Robert Hazen is on the right track. Function is sequential and when we examine any system, there are specific functions in a specific sequence. This is very old information in the study of time. There are no new patterns, there are changes in the way mathematicians are thinking.
You don't want to say they 'discovered' it because it's been there all along? Isn't that exactly why we use the word discovered, rather than 'invented'?
"Dis" meaning to remove or negate. "Cover" meaning to conceal, hide, or obfuscate. Discover literally means finding something that is already there, but you didn't previously know about.
The thing is it is not “hiding”, and the term we use that she should have used is that it was “hiding in PLAIN SIGHT”.😊. So nothing uncovered just “unnoticed”.
It's a tricky nomenclature. You can discover a new species that was there all along but unbeknownst to us, however you don't discover a fractal shape of a fern, you discover an underlying mathematical relation or a mapping that generates such shapes. I guess inventing applies to none of the above.
I think Sabine must accept that a lot of nerds watch her videos. Since nerds often have mental problems that make it difficult for them to perceive anything but the most blunt humor(usually jokes that a 10 year old would get, and of course, puns)...and makes them very argumentative and pedantic about the most irrelevant of things... I hope she will be understanding about her nerd viewers. Further, I hope she doesn't argue with any of her nerd viewers about how irrelevant their pedantic views are because they will furiously defend their irrelevant viewpoint... because honest discussion and growth doesn't interest them. Winning an argument is one of the most important goals in life to a nerd.
"Using rounded corners might reduce pressure and increase stability or flexibility" - that's the cause and effect backwards. Cells start off soft, so they are flexible, and the pressure simply leads to rounded corners, because everything's trying to become a circle or a sphere.
Right! It is a combination of surface area reduction and space-filling. Cavitation causes structural and immunological issues in biological systems. The sharp corners most likely originate from where the cell started. Imagine a bunch of steel cables from one side of the cell to the other. This is how the structural proteins often organize. If they originate from the same location, because there was something already there that they could anchor to, that will be the corner. Then it will form these shapes by default.
But what about plant structures having these same shapes? Aren’t they supposed to be more rigid due to having a cell wall? I think the fact that it occurs so often is less because things deform to become curved and more because it’s the best course otherwise I’d suspect nature would make things rigid enough to not deform. Also the fact that they do deform might be “intentional” as in they are supposed to deform to the most optimal shape for their environment after being made. Perhaps too much rigidity would also limit the plant or whatever in terms of movement and nutrient transportation so it could also be a balance of structure and movement, like how we want buildings to sway in the wind. I’m not a biologist, but I do think these mathematicians have a point when studying these shapes, I’d like to see them compared to rigid shapes in some simulations in the future.
In nature these types of geometries have been extensively studied, see "space-filling fractals." The novelty here seems to be that the authors define a mapping between a polyhedron packing/tiling and one in which there are no gaps (i.e., the soft cell equivalents). The paper is interesting, but not a paradigm shifting breakthrough.
Interesting. I learned from an apiarist that bees build their honeycomb cells in cylindrical shapes but the contact between adjacent cells and the malleability of wax allows them to make straight lines at their interfaces (like bubbles do) and tesselate into hexagonal shapes. Surely this is a universal phenomenon at both macro and microscopic scales depending on the substances and the forces between them + gravity, hydrogen bonding, etc.
And the cells on either side of the honeycomb are offset from each other and meet at the tetrahedral angle of 109.47 degrees (tetrahedron - one of the Platonic solids), which also appears in trabecula bone - geometric patterns and shapes don't just appear out of nowhere.
Agreed - Copied from above and pasted here I claim it is only a consequence of geodesics associated within growth constraints within a growing organism. As an aside nestings of nestings of ... of nested things is really a wonderful concept than humanity also needs to consider. Perhaps sooner rather than later. It seems to help with quanta as they definitely seem constrained by their geodesics. Anyway ... If we take growing organisms as crystals forming in suitable solution with suitable external influences you will see, not surprising really, crystals taking a crystalline structure. This indicates a "standard" geodesic has been followed leading to "standard" recognizable shapes shaped or imagines in 2d or 3d, and if our thinkspace allowed it, 4d and higher dimensions. Such are the wonders and limitations of humanity no? What is surprizing is that similarity of processes in organic and inorganic things has not been identified earlier and merely speculated in these Comments sections in wonderful RU-vid channels such as Sabine's
Personally I think that fruit and vegetables are round due to the way they are created rather than the optimization of shape. When something grows from the center, normal distribution shows that it will grow out evenly from all directions when the cells split rather than the fruit being sentient and picking the most optimal shape.
The fruit is obviously not sentient but if it was worthwhile to use a different shape evolution could have found it. Like the whole honey comb thing. Sure we could argue that it's completely coincidental but alternatively we could also argue that evolution found a local optimum where hex shaped honey combs are useful.
It is good that identification of growth patterns between organic and inorganic things start to appreciate governing factors linked to geodesics of their differential relations of growth factors. Add a "decay disease" parameter into the differential form and Bingo! a new shape has been decsribed. Besides I always thought that rounded corner rectangles looked sorta organic and taking rounded corner rectangles with/without skew seems to turn seedlike shapes into crystal-type shapes apparently linking organic and inorganic growth patterns
That video of Richard Feynman discussing how he as a scientist appreciates beauty so much is really prescient. I appreciated the beauty of nature so much I went from physics into nature photography, and still consider myself to be more artist than scientist, but both have always been so intriguing to me!
These are Gielis shapes. They are made with the Gielis formula, previously known as the super formula. The formula has been tested on over 60,000 organic shapes. It was discovered by Johan Gielis and is used in various fields such as economics, biology, ecology, and physics. There are some really cool 3D modelers out there online.. So I would say these are Gielis paterns. Really cool
@sabinehossenfelder I think you forgot to state the Einstein shape is the 1st shape that can fill a plane "without repeating a pattern" You're gonna find your comment section filled with people pointing out hexagons & squares(& probably some strange shape ive never seen) can fill planes without gaps.
A fundamental of mechanical engineering design is avoiding excessive stress concentration which occurs in sharp corners, and can result in failure under cyclic loading.
I felt like, in this vid, we were being mocked a bit. Rounded off shapes being more stable under pressure, is very well known and widely used. And now we even "discovered" what skittles and M&M chocolate knew all along and you could have googled: the rounded shapes with the sharp edges, fill the bags better because they don't leave gaps between each other. 🤡
Very interesting. It's worth pointing out, tiling is considered a branch of math. Why? Because it's based on the question, how can you cover a given finite space with one congruent tile. That is, a tile that can be rotated and changed to form congruent shapes. In graduate school (long ago, unfortunately) I did attend a few seminars on tiling and do occasionally read papers on it. The reason that this "new shape" is seemingly so significant is because finding how congruent shapes fit together, geometrically, in, say, two-dimensional space is a complex problem. A new shape that can be used for many congruent shapes and that can "fit" algebraically in many complex, two-dimensional spaces would help to solve, or at least, understand how tiling problems are solved. (Btw, because I'm an academic, I downloaded the paper and going to read it when I have time. So, what I've stated is based just on a skim reading of the paper and not on Sabine's video.)
Really instructive graphics on your examples.I think you omitted the important property of the Einstein tile: aperiodicity. For us non german speakers you might point out it isn't related to work of your nodding friend Albert.
Indeed, Einstein in this context just means (German writing) Ein-Stein. Wich means One-Stone, or in this context One-Tile, wich is the official name for the riddle only use one 2D pattern in multiple arrangement without producing repetitive Super-patterns. You could have done that explanation as well.
Corners are high stress points that cause a lot of fracturing and when the fracturing occurs most of the time it's just the corner that falls off leaving a rounded edge. Why build something that's going to fall off? Have you noticed that this is the shape of teeth?
Not sure who you mean with "those people" and there are more then enough paper (or if you prefer even videos) out there that describe where in nature it is showing up (from flowers to growing processes to quasi crystals).
@@Techmagus76 Yes, he's talking about the people who see it _everywhere_. Look up "golden spiral" on an image search. You'll see plenty of logarithmic spirals.
It's due to π's geometrical relationship to its containing square of 1. Take 4r² = 1. Remove A = πr² = π/4 such that 1 - A = 4њr² = њ for r = 1/2. The "soft cell" region(s) њ "nye" is all area outside the circle but inside 1, a natural soft cell(s). This region њ is also represented inside π as a soft cell surrounded by 2(π/4 - 1/2): 2(π/4 - 1/2) = π/4 - њ π/2 - 1 = π/4 - њ π/2 - 1 + њ = π/4 π/2 - 1 + 2њ = 1 π/2 + 2њ = 2 π + 4њ = 4 π = 4 - 4њ etc. The soft cell maximizes efficiency & relates to L² ≥ 4πA with equality holding only for perfect circles.
I think the advantage of softened corners might be relatively straight forward: stress lines tend to concentrate on inside corners. If you can eliminate an inside corner, you strengthen the material. But if you eliminate an inside corner, thus creating an inside curve, you need an OUTSIDE curve to fill in the gap.
Macchu Picchu comes to mind immediately, there are few if any straight edges or corners, everything is rounded and shaped to fit perfectly in the space although the blocks are all different sizes. Does this make it stronger and is this why it's so well preserved?
Many great scientists were also philosophers and artists at heart, and many others have spoken about the necessity of teaching science like arts. Groundbreaking science is about beauty, intuition and creativity too.
Sabine, in the past you've often looked at stories about data integrity and honesty in the science community. So I was wondering if you'd investigate the doctor who's been accused of faking literally decades worth of Alzheimer's research data. My mother died in 2019 with Alzheimer's, and if true this makes me so angry. It's a colossal scandal that should be exposed.
Tiles with hard and round edges have been used for a long time. Florentine pattern was very much in fashion in the 60th for making jewellery and in the 70th as ceramic tiles for kitchens and bathrooms.
0:36 "The first single shape that can fill a plane without leaving gaps" w... WHAT?? uhh... What about hexagons? What about, you know, squares? If you wanna allow for rotations or mirrorring, triangles?
I think it was an unintended mistake in her script. And, to be even more precise, that was the first known shape capable of tiling the plane ONLY in a non-periodic way, that is, there is no periodic tiling using this shape.
The British discovered this with their windows in airliners at altitude. The Square windows cracked, with dire consequences after repeated exposure to pressure. By the time the solution was found, rounded corners, fatal damage had been done to confidence in the British aerospace industry.
Honey combs arent really hexagons. Its a bunch of circles. The leftover spaces give us the illusion of hexes nestled together. The giants causeway is actual hexagons.
Any knife smith can tell you corners create stress points. Same goes for Castle construction and 3d printing. Rounding off corners with a circular radius will improve your structural integrity.
As a graduate student at Princeton in the 1980s, I had the good fortune to run into Eugene Wigner in the Jadwin library. He was sitting at a table near where the University had set up a display with his Nobel Prize. I was going to approach him but I noticed he was writing a letter. It was to a local car dealer with whom he was not satisfied about the quality of his purchase.
It's the first single shape that can fill a plane without leaving gaps, *but can do so only non-periodically*. Omitting the latter part was an obvious slip of the tongue.
This really is nothing new. Fundamental forces like fluid dynamics and surface tension easily explain how these kinds of forms arise by following the path of least resistance. A primordial fluid containing amino acids and proteins, when agitated, will form into a foamy structure of protobiological cells. As biological structures grow, they generally follow the proportions of the Fibonacci sequence which allows room for branching forms to stay out of each other's way. This gives rise to the familiar spiral patterns that we often see in nature.
RE "feeling that the question of just why that is so will still be discussed in 1000 years, hopefully in the interstellar metaverse, in a virtual room that has only soft edges." so fun.
… The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/)[1][2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers
But it's not likely to be previously unseen or unknown. If I find a waterfall in the forest that was previously unknown to me, sure I discovered it for myself, but to say I was the first would likely be wrong. So it's a discovery in one way but not another, which I assume is why she is hesitant to say discovered.
@@Ivan.Wright Yeah, if I walk into Walmart for the first time have I discovered it? I really don't understand what the "discovery" is. That these shapes exist? Because everybody knows. And it's just not clear to me what new mathematical understanding here is or any other reason to think anything has changed.
@@beatrixwickson8477 You have discovered it if you went to it on your own. Even if you knew what a Walmart was but you found one in a location that you didn't previously know about. The word in that context has the definition something like "The act of adding new information to your knowledge base". I think the ambiguity has to do with scale from an individual to all individuals. In between the limits there are communities. I'd say the definition of a community in that sense would be a function of unique knowledge compared to other communities and the ability to transfer that information. Some information is shared between communities but most of the "information flux" is within the boundaries of the community. All that to lay the groundwork to say; within the scientific community it may be considered a discovery but in actuality it's just a formalized description (which is arguably the abstracted idea of what a discovery is but with the benefit of removing the prescription of foundership or ultimate originator). That's just my attempt to say what I'm seeing though, I hope it makes enough sense.
Geometry as most people know means earth measure. The Ancient Greeks were obsessed with it which is why school children today still learn about Euclidean Geometry. Since the structure of matter depends upon molecules and atoms , there seems to be little doubt that there is some fundamental order to the universe. I would take issue with a couple of points Sabine made, but these are not to contravene the overall thesis. Sabine says “they didn’t discover anything, it was already there “. Discover is to find something which it sounds like this is exactly what they did. They may have invented a pattern or equation, but they discovered a match in nature. The second thing I noticed is the example of three dimensional patterns with no hard corners does appear to have hard corners, at least from the illustration. They look like mail boxes stacked on top of each corner curved on top and bottom but meeting at their outside corners in the vertical plane. I may be wrong on this point, but it would be helpful if the illustration were animated and rotated to confirm or refute this. As one who doesn’t believe in a higher being or a creator, I like science as a more objective explanation of how the universe works. But I also am in awe of this experience we call consciousness and the paths of speculation it can take us down. One thing that fascinates me is the human obsession with right angles. This is the most pervasive shape in our world from our street layouts to home construction or the shape of documents and furniture, and yet there are very few right angles in nature. The only thing that comes to mind is the salt crystal, but I think it’s highly unlikely that this was the inspiration for its pervasiveness in our world.
I think that if you zoom out and examine human society, you see a growth distribution pattern that is equivalent to the growth distribution of viruses. Whether a natural resource for humans, or nutritional resource for a virus, the growth looks identical.
All of these shapes can be described using fractal geometry. The scientific community should immediately revisit the work of Benoit Mandelbrot to spark the next scientific revolution. I don’t understand why it takes so long.
“The same energy that drives a galaxy through space makes small man’s heart beat. “ That’s a quote , as best as I can remember it, from a ‘Perfect of Saturn’ .. an incredibly advanced extra terrestrial intelligence, living on.. Saturn. The Aetherius society has the full audio recording from his transmission.
YES - as above so below- perhaps in the depth the "in-betweens" behind the overlaps "holes" interconnect the scaffolding of the 3 dimensional layers spinning our expansion concealing mass as reflected into a flat plane of perception...
The inclusion of the zebra pattern is, I think, a clue. We actually already have characterized systems that give rise to those exact patterns. Alan Turing did work on this. They arise from reaction-diffusion systems. See his paper from 1952 titled "Chemical Basis of Morphogenesis". The patterns produced are called Turing patterns. I would say, however, Sabine, that we have hard proof that the reason mathematics is not as useful outside of the natural sciences and applied to emergent systems is absolutely not because we haven't found the right maths to describe them yet. We know that they are intractable. We know, and can prove in the most mathematically rigorous way, that it is impossible not just for our mathematics but ANY mathematics to address these sorts of things with the same rigor applied to linear systems. There are fundamental limits to computation in the universe, there are chaotic dynamics in even very simple nonlinear systems, and being "close" doesn't exist. Your error term will always grow faster than any computable function, which makes it indistinct from nonsense. My suspicion is that we will have to develop complex systems ourselves to simulate/predict such nonlinear systems which are themselves mathematically intractable. Mathematics might try to lay claim to that, but it'll really have to be something philosophically separate.
From WC USA, Building houses out of cement blocks are a lot stronger, like they do in Germany, and some in Florida, like one we had when we lived there, and they do hold up a lot better to Hurricanes and Tornadoes than houses built with 2 by 4’s!
The irony here, Sabine, is nature uses shapes to remove excess space, yet atoms, something everything is composed of, are very little more than empty space!
Nature prefers the simplest patterns (e.g. shapes that can combine with no gaps), or the patterns that require fewest rules (i.e. Occam's razor). To suggest that this happens by "pure chance" is to multiply explanations (because nature is driven not by "pure chance" but by the constraints of the various systems that it embodies). The onus is on those that multiply explanations to provide a proper alternative hypothesis...
So if you make a shape "squishy" you can cram it into a space? The fact that they show up in things that grow is may be different since those shapes we see are made up of other shapes and grow to fill in those spaces, they are not made up of those shapes like building a brick wall. I likely just don't understand enough about it, but it seems like trying to explain something a little too hard. Will look into it more so I can understand what I'm missing...if I am.
Very nice. At 4:50: I prefer to appreciate natural structures as elucidating the interactions of processes, not as governors of processes. And, for me, to say that a natural process is meaningless just means that I lack enough understanding of the process to make any meanings.
I've painted this pattern a thousand times because it's everywhere. Maybe, because it is so common, everyone thought it had already been studied before.
Re 'the social sciences,' perhaps Sabine, you could connect this paper with the one on patterns characterizing perceptual function (the one against Musk's brain implant for sight). Thanks, as always 👍
The "social sciences" are not even "scientific" at all as they're not based on material analysis. Neoclassical economics for example is better described as an idealist philosophy rather than a science as it treats the economic system as strongly emergent from the free will of human ideas, and describes the laws that govern the movement of those ideas as irreducible and incapable of being influenced by anything else. Those laws also were not even discovered through experiment, as if the foundations of neoclassical economics was formed through intense psychological examination to figure out how human ideas change over time, but rather were just invented in someone's armchair one day. Too many people view math and science to be equivalent, so if you fill up a textbook with mathematics it must be science, but the problem is in neoclassical economics the mathematics is based on non-scientific foundations so nothing derived from it has any real-world meaning. This is a general problem throughout the social sciences.
the elongated s-shape that seems to be almost everywhere in nature and beyond is what fascinates me. always felt if string theory is a thing the strings wouldn't be closed but more like a stretched-out S.
The simplest way to ensure your room tiles fit despite being weirdly shaped is to fix square tiles and hammer them lightly till they crack. Viola you have a perfect fit of weirdly shaped tiles😅
Years ago I was intrigued by how nature's pattern is the rule of the universe. Sometimes the answers are so simple and so obvious and right in front of you . Just a thought.
See also: A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science A book that helped this artist appreciate math for what it actually does; describe the universe. Rather than: Make my head feel like it's full of straw and cotton wool.
This is interesting because it has actual implications with meaning. I was half worried this would be one of those "patterns in nature" ideas where someone gets excited because they found a magic number in like seven things, completely ignoring the fact that there is an infinite number of things to notice, and an infinite number of things to notice about those things, and finding seven instances in a set that includes two orders of infinity of possibilities means precisely nothing. I guess, as always, I should have had more faith in Sabine
When a phenomenon or property emerges by complexity from a lower level there are subtle differences in the firmness of the property which allow predictions about the more foundational level. This nonrandomness coupled with construction over time gives rise to irrationally correct answers in neural networks and unexpected successes in evolution. This nonrandomness is reenforced by emergent properties impacting other nonfoundational properties. For example, no one can think hard enough to change spacetime but they can think themselves emotional enough to impact their own memory or cause physical illness.
That's a good one. If I'd known nobody had every tried to curve corners of tiles and still keep them tile-able I'd have codified it!? DOHHHH!.. so simple. I've looked into tiling quite a lot so no real excuses, other than being more concerned about 'number of edges' than anything else.
Isn't this just a result of how these structures grow so the tiles can fill up a plane that is expanding with them? So it wouldn't be that nature has preselected arrangement of tiles that magically works out. But rather as the tiles grow they are pushing and expanding on surrounding tiles that causes them to take on these patterns.
Perhaps when the mathematics of soft edges is perfected it can be applied to remediate infinity results that are the bane of theoretical physics? Perhaps the infinities of black hole singularity can be remediated with some soft edge mathematics?
"The problems with beauty in science arise once we start using it as a guide rather than a motivation". I made a note of this, whch I find very true also outside science
Y'all know that bubbles of the same size as foam in a volume would make one of such shapes naturally, right? One drop of soap in a percolating water device and Bob's your uncle
The eye is attracted to circular and radial compositions naturally, behavior and pattern recognition. Beyond that, look at the shape of eyes in nature and reproductive organs these are everywhere you look.
