Prof Norman, on slide 8, I saw the Golden Ratio. I was exited to see the Golden Ratio. I sense that you are not exited to see the Golden Ratio. It pops up in many places. See Wikipedia.
He does not accept the "real" number system as it stands, so I suspect is less thrilled (and its appearance in the icosahedron is welll-known). However (!), I think it's important to note when familiar irrationals appear -- hinting at something deeper within, really. I believe he's in that line of thinking, too.
@@Cid2065 Myself, I am not qualified to discuss "real" numbers. As a software engineer, I have never multiplied real numbers. I view them as mysterious objects that are beyond our capability to understand. To me, the Golden Ratio is a mysterious and interesting object, even beautiful. Leonardo Da Vinci used the Golden Ratio in his artwork. It also appears in nature, in the spiral aloe as well as in the sunflower.
Professor, I've been watching your videos for years (I found your channel in high school) and have learned so much invaluable information. Now with this series I'm getting the sense that I will never run out of things to learn... I am SO grateful. Thank you for devoting so much of your time to education. You are an inspiration and a blessing.
I just want to break down with tears of joy. Your TH-cam channel is exactly what I need at this exact moment in my life. Thank you, thank you for making and sharing your videos with us. :)
It's a pleasure to seeing you well and back to your great work on one of the most smart ways of looking at mathematics. Thank you dear professor Wildberger.
That was a wonderful and insightful presentation. I’m a physicist and can see how such structures should be more widely known about and explored. Thanks so much!
This is, for me, a wonderfully clear exposition that seems to be aimed just at my level of understanding of groups, graphs and polyhedra. Somehow this talk speaks directly into my brain. That is rare and I really appreciate it.
Video Contents 0:00 Intro 0:07 Overview 0:38 Two Types of Symmetries 2:30 Though Experiment: Gas Diffusion 5:10 History of the Subject 9:03 An Initial Example 16:33 The Multiset of Neighbors 23:43 Fusion Rule Algebra 31:00 The Probabilistic Variant 38:34 Character Table of an Alternating Group 43:17 Character Table of a Hypergroup
It's such a happy coincidence that I have been working with representation theory and knot invariants now and there's a lot of fusion algebra - maybe not a coincidence now I watched your video.
Be aware that the diffusion of a gas is more complicated than depicted here. In a gas the velocities follow the Maxwell-Boltzmann distribution, implying fast and slow molecules traveling in all directions. The wave front blurs so to speak on its way to the boundary of the box, and the shape quickly randomize due to the elastic collision of the gas molecules.The bouncing at the wall will not be perfect symmetrical due to the surface roughness and the interaction between gas and wall molecules. The shape of the boundaries is therefore not important. The position of the molecules is always random, which gives no symmetry. The mathematical picture here is more that of tennis balls travelling with the same speed from a hypothetical center. In that case radial symmetry remains until the balls hit the wall and bounce perfectly back. In that case symmetry and shape between the balls remains and changes to a new symmetry and shape upon bouncing.
"Randomness" is the mathematically the least coherent and interesting interpretation of the phenomenon in question. Phenomenologies of the external senses and internal sense of intuitive pure geometry are different categories, and their interrelations is a fascinating topic of research. Bohm started from studying Brownian motion, and ended up with theory of holistic causation called 'Ontological interpretation' by Bohm and Hiley. From geometric point of view, the process ontology of Holomovement and active information of quantum potential implicate that platonic geometric forms can be constructed as and exist only as interactive animations - which is what also Euclid is actually saying. Thus, holographic informing from whole holomovement to parts is not ontologically random, but it implies that "snapshots" of the constructive continuously animated computing is mostly beyond our epistemic limitations, and thus mathematically not very meaningful. Especially when our mathematical foundations generally start from the bottom-up direction of constructing instead of the top-down. So the phenomenal "variation" from our nth logarithmic perspective of the animated pure geometry of holomovent appears as fractional "degrees of freedom" compared to platonic circles and solids etc. Quadratics, as mentioned by Wildberger, offer periodic "constants of movement" that can be associated with the form preserving scale independent property of the Bohmian quantum potential in which the "probability amplitude" is both in the numerator and denominater, and can thus be given numerical anatomy of 1/1, 2/2, 3/3 etc. on the simplest level of "unitary" fractions.
Wow that was really nice! I would love to see more on diffusion symmetry and how it might relate to ohysics I'm thinking about Feaynman graphs for example, but who knows, many unforseen applications aswell!) Anyway, I will check out more of yours stuff :)
Diffusion Symmetry will give huge possibilities to make new decissions for biomedicine, for diagnosis and treatment COVID, for correction personal psychological, emotional pecularities.
APL (A Programming Language by fellow Canadian Ken Iverson) would be ideal for expressing problems of this nature: unlike most modern languages, APL is array-based, as in arrays are a fundamental type, like integers being a fundamental data type in, say, C. The result is that operations over all elements of an array (n-dimensional, by the way, only limited by system memory) either element-wise, or cumulatively are at the heart of the language. Basically, working with graphs, poly-numbers, or vectors etc. of any size is what APL does. Also the syntax is extremely concise, based highly on mathematical notation. Check out Connor Huykstras videos (another Canadian) for some nice examples comparing APL solutions to more common languages. APL seems like the perfect tool for this approach. Can't wait for the next video!
I think that Matlab took this space. What mainly killed APL was the need for special keyboard because it used a bunch of “special” characters to represent matrix operations.
Love the phrase "Diffusion Symmetry" The 26 + 1 Finite Non-Abelian Simple (non-parametric) Sporadic Groups and 17 Finite Non-Abelian Simple Lie Group Families (one or two parameteric) plus the Prime Group Family (obviously Finite Abelian) as a Framework for understanding more properties of this "diffusion symmetry" -- eg, Gyrosymmetry, Axisymmetry. Better understanding the missing Lithium 7 problem, the fine structure constant, proton/electron mass ratio. Seen all of your public videos (except the Go stuff, gave up on Go in the 70's)
Interesting use of multisets. I wonder what other uses there are for multisets. May be a powerful approach in computer science! For example in graph theory I now found that multisets are used.
@Anders As we proceed we'll see that multisets are flexible and powerful data structures that are particularly suited for studying group theory. Set theory is just too limiting, it turns out.
To create group create a multiplication table (what a Group is) In higher dimensions understanding groups is a tough, hard to get ones head around Character table by forbeneus: Permutations values of characters(or rows), evaluated on Conjugacy classes (character theory) Alternatipn group of 5 things has cycle notation Character table Useful computation Device to understand Sqrt cannot compute to an infinite precision Look at an Infinitesimal with skeptecism! Diffusion view Look at hypergroup object little c probabilities 5-1 =4 objects of different circles around point It's commutative harmonic analysis so definitons are simpler than in group theory case . Thus this mimics process of ccommutitive geoups -r5 quadratic extension function (of rational number)(to build from rational numbers) Can deal with as matricies (watch Dihedrons complex number on quadratic extensions) Q. How you deal with complex numbers but at the same time only able to acknowledge rationals? Isn't the gap between those 2 already huge to be compared with? Finally r5 squared=5
I like the idea of finding new ways of looking at symmetries. However, to compare this particular graph with an icosahedron? It seems to me that an icosahedron is a very atypical example of this a graph. A graph doesn't say anything about the separation of the nodes nor of the dimension of the vertices nor whether the points lie in a plane or in some other space. However, there is a lot of symmetry about. So what is it and how can it be used, I am curious about what Norman has discovered.
With some unusual notation, your C's are morally equivalent to matrix elements of powers of the graph adjacency matrix, and the diffusion operators to normalized adjacency matrix, or 'graph laplacian' used heavily in probability theory and physics. There are mistakes in your calculations--- the one-step followed by three step motion on the graph does not have to lie at distance 2, it can lie at distance 1, 2 or 3, because you can walk around a closed triangle to come back where you started, so there is no parity, but I'm ignoring the calculation errors. The graph laplacian eigenvalues determine the deviations from 1 in your diffusion process, and these diffusion processes do show up in physics a lot, for example, Polyakov uses continuous diffusion processess at small times and large times to give a physics proof of the Atiyah-Singer index theorem.
I'm following you since years on many of your playlists, and I really love all your lectures, especially the ones related to WildTrig, which lead me to remove entirely cos/sin from my code (I'm a Math developer in my daily job). Currently I'm a Math addict devoted to the study on Mathieu groups and the Monster, passing through the MOG & Co. In studying that, I faced many times the ADE classification, trying to attack it from many POVs, like Coxeter groups, Weyl groups, etc but still struggling in getting a unified vision of it. I saw also some lectures from yours about ADE, which were quite enlighting. It would be really interesting if you'd be so cool to create a dedicated spin off about ADE theory, using may be some new "cool" approach" on that as you did here for the diffusion symmetry. Is there any hope to get something similar in future? Tx for all your efforts, Prof. Wildberger.
You are in luck! At my Wild Egg Maths channel, I have a series exactly on a novel approach to ADE graphs. The series is called Dynamics on Graphs, and gives an approach to ADE graphs via two remarkable games on graphs. Currently there are about 10 videos, but I intend on adding many more, as it is a very rich subject. Here is a link to the Playlist, which however is available only to Members of that channel:th-cam.com/play/PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM.html
@@njwildberger Thanks Prof, I just joined the channel becoming a new member. That's really cool, you read in my mind remotely...always ahead. Thanks a lot.
Prof Norman, on slide 8, I saw the Golden Ratio. I was exited to see the Golden Ratio. I sense that you are not exited to see the Golden Ratio. It pops up in many places. See Wikipedia.
He does not accept the "real" number system as it stands, so I suspect is less thrilled (and its appearance in the icosahedron is welll-known). However (!), I think it's important to note when familiar irrationals appear -- hinting at something deeper within, really. I believe he's in that line of thinking, too.
@@Cid2065 Myself, I am not qualified to discuss "real" numbers. As a software engineer, I have never multiplied real numbers. I view them as mysterious objects that are beyond our capability to understand. To me, the Golden Ratio is a mysterious and interesting object, even beautiful. Leonardo Da Vinci used the Golden Ratio in his artwork. It also appears in nature, in the spiral aloe as well as in the sunflower.
Professor, I've been watching your videos for years (I found your channel in high school) and have learned so much invaluable information. Now with this series I'm getting the sense that I will never run out of things to learn... I am SO grateful. Thank you for devoting so much of your time to education. You are an inspiration and a blessing.
I just want to break down with tears of joy. Your TH-cam channel is exactly what I need at this exact moment in my life. Thank you, thank you for making and sharing your videos with us. :)
It's a pleasure to seeing you well and back to your great work on one of the most smart ways of looking at mathematics. Thank you dear professor Wildberger.
That was a wonderful and insightful presentation. I’m a physicist and can see how such structures should be more widely known about and explored. Thanks so much!
This is, for me, a wonderfully clear exposition that seems to be aimed just at my level of understanding of groups, graphs and polyhedra. Somehow this talk speaks directly into my brain.
That is rare and I really appreciate it.
Thank you Professor. I'm here since 2008 and it is always a pleasure watching your videos.
Video Contents
0:00 Intro
0:07 Overview
0:38 Two Types of Symmetries
2:30 Though Experiment: Gas Diffusion
5:10 History of the Subject
9:03 An Initial Example
16:33 The Multiset of Neighbors
23:43 Fusion Rule Algebra
31:00 The Probabilistic Variant
38:34 Character Table of an Alternating Group
43:17 Character Table of a Hypergroup
It's such a happy coincidence that I have been working with representation theory and knot invariants now and there's a lot of fusion algebra - maybe not a coincidence now I watched your video.
Can you share your work?
Been really liking your content and your approach. Thanks for the work you do!
This is absolutely amazing! I hadn't heard of hypergroups before
Norman's got his white board and is wearing a suit! This is going to be good
I always appreciate your dihedrons.
I ’d like to see diffusion symmetry applied to large seemingly random graphs. There may be a method to computationally reduce after all. Nice work.
Computing the powers of the normalized adjacency sparse matrix is quite efficient. And used all over the place in data science.
Be aware that the diffusion of a gas is more complicated than depicted here. In a gas the velocities follow the Maxwell-Boltzmann distribution, implying fast and slow molecules traveling in all directions. The wave front blurs so to speak on its way to the boundary of the box, and the shape quickly randomize due to the elastic collision of the gas molecules.The bouncing at the wall will not be perfect symmetrical due to the surface roughness and the interaction between gas and wall molecules. The shape of the boundaries is therefore not important. The position of the molecules is always random, which gives no symmetry.
The mathematical picture here is more that of tennis balls travelling with the same speed from a hypothetical center. In that case radial symmetry remains until the balls hit the wall and bounce perfectly back. In that case symmetry and shape between the balls remains and changes to a new symmetry and shape upon bouncing.
I see it is interesting also.
"Randomness" is the mathematically the least coherent and interesting interpretation of the phenomenon in question. Phenomenologies of the external senses and internal sense of intuitive pure geometry are different categories, and their interrelations is a fascinating topic of research.
Bohm started from studying Brownian motion, and ended up with theory of holistic causation called 'Ontological interpretation' by Bohm and Hiley.
From geometric point of view, the process ontology of Holomovement and active information of quantum potential implicate that platonic geometric forms can be constructed as and exist only as interactive animations - which is what also Euclid is actually saying.
Thus, holographic informing from whole holomovement to parts is not ontologically random, but it implies that "snapshots" of the constructive continuously animated computing is mostly beyond our epistemic limitations, and thus mathematically not very meaningful. Especially when our mathematical foundations generally start from the bottom-up direction of constructing instead of the top-down. So the phenomenal "variation" from our nth logarithmic perspective of the animated pure geometry of holomovent appears as fractional "degrees of freedom" compared to platonic circles and solids etc.
Quadratics, as mentioned by Wildberger, offer periodic "constants of movement" that can be associated with the form preserving scale independent property of the Bohmian quantum potential in which the "probability amplitude" is both in the numerator and denominater, and can thus be given numerical anatomy of 1/1, 2/2, 3/3 etc. on the simplest level of "unitary" fractions.
Wow that was really nice!
I would love to see more on diffusion symmetry and how it might relate to ohysics
I'm thinking about Feaynman graphs for example, but who knows, many unforseen applications aswell!)
Anyway, I will check out more of yours stuff :)
Great video. This can be linked to the work on the theory of representations made by J.Q. Chen. Congratulations
Diffusion Symmetry will give huge possibilities to make new decissions for biomedicine, for diagnosis and treatment COVID, for correction personal psychological, emotional pecularities.
APL (A Programming Language by fellow Canadian Ken Iverson) would be ideal for expressing problems of this nature: unlike most modern languages, APL is array-based, as in arrays are a fundamental type, like integers being a fundamental data type in, say, C. The result is that operations over all elements of an array (n-dimensional, by the way, only limited by system memory) either element-wise, or cumulatively are at the heart of the language. Basically, working with graphs, poly-numbers, or vectors etc. of any size is what APL does. Also the syntax is extremely concise, based highly on mathematical notation.
Check out Connor Huykstras videos (another Canadian) for some nice examples comparing APL solutions to more common languages.
APL seems like the perfect tool for this approach.
Can't wait for the next video!
I think that Matlab took this space. What mainly killed APL was the need for special keyboard because it used a bunch of “special” characters to represent matrix operations.
Will definetly join the WildEgg members. Might also reenable my Patreon support. Great videos
Beautiful! Thank you very much (from a retired physics teacher walking behind you).
Love the phrase "Diffusion Symmetry"
The 26 + 1 Finite Non-Abelian Simple (non-parametric) Sporadic Groups and 17 Finite Non-Abelian Simple Lie Group Families (one or two parameteric) plus the Prime Group Family (obviously Finite Abelian) as a Framework for understanding more properties of this "diffusion symmetry" -- eg, Gyrosymmetry, Axisymmetry. Better understanding the missing Lithium 7 problem, the fine structure constant, proton/electron mass ratio. Seen all of your public videos (except the Go stuff, gave up on Go in the 70's)
I do think you should pick up Go again!!
What do you do with all the chess players then
I think those nice properties of the icosahedral graph are probably related to the fact that it's distance-regular.
Interesting use of multisets. I wonder what other uses there are for multisets. May be a powerful approach in computer science! For example in graph theory I now found that multisets are used.
@Anders As we proceed we'll see that multisets are flexible and powerful data structures that are particularly suited for studying group theory. Set theory is just too limiting, it turns out.
@@njwildberger Great, looking forward to that.
To create group create a multiplication table (what a Group is)
In higher dimensions understanding groups is a tough, hard to get ones head around
Character table by forbeneus:
Permutations values of characters(or rows), evaluated on Conjugacy classes
(character theory)
Alternatipn group of 5 things has cycle notation
Character table Useful computation Device to understand
Sqrt cannot compute to an infinite precision
Look at an Infinitesimal with skeptecism!
Diffusion view
Look at hypergroup object little c probabilities 5-1 =4 objects of different circles around point
It's commutative harmonic analysis so definitons are simpler than in group theory case . Thus this mimics process of ccommutitive geoups
-r5 quadratic extension function (of rational number)(to build from rational numbers)
Can deal with as matricies (watch Dihedrons complex number on quadratic extensions)
Q. How you deal with complex numbers but at the same time only able to acknowledge rationals?
Isn't the gap between those 2 already huge to be compared with?
Finally r5 squared=5
I like the idea of finding new ways of looking at symmetries. However, to compare this particular graph with an icosahedron?
It seems to me that an icosahedron is a very atypical example of this a graph. A graph doesn't say anything about the separation of the nodes nor of the dimension of the vertices nor whether the points lie in a plane or in some other space. However, there is a lot of symmetry about. So what is it and how can it be used, I am curious about what Norman has discovered.
With some unusual notation, your C's are morally equivalent to matrix elements of powers of the graph adjacency matrix, and the diffusion operators to normalized adjacency matrix, or 'graph laplacian' used heavily in probability theory and physics. There are mistakes in your calculations--- the one-step followed by three step motion on the graph does not have to lie at distance 2, it can lie at distance 1, 2 or 3, because you can walk around a closed triangle to come back where you started, so there is no parity, but I'm ignoring the calculation errors. The graph laplacian eigenvalues determine the deviations from 1 in your diffusion process, and these diffusion processes do show up in physics a lot, for example, Polyakov uses continuous diffusion processess at small times and large times to give a physics proof of the Atiyah-Singer index theorem.
C_k does not mean the kth power of C. It should only contain the vertices whose *shortest* distance is k.
Please send me a link where I can read Polyakov's proof.
Is Polyakov's a rigorous proof?
I'm following you since years on many of your playlists, and I really love all your lectures, especially the ones related to WildTrig, which lead me to remove entirely cos/sin from my code (I'm a Math developer in my daily job). Currently I'm a Math addict devoted to the study on Mathieu groups and the Monster, passing through the MOG & Co. In studying that, I faced many times the ADE classification, trying to attack it from many POVs, like Coxeter groups, Weyl groups, etc but still struggling in getting a unified vision of it. I saw also some lectures from yours about ADE, which were quite enlighting. It would be really interesting if you'd be so cool to create a dedicated spin off about ADE theory, using may be some new "cool" approach" on that as you did here for the diffusion symmetry. Is there any hope to get something similar in future? Tx for all your efforts, Prof. Wildberger.
You are in luck! At my Wild Egg Maths channel, I have a series exactly on a novel approach to ADE graphs. The series is called Dynamics on Graphs, and gives an approach to ADE graphs via two remarkable games on graphs. Currently there are about 10 videos, but I intend on adding many more, as it is a very rich subject. Here is a link to the Playlist, which however is available only to Members of that channel:th-cam.com/play/PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM.html
@@njwildberger Thanks Prof, I just joined the channel becoming a new member. That's really cool, you read in my mind remotely...always ahead. Thanks a lot.
@@riccardoventrella Great to have you join! There are quite a few other interesting Playlists there also I think ...
This one has left my mind bouncing around all over the place. Can't put my finger on it.
Very cool, thank you
Pure diamond. Lovely
Thankyou.
This is genius.