Channel Title: Joseph Newton Channel Description: Videos about maths! <video> Honours presentation excerpt (sus) Description: amogus </video> <video> Playing tricks with 2-variable limits Description: Oh hey look, a function! 0:00 Intro 1:57 Counterexamples to checking some straight lines 3:40 Counterexample to checking all straight lines 4:39 A more 'natural' counterexample (and touching grass) 6:54 Outro The graphing program used in this video is the Grapher app that comes with Mac computers. </video> <video> The Coxeter Classification 1/2: Combinatorics is hard Description: A very hands-on approach to Coxeter systems. Video 2: https://www.youtube.com/watch?v=NrtN-l9ZDtU #SoME3 00:00 Opening 02:23 Part 1: Introducing Coxeter systems 10:14 Part 2: Proving things are infinite 15:23 Part 3: Symmetry groups of polyhedra 25:00 Part 4: The exceptional systems 32:18 Conclusion Some casual overviews of finite groups, if you want a primer for this video: From 3Blue1Brown – https://www.youtube.com/watch?v=mH0oCDa74tE From Numberphile – https://www.youtube.com/watch?v=jsSeoGpiWsw If you're looking for more detailed proofs of results like the Word Property, I recommend the textbook Combinatorics of Coxeter Groups by Björner and Brenti which covers these in the first 3 chapters. Small corrections thanks to commentors: *The word 'prism' at 19:16 should be 'pyramid', or equivalently an asymmetric prism so that reflecting through the length is not a symmetry. </video> <video> The Poynting Vector in a DC Circuit Description: Energy in a circuit flows in the electric and magnetic fields around the wires. Here's a fully-worked example of how. Veritasium posted a followup video to the one mentioned here, which I'd highly recommend. It looks more into the dynamics whereas this video focuses on the statics. https://www.youtube.com/watch?v=oI_X2cMHNe0 Mathematica notebook for calculating the Poynting vector in a circuit: https://drive.google.com/file/d/1yf_RgQlE5KzqOlFYB8yMXUFiP6O_dZfg/view?usp=sharing PDF version: https://drive.google.com/file/d/14yTFRSe7wwACzLPqXQdEHhAu0W2N-zoe/view?usp=sharing Veritasium's video: https://www.youtube.com/watch?v=bHIhgxav9LY ElectroBOOM's video: https://www.youtube.com/watch?v=iph500cPK28 Animations were made with Manim: https://docs.manim.community/en/stable/index.html I tried to keep this video reasonably short but I know lots of people are going to have lots of questions, so here's a long FAQ to cover some things that I missed in the video: Q: In the wire between the capacitor plates, why does the energy come from far away? A: This is a quirk of using infinitely big plates, as we just assume the plates have a voltage difference without making it clear where the source is. We could picture the plates as very big disks, and at their edges are a ring of batteries connecting them providing the voltage difference. Q: Doesn't that formula for the magnetic field only work for an infinitely long wire? A: Yes, I had to sacrifice exactness for simplicity. The calculations for the square-shaped circuit make no such approximation, although they do assume infinitely thin wires, and this assumption needs to be treated as an approximation when calculating the total power. Give me a break, I'm trying my hardest! Q: If electric and magnetic fields are produced by charges and currents, then surely it is correct to say that energy is carried in the wires? A: This is really a question of semantics and there are many arguments to be made either way. Fields can exist without charges and currents in the form of electromagnetic radiation, although you could argue that the radiation had to come from particles originally. Special relativity and quantum mechanics makes these philosophical questions very difficult. Q: I heard that charges move through a circuit very slowly. If that's true, then how do charges build up on the surfaces of wires? A: Individual charges don't have to move very far because there are many more positive and negative charges filling the wires. The battery pushes in charges at one end and everything shifts over very slightly, resulting in more charges coming out at the other end and on the surface. Think of it like squeezing water into a teabag; the water doesn't flow through very quickly, but a decent amount still oozes out the sides. Man, that's a disgusting analogy. Q: Why do the wires in the two examples in the video behave in such different ways? A: In the parallel plates example, the wire is resisting and taking up the entire voltage drop across the circuit. That means energy is flowing into the wire and being turned into heat. For the battery and lightbulb, the wires are ideal and so no energy flows into them. Real wires with low resistance are somewhere between these, with the Poynting vector directed mostly parallel to the wires but also slightly into them. Q: So... ideal wires are kind of weird? A: Yes. If there is a voltage source across an ideal wire, the current is theoretically infinite, which doesn't make sense. The calculations only make sense if at some point in the circuit there is something to take up the drop in voltage between the terminals of the battery, e.g. a light bulb. But this isn't hard to see for yourself: if you attach a conducting wire to both ends of a battery, you'll find that it heats up very quickly. A lot of current is flowing through because of the low resistance. Be careful if you try this at home, because it's a very fast way to burn yourself. Q: Circuits use measured quantities like resistance, but electrodynamics also has measured quantities like resistivity and permittivity. Where do these quantities come from? A: The electric properties of a material are determined by the material's atomic structure, and studying this requires a looooot of quantum mechanics. This area of physics is called 'solid state' or 'condensed matter' physics, and it provides quantum-mechanical explanations for things like electrical properties, colour, transparency, elasticity and brittleness. 00:00 Introduction 01:20 A wire between plates 05:58 A simple circuit 09:41 Electrodynamics versus circuits 13:23 Conclusion </video> <video> Steiner's Porism: proving a cool animation #SoME1 Description: Strange circle stuff. (Some people have commented that the audio is really low. Unfortunately I haven't found a way to fix it without re-uploading the whole video, but your feedback will be taken on board for the next video! Also to everyone begging for more content, I’m currently in the middle of university study, but I’m hoping to do my next video around the end of November.) This video is my submission for 3blue1brown's Summer of Math Exposition 2021: https://www.3blue1brown.com/blog/some1 For more videos created by others for SoME1, here's a playlist: https://www.youtube.com/playlist?list=PLnQX-jgAF5pTkwtUuVpqS5tuWmJ-6ZM-Z Some more detail on the stereographic projection: http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node33.html https://www.cut-the-knot.org/pythagoras/StereographicProjection.shtml http://www.ams.org/publicoutreach/feature-column/fc-2014-02#theorem2 Animations were made in Manim: https://www.manim.community </video> <video> Cursed Units Description: A collection of cursed scientific units. Cursed Units 2 is now out! Watch it here: https://www.youtube.com/watch?v=Zg7xe8MkJHs What If? article: https://what-if.xkcd.com/11 Dimensional analysis: https://www.youtube.com/watch?v=SUnAvL-ThMs Distance in astronomy: https://www.youtube.com/watch?v=vh5kZ4uIUC0 Pulsar scintillation: https://arxiv.org/pdf/astro-ph/0111034 Background music is me improvising on a piano. More info: https://www.joenewton.au/music#piano Some small corrections: the distances between the planets in the graphics should be between their centres, and the wording when describing the pulse delays is backwards at one point. Some commenters explained that the 2 astronomical units is because the distance between the measurements is twice the distance from the earth to the sun. I know that; I’m jokingly questioning why the definition of the parsec doesn’t have the factor of 2 included (I also know why it doesn't, this joke wasn't communicated very well). If you know any other cursed units, share them in the comments! Some of these are in Cursed Units part 2, but here's some more... Noise in a voltage signal is measured in V/√Hz (volts per square root hertz). You would expect the amount of data produced by a particle collider over a time period to be measured in bytes or number of collisions, but apparently it's measured in fb^-1, inverse femtobarns, a barn being roughly the cross-sectional area of a uranium nucleus. Multiplication and division of units is all well and good, but it breaks down when you're working with units on logarithmic scales, like pH, decibels, or stellar magnitude. Sheet resistance, the electrical resistivity of rectangular sheets of materials, is measured in Ohms per square. Not square metres, not square inches, just square. This is because resistance across a rectangle is proportional to length and inversely proportional to width, so it depends on the aspect ratio but not the size. Other bizarre combinations of units mentioned by commenters: thermal effusivity is measured in J/(m^2 K s^(1/2)), a Jansky in radio astronomy is 10^(-26) W/(m^2 Hz), and electrical capacitance is measured in Farads which are kg^(-1) m^(-2) s^4 A^2. A lot of commenters suggested various imperial units. To me these aren’t cursed in a particularly deep way, since they just involve a conversion factor. I’ll give a special shout-out to acre-feet though. </video> <video> Suika Game is about binary counting Description: A video about Suika Game, Threes!, 2048, binary counting and scoring systems (and definitely nothing else). ======================== Articles referenced in the video ======================== The Verge article revisiting Threes! and 2048: https://www.theverge.com/22914955/threes-2048-ketchapp-copycats-clones-mobile-games Articles on Suika Game and Merge Big Watermelons: https://automaton-media.com/articles/newsjp/2023917-264866/ https://www.cyzone.cn/article/618919.html Articles on Threes!: https://www.wired.com/2014/05/threes-game-design/ https://www.theverge.com/2015/6/11/8722909/threes-free-to-play-launch-ios-android https://www.polygon.com/2015/6/11/8764835/why-it-took-a-year-to-release-a-free-version-of-threes Articles on 2048: https://www.buzzfeednews.com/article/hillaryreinsberg/why-this-free-puzzle-game-is-the-most-addictive-thing-on-the https://www.independent.co.uk/tech/2048-game-how-to-play-the-addictive-successor-to-flappy-bird-9207556.html https://www.wsj.com/articles/BL-DGB-33656 Open letter from Threes! developers addressing clones: https://asherv.com/threes/threemails/ The music used in this video is the music from Suika Game (developed by Aladdin X) and Threes! (music composed by Big Giant Circles) </video> <video> A Finite Game of Infinite Rounds #SoME2 Description: A short video about a random variable with no expected value. Made for the Summer of Maths Exposition 2. 0:00 Let's play a game 2:33 A better-behaved example 4:49 Working through the maths 7:08 Does the game always finish? 9:34 Discussion, and another example 11:04 A challenge problem Correction: the sum at 4:32 should be P(1)+2P(2)+3P(3)+… Some extra details I found later after some discussion in the comments: The Cauchy distribution has a cool property that intuitively explains why it behaves so weirdly: the probability distribution of the average of a number independent Cauchy-distributed variables is the exact same Cauchy distribution! No matter how many runs you average, the distribution doesn't narrow, making the Cauchy distribution a counterexample to the central limit theorem. Why do the simulations at the start of the video seem to go up logarithmically? Let's say you run N games and average the results. The mean is infinite, but we could shoddily get around this by assuming no game runs for K rounds or longer, the probability of which is (1-1/K)^N. Intuitively K should vary with N, so let's fix this to some chosen threshold probability p (e.g. p=0.05) and rearrange to get K=1/(1-p^(1/N)). We can now calculate the expected value only summing up to the K-th term, and after approximating the harmonic numbers with a logarithm and using a small Taylor expansion I got an expected value of ln(N) - ln(-ln(p)) + γ - 1. So yes, by some weird metric it is logarithmic with the number of games! However I should mention that I actually rerecorded the programs a couple of times before I got results that fit with the flow of the video; in reality the average tends to jump around a lot more. </video> <video> The Coxeter Classification 2/2: Who cares about Representation Theory? Description: A more elegant approach to Coxeter systems. Video 1: https://www.youtube.com/watch?v=BV5mYjh8m4E 00:00 Opening 01:49 Part 1: The geometry of reflections 05:38 Part 2: The general geometric representation 14:44 Part 3: When is a "reflection" actually a reflection? 26:17 Part 4: Completing the classification (again) 36:04 Conclusion, and connections to other things The textbook mentioned in the video is Reflection Groups and Coxeter Groups by J. E. Humphreys. The book covers everything in a lot more detail, but the language used and the order in which things are proven is quite different. Music is made by myself, except for the bit at 19:12 which is by Kevin MacLeod. </video> <video> How good is Advantage in D&D? Description: Can we equate rolling with advantage in Dungeons and Dragons to a flat modifier? Let's ask this kenku we pulled off the street. Oh no, they're going on a long-winded tangent on probability... (Note for commenters: I am aware that rolling a 1 is automatically a failure for attack rolls; however in 5th edition fail-on-1 for ability checks and saving throws is an optional rule, not an official one.) This video was made for the Summer of Math Exposition π! The organisers intended to break for a year between SoME3 and SoME4, so there is instead a community-run version for this year. More information: https://some.3b1b.co 0:00 Modifiers vs advantage 2:05 Equating advantage to a flat modifier 4:51 Other questions about interpreting probabilities 6:42 Odds as a way of comparing probabilities 9:06 Conclusions </video> <video> An Overly Sophisticated Proof of a Disproportionately Simple Fact Description: Proof that the cube root of 2 is irrational. This proof has been around for a while, but I originally saw it here: https://mathoverflow.net/a/42519 The best thing about this proof is that it doesn’t even work for the square root of 2. </video> <video> Cursed Units 2: Curseder Units Description: The long awaited sequel. See the original here: https://www.youtube.com/watch?v=kkfIXUjkYqE Corrections: the force F in Ampere's law should be force per unit of length, and the denominator factor should be 2 instead of 4. The stack exchange post referenced at 1:05: https://electronics.stackexchange.com/questions/211843/why-is-the-power-consumption-of-electrical-lamps-specified-in-kwh-1000h Maxwell's "A Dynamical Theory of the Electromagnetic Field is available on the Royal Society's website Other videos briefly mentioned at 18:50 which you should check out: https://www.youtube.com/watch?v=-scs_yF59YE https://www.youtube.com/watch?v=KmfdeWd0RMk Background music is me improvising on a piano. More info: https://www.joenewton.au/music#piano </video>

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