point or someone will publish similar research!

Changing the values of two of the three elements in the growth law produces markedly different shells. The starting curve at the growing edge of the shell sets the shape of the shell's opening, while the scaling factor, which the researchers call lambda (λ), determines how fast the shell expands. By changing those two factors in the model, Bhattacharya and Liu were able to generate the shape of nearly all known molluscan shells.

If only someone had a time machine to whisper this truth in the ear of Italo Calvino before he wrote his Cosmicomics series of stories!!

quote from paper abstract https://www.pnas.org/doi/10.1073/pnas.2614478123

We are guided by two principles in the spirit of D’Arcy Thompson. First, the growth is governed by the repeated and continuous application of a fixed growth law, even as the shell evolves in overall shape, without any complex biological machinery to monitor and control the growth. Second, the growth law depends solely on local geometry at the shell’s growing edge. The first principle naturally leads to the mathematical statement that the shape of the shell is generated by the action of a Lie group on a protoconch [fuckin duh bruh]. The second naturally leads to a particular representation of the Lie group. We use this representation to show that the shapes of nearly all known molluscan shells can be described by essentially three parameters: a scalar (scaling), a vector (orientation), and a curve (edge of the protoconch). We relate these parameters to the phylogenetic tree. In addition to the morphogenetic insight, our results potentially point to a alternative growth-based approach to engineering complex structures.

Wow what a strikingly beautiful thing, it is also the tiniest microdose of haunting for me because I distinctly remember remarking to a geology professor in college that I wondered if you could do a fourier transform or some kind of mathematical analysis of shells to determine if there was unifying patterns that hinted at mechanisms in some productive way and he responded with something along the lines of "but why?? what would that answer?" and all I could do was shrug and say "I have no idea, I was just wondering."

also see https://www.caltech.edu/about/news/the-mathematics-of-growing-a-shell

paper on related subject from 2019, some great graphics https://arxiv.org/html/1901.00497v3

We note that previous work by some of the authors [36] already has highlighted the role of a spatial variation in material properties in shaping the sharp spines seen on the shells of bivalves, cephalopods and gastropods. The present communication adds to this emerging picture of the influence of physics, by shifting the focus to geometry, which, driven by morphoelastic growth, acts through the mechanism of surface growth and a parametric dependence on curvature.

Of particular interest is the further coupling of this framework with pattern formation by a range of reaction-transport equations. This would make accessible well-studied developmental milestones such as the patterning and morphogenesis of limbs and digits. Other problems in morphogenesis, such as the formation of skeletons are also within reach of our computational framework. Finally, we note that we have modelled antimarginal ornamentation events via bursts of growth. It is more likely that the growth rate does not change significantly at the location of ornamentations, but rather that the thickness of the mantle decreases, creating an increase in the length of the active mantle strip and a corresponding decrease in stiffness without requiring a strong increase in material. This effect would be interesting to incorporate in future studies, as the decrease in stiffness would further amplify the amplitudes of transverse deformation, and also may be non-uniform along the length of the mantle strip.

^ this is basically what the linked article/paper claims to have described

“A lot of the reason why people do pure math, even though it pays so much less than industry, is because you get to author your own creative contributions and then give them to a community that appreciates them for their intellectual beauty,” Chu said. “This environment where mathematical discourse is centered around machines talking and humans listening, I don’t think it’s a very attractive one.”

And AI is itself undermining the kind of education that would be necessary for that role. This came up at the Berkeley colloquium, too. “There’s a serious risk that 20, 30, 40 years from now, there will be no people around who have the same classical mathematics training,” one student said. “I’m worried about a world where the full body of mathematical knowledge is inaccessible.”

The result could be a lost generation. Goh is one of many young mathematicians struggling with the angst such a future is producing, and with what he sees as a lack of effort on the part of working mathematicians (at least until very recently) to remake the incentives to avoid it.

“There’s something sick about the culture,” he told me. “I didn’t think so, but now I do.”

Goh is alarmed by how some have taken up AI without enough critical thought, and he feels pressured to use it so that he doesn’t get left behind. “I was just very depressed about that,” he said. “That doesn’t feel inspiring to me. … It didn’t have to be this way.”

He likens it to Saruman from The Lord of the Rings, corrupted by the promise of unlimited knowledge and power. Goh and a growing number of mathematicians are starting to consider themselves AI “vegetarians,” (opens a new tab) consciously measured in their use of the technology. There’s nothing rewarding or interesting or fun about offloading their cognition to a machine, they argue.

also see this good podcast about the AI "solving" the Navier-Stokes equation for fluids.

https://www.newscientist.com/podcasts/ai-just-solved-one-of-the-hardest-problems-in-mathematics/

TL;DL All scientific advance6 is people building on people, the great man hypothesis reflected into computers into a great computer hypothesis is what this really is, the AI was able to connect the dots between converging research faster than a human could... but ok....? True scientific advance involves a one-two punch to your face, first you get the jab of proving something novel or that has stumped scientists for a long time up until that point and then you hit them with a full power punch of insight to seal the deal. This is only the first part, and to say "AI did it" is to again lean into the "great man hypothesis" but you have replaced the great man with a great computer...

Kill the King is proof that T.I. - and the rest of his crew - are of the greatest set theorists, and hyperbolic geometrists, of our time.

So i fell upon the following differential equation : X''(t)X⁴(t)=f(t) I couldn't find anything similar on internet... Thanks in advance !

https://epoch.ai/frontiermath/open-problems

Crossposted from https://piefed.social/c/science/p/1406456/first-shape-found-that-cant-pass-through-itself

In 1876, Peter Guthrie Tait set out to measure what he called the “beknottedness” of knots.

Tait had an idea for how to determine if two knots are different. First, lay a knot flat on a table and find a spot where the string crosses over itself. Cut the string, swap the positions of the strands, and glue everything back together. This is called a crossing change. If you do this enough times, you’ll be left with an unknotted circle. Tait’s beknottedness is the minimum number of crossing changes that this process requires. Today, it’s known as a knot’s “unknotting number.”

If two knots have different unknotting numbers, then they must be different. But Tait found that his unknotting numbers generated more questions than they answered.

If Tait missed something, so did every mathematician who followed him. Over the past 150 years, many knot theorists have been baffled by the unknotting number. They know it can provide a powerful description of a knot. “It’s the most fundamental [measure] of all, arguably,” said Susan Hermiller (opens a new tab) of the University of Nebraska. But it’s often impossibly hard to compute a knot’s unknotting number, and it’s not always clear how that number corresponds to the knot’s complexity.

To untangle this mystery, mathematicians in the early 20th century devised a straightforward conjecture about how the unknotting number changes when you combine knots. If they could prove it, they would have a way to compute the unknotting number for any knot — giving mathematicians a simple, concrete way to measure knot complexity.

Researchers searched for nearly a century, finding little evidence either for or against the conjecture.

Then, in a paper posted in June, Hermiller and her longtime collaborator Mark Brittenham (opens a new tab) uncovered a pair of knots that, when combined, form a knot that is easier to untie than the conjecture predicts. In doing so, they disproved the conjecture (opens a new tab) — and used their counterexample to find infinitely many other pairs of knots that also disprove it.

The result demonstrates that “the unknotting number is chaotic and unpredictable and really exciting to study,” she added. The paper is “like waving a flag that says, we don’t understand this.”

I'm not very advanced in math or coding but I do enjoy some of the low level challenges. Anyone else do these? Link

@math Hi! I am currently working on an open Diophantine equation listed in a publication by Dr Bogdan Grechuk: y^3 + xy = x^4 + 4. The question if any integer solutions exist eludes brute force attempts even with scripts in systems such as SageMath. Empirical work suggests there may be no solutions. I have been trying to prove by various approaches that no integer solutions exist. Dr Gerchuk gave critical feedback on my first two approaches but I am hoping to get more informal feedback or even collaboration on this before I send it to him a third time.Here is the Overleaf link to the project and I am open to also share a link to edit and possibly coauthor if anyone is interested in helping get this case closed or to lay it to rest.overleaf.com/read/ncqwpfxzkqfr

Part (b). So, this is from the introduction of a discreet math book, How to Prove it, idk where to even start with this, and figure that since it's part of the 1st question in the intro that I should know how to do it. But, alas.

It's based on a fairly common interview question but spiced up a bit.

It's based on this but I spiced it up a bit: https://medium.com/@shelvia1039/brain-teaser-2-tiger-and-sheep-9293acd97012

https://goatmatrix.net/c/MatrixFun/F71YGTSzZg

Project website of the guys who proofed the Einstein tile tiles the plane non-periodically.

Random thought on magic squares:

If I view the smallest possible non-trivial magic square

2 7 6
9 5 1
4 3 8

since its rows and diagnoals sum up to 2+5+8 = 2+7+6 = 4+5+6 = 2+9+4 = … = 15

in the article as a 3x3 Matrix, its determinant is Δ = -360 . Its inverse:

-37/360 19/180 23/360
17/90 1/45 -13/90
-7/360 -11/180 53/360

note how this is a magic square, rows and diagonals sum up to 1/15.

https://matrix.reshish.com/inverse.php

Now if you are really bored (I can not do this): proof that for any non trivial magic squares the inverse …

• exists (i.e. every non-trivial magic square has an inverse)
• is a magic square.

Two students who discovered a seemingly impossible proof to the Pythagorean theorem in 2022 have wowed the math community again with nine completely new solutions to the problem.

While still in high school, Ne'Kiya Jackson and Calcea Johnson from Louisiana used trigonometry to prove the 2,000-year-old Pythagorean theorem, which states that the sum of the squares of a right triangle's two shorter sides are equal to the square of the triangle's longest side (the hypotenuse). Mathematicians had long thought that using trigonometry to prove the theorem was unworkable, given that the fundamental formulas for trigonometry are based on the assumption that the theorem is true.

Jackson and Johnson came up with their "impossible" proof in answer to a bonus question in a school math contest. They presented their work at an American Mathematical Society meeting in 2023, but the proof hadn't been thoroughly scrutinized at that point. Now, a new paper published Monday (Oct. 28) in the journal American Mathematical Monthlyshows their solution held up to peer review. Not only that, but the two students also outlined nine more proofs to the Pythagorean theorem using trigonometry.

Calculator: https://www.omnicalculator.com/everyday-life/dilution-ratio

If I type in the dilution ratio and final volume it calculates the concentrate amount and water amount but I don't know how it does that and want to find out how it does that