Thanks for the shout-out! Where would you place the icosahedron? I think it deserves an S-tier spot since it has extreme importance to whether or not my DnD antics are successful
An icosahedron would be B Tier for me, since it can't tessellate euclidean 3 space, unlike the rhombic dodecahedron, but still very pretty! What about the brachistochrone?
I would definetly rank the double cone higher, since the conic sections are so fundamental. Just think of how beautifully and naturally the trajectories of celestial bodies are hidden in the cone...
Not to mention its application to optics. Primary, secondary and even tertiary reflectors in optical systems (using what we call "light" and "radio" waves) are all derived from conic sections.
I dont know jack about math but iirc the mandelbrot as well as other fractals, (idk about all fractals), can be defined as infinitely space filling curves which has some applications in logistics and packing. As well as something or other to do with chaos theory. The golden ratio can also be derived from it and as a result the fibonacci sequence.
@@6ic6ic6ic The term fractal is ill defined, but not that the mandelbrot is has an infinite boundary (perimeter) while keeping a finite area. It also contains copies of itself
"If mandelbrot isn’t in S we riot" Based, I completely agree with you. I am happy that the Mandelbrot Set was in S-tier, because it genuinely deserved to be in S-tier. She saved the best for last, because the Mandelbrot Set is the best math shape of all time in my opinion.
Math is big enough that this video could probably have gone on forever. That said, here are some more interesting shapes that happened to not appear in the video: The Klein Quartic: This "shape" is a surface with genus 3 that has the maximum amount of symmetry of it's genus. That is, it's another record setter. It has deep connections to many other areas of mathematics including hyperbolic geometry and Fermat's last theorem (apparently). The Fano Plane: This is the smallest projective plane. A projective plane is a set P of "points" and a set L of subsets of P which we call "lines", though these need not correspond to the usual points and lines of Euclidean geometry. Further, in a projective plane there is a line between any to points and a point shared by any two lines. The Fano plane has 7 points, 7 lines, and a huge amount of symmetry. In fact, the automorphism group of the Fano plane is isomorphic to the group of the Klein quartic. The 11-cell and 57-cell: These are exceptional abstract 4-polytopes. Recall that the 6 regular 4-polytopes have names like 8-cell (tesseract) and 16-cell. The 11- and 57-cell are like those, but they're built of hemi-icosahedra and hemi-dodecahedra rather than ordinary platonic solids. As a result these shapes can't be embedded into real space in the usual way, instead they live as "abstract polytopes": structures which keep track only of incidences between vertices, edges, faces and so on. The Paley Biplane & The Kummer Configuration: These two are structures called Biplanes, they're like projective planes except that there are two lines between each pair of points and two points shared by each pair of lines. Only 18 such structures are known, and these two are two nice examples. The Paley biplane has 11 points and 11 lines and has an automorphism group closely related to that of the 11-cell. The Kummer configuration is the most symmetric of the three 16-point planes which exist, and gets extra points for having a really nice description in terms of a 4x4 grid of points. The Petersen Graph: This is a very nice graph with 10 vertices and 15 edges. It is cubic, meaning each vertex has exactly 3 neighbors, and it is distance-transitive which is a symmetry condition meaning that if two pairs of vertices {a,b} and {c,d} have the same distance D(a,b)=D(c,d) then there's an automorphism of the graph with f(a)=c and f(b)=d. There are only 12 finite graphs with both these properties, including the skeletons of the tetrahedron, cube, and dodecahedron. In fact, this graph is the skeleton of the hemi-dodecahedron I mentioned earlier. The Leech Lattice: This is an exceptional 24-dimensional lattice of points (in Euclidean space, no worries). I personally don't entirely understand this structure, but it's worth noting for being deeply connected to the error-correcting "binary Golay code" and to the Conway groups which one may recognize as 3 of the 26 sporadic groups. The Pair of Pants: Like the mobius strip, the pair of pants is an important surface in topology. It also has a very funny name. Pants are used to study topological surfaces via "Pants decompositions", wherein one chops up a surface into pairs of pants to derive information about it. For example, the Klein Quartic above can be decomposed into 4 pairs of pants. Not sure how I'd rank these, but I thought they would be worth mentioning.
In a sense the sphere is the very opposite of a triangle- smooth versus spiky. In computer graphics, the more triangles you add to the tesselated approximation of a sphere, the smoother the result will be.
I recently found your channel, and I could not be more glad that I did. Your voice is so soothing and relaxing it's actually unbelieveable. You are my go to when I need somebody to accompany me during my studies! Thank you so much.
I liked the video, but there are a couple of shapes I would have liked to see: - The Meissner Tetrahedron is my favorite solid of constant width, and there's even a (little-known) symmetric version of it where all edges are rounded in the same way. - The Gömböc is one of my favorite shapes, because it is the answer to the long-standing question of whether there is convex mono-monostatic body with uniform density. - Also there are some pretty fascinating shapes for 3D-honeycombs, like the Triakis Truncated Tetrahedron or the two shapes of the Weaire-Phelan structure. Also I wouldn't call a soccer ball a sphere as you did in the beginning. Instead it's clearly a truncated icosahedron. ;)
Thank you so much for introducing me to my now favourite mathematical object of all time, the Buddhabrot. Not because it's such an interesting shape (which it definitely is, by the way). Oh no, what makes it special to me is the name, which forms an absolutely hilarious word play in German, and works best for people with a northern German accent. "Buddhabrot" sounds very similar to "Butterbrot", literally butter bread, meaning a slice of bread with butter spread on it, a staple food dock workers (and most other working-class people too) used to pack for their lunch breaks, and which every school child got in their lunch box at some point or another.
There was a program on our elementary school computers back in 1999 that would generate shapes after typing numbers and letters into it, never knew how it worked but I’d always type in random things and get pretty interesting shapes
Interesting point about the dodecahedron: its symmetry group (well, proper symmetry group, which doesn't include reflections) is A_5, the alternating group of degree 5, which is the smallest finite nonabelian simple group. For four dimensions, I'd advocate the 24-cell, which is a platonic hypersolid with no three-dimensional analogue (as opposed to, say, the tesseract, which is analogous to the cube).
I feel the double cone deserves to be B tier alongside the sphere. It's fundamental to a whole class of wildly useful curves -- the conic sections! The double cone when intersected with a plane can produce a point, a line, a circle, an ellipse, a parabola, or a hyperbola, just depending on the positioning of the plane! Cones also see use in a variety of industrial contexts, such as drills, funnels, speakers, ice cream service, traffic engineering, and so on.
Flex-a-hex-a-gon B teir, it's a cool thing you can make with what, 3+ sides? Kelin bottle, which is a 3-d representation of a 4-d shape that doesn't self intersect but I don't remember why it's special. Thinking about it, it kinda seems like a container that would go around a mobious strip but not quite.
I am putting in for the humble triangle. The only 2-D shape for which SSS guarantees congruence (SSSS does not for a quadrilateral), its several centres and the existence of teh Euler line, its historical importance in map making (triangulation), and we wouldn't have sine or cosine without right triangles. This gives you the whole of Fourier Analysis and theory of functions, and remember that this was used in the mechanism where 12-or-so real telescopes were used to make a virtual telescope the size of planet Earth, which in turn gave us the first "photograph" of a black hole!
Nice video Toby, thanks. And great that you included the Mandelbrot set. I have been trialing Brilliant recently (introduced to me by another channel), and was a bit disappointed with some of the material in the introductory "Scientific Thinking" course. For my online learning I will be sticking with other streams such as MIT's Open Courseware offerings, 3Blue1Brown, etc.
I've been watching way too many videos from PBS Spacetime in the last couple days, so I'd like to make a minor correction to the bit about Flamm's paraboloid. The event horizon is the boundary where the escape velocity of the black hole is equal to the speed of light, not where spacetime stops working. Where that idea comes from is the view of an observer, which is what Einstein's models: the path of an object in reference to the oberver. As you get closer to the event horizon, the gravitational pull on light becomes closer and closer to its velocity, so its net outward velocity goes to 0 as it reaches the event horizon. Because of that, as an observer sees it, the time light takes to come back as it gets closer to the black hole goes to infinity. And so, to an observer, space seems to break (or rather, freeze) as a viewed object or light gets closer the event horizon. To the light or object falling in, space is perfectly fine (well, as fine as crushing density and insane gravitational forces can be I suppose), it only gets broken for the viewer. But space really does break at the point of the singularity. Singularities have infinite density so it infinitely warps spacetime around it and makes itself infinitly are away from normal space, so whatever falls into the singularity (from it's view) will never end up actually getting to the singularity (which is also why Einstein-Rosen bridges arent traversable). Or I'm pretty sure, at least; I'm not a physicist. Gotta love general relativity and how needy it is about the reference point lol
As you say; for an object falling into a black hole, space is perfectly fine apart from the crushing density and being spaghettified by the insane gravitational forces.
A plasmid expert in the biology department at my university was describing a problem with either replication or recombination of plasmids (circular DNA) which resulted in a Möbius strip topology that has to be resolved or avoided. I think it makes sense the topology would show up in the interaction between two circular pieces of DNA. It was a neat real world example of the Möbius strip
Interesting Video Toby,. When very Young I was interested in these fractals, thinking they were part of a simple computer graphics technique. They were very fashionable and Mandelbrot is perhaps involved in high Art and high mathematics and data science.
The solutions to Newtons methods eg on f(z)=z^{3}-1 are like the Mandelbrot but obviously what is going on tells us some very deep things about convergence of that method. Calabi-Yau manifolds for their association with string theory and methods to fold up compactified dimensions. Lorenz's strange attractor.
Aww you’re such a sweetey soft spoken lovely doll 🌟 When we were young we used to make mœbius strips of paper. I remember one day I cut one along the line you drew through its middle and it produced two shapes caught in a chainlike formation..
The hyperbolic paraboloid shape is a good one. That's the shape that Pringles chips look like. The guy who invented Pringles had the task of making a potato chip type snack that didn't have all kinds of broken up chips in the bag, and he succeeded.
Thank you very much, as always. My favourite is the Klein Bottle. Maybe due to my vast ignorance of maths. And the beauty of simplicity. K.I.S.S.. They never taught us topology at the Uni, sigh. Even did not mention.
Check out 3Blue1Brown's channel - he does a *lot* of visualising to help promote self-discovery of mathematical principles. Some of his videos have changed the way I think about certain mathematical concepts I have been using in my work for years.
Nice and very interesting. My friends fractals (2D, 3D, 4D, etc) have one question. How to write a complex object / drawing into a single mathematical pattern(for ex. z = z2 + x2 + c) ? Well thank you.
There are spaces with one “hole” that aren’t topologically equivalent. However, every two compact orientable surfaces without boundary of the same genus are topologically equivalent.
As an engineer usefulness weights so much in any decision it was kinda funny to see things going tiers so much higher or lower than I thought they would.
Please rank the hyperbolic surfaces called the Bolza surface, the Klein quartic and the Bring's curve! These hold unbeatable records for the number of symmetries of compact hyperbolic (orientable) surfaces of genus 2, 3 and 4 respectively. The Klein quartic especially holds the largest theoretically possible symmetry group for its genus (theoretical max is 84*(g-1) where g is the genus, although this is only attainable for some genuses, not all, even though there are infinitely many where it works). These surfaces may also be tiled accordingly to their symmetries, in order to become hyperbolic Platonic solids!! Bolza surface consists of 6 regular octagons, three at each (regular) vertex, or dually of 16 regular triangles, eight at each vertex, for a symmetry number of 48 = 48*(2-1). Klein quartic surface consists of 24 regular heptagons, three at each vertex, or dually of 56 regular triangles, seven at each vertex, for a symmetry number of 168 = 84*(3-1). Bring's curve (really a surface despite the name) consists of 24 regular square-angled pentagons, four at each vertex, or dually 30 regular non-square tetragons, four at each vertex, for a symmetry number of 120 = 40*(4-1). The weirdness of the angles is similar in Spherical geometry were the usual (spherical) Platonic solids actually live, thus e.g. the cube consists of non-square tetragons with angles of 120 degrees. And there are dihedra and hosohedra that are also legitimate Platonic solids, in the Spherical plane. It is probably possible to deform hyperbolic solids into shapes in 3d Hyperbolic space so that they get sharp edges and corners and non-constant curvature, so their faces readjust to their usual Euclidean angles, similarly to how this is possible for spherical solids in 3d Euclidean space. Also genus 1 surfaces may be tiled so they become Euclidean Platonic surfaces, like e.g. 5 squares, four around each vertex, 7 hexagons, three around each vertex, or 14 triangles, six around each vertex. There is no maximum for symmetries of compact (Euclidean) genus 1 surfaces though, they may have any cyclic group or direct product of two cyclic groups as symmetries. Also there are non-orientable compact tilings/Platonic solids, like the hemicube, hemioctahedron, hemidodecahedron, hemiicosahedron, hemi-Bolza, hemi-Klein, hemi-Bring (either of the pair of dual solids of each genus-symmetry class), hemi-square-torus, hemi-hexagonal torus etc (the last two being regular tilings of Klein bottles). Each non-orientable solid/tiling has half the number of faces, edges and vertices of the corresponding orientable solid/tiling, and not every orientable tiling can be folded into a non-orientable one. In Hyperbolic space there are also non-compact surfaces that are heavily curved and may be regarded as tilable "spheres", like the pseudo-Euclidean horospheres and the pseudo-Hyperbolic equidistant surfaces/"hyperspheres". These look somewhat similar to the compact pseudo-Spherical spheres, that also exist in Euclidean space. In Spherical space, only the pseudo-Spherical spheres exist, plus the flat planes are Spherical of course.
With respect to the trefoil knot, I would like to point out its popularity in cultural history (I know this is no direct application in science/engineering, but apparently, people the world over have felt quite intrigued by it😃).
Nice ranking! I would place sphere in A because of its overwhelming symmetry and the cone in S because it is simply too versatile (physically speaking, I think it is much much more important than the paraboloid), not to mention its algebraic importance. On the other hand, the paraboloid is not so gigantically meaningful in my opinion, although I agree that the concept of event horizon is. But then again, the concept of event horizon has to do more with light-cones than with paraboloids, so another point in favor of cones in my opinion. So I'd say cone S, paraboloid C. At the end of the day, parabolas are just intersections of a plane on a cone :P
I think the sphere should be higher, since it in many ways is the most stable shape. Especially when you look at it from an astronomical or cosmological point of view.
Hi all, Thanks for an informative video.Various ideas of knot theory have been used recently in multipartite entanglement studies in quantum information e. g, there is a nice connection between Borromean rings and GHZ state (entangled state of 3 qubits). Thanks
