Summer of Math Exposition

Discover the beauty of mathematics in the shortest proof of the Pythagorean Theorem. By blending geometry and algebra, we explore the universal truth behind right triangles and reveal how their proportional relationships define this timeless theorem. Through the proof, we conquer the infinite and witness how geometry gives us a window into the symmetry of shapes.

This video analyzes Donald Knuth's semi-serious paper "The Complexity of Songs" and what happens when it's interpreted as a way to pick which songs to memorize. Join me on this surreal journey in which the French are out for (more) blood, modern drugs lead to demands for less memory space, and Stick Figure Donald Knuth is proven a liar on more than one occasion.

To estimate the compound growth rate of anything by doing simple division. Exploring the idea behind using linear estimation for a fundamentally non-linear relationship. Includes step-by-step guides so that readers can play along and create something similar themselves. Long link for the video in case the short link does not work: https://www.youtube.com/watch?v=aCHFgOCX1hU

Why are linear differential equations called linear in the first place?. From function spaces to Wronskians, I present a linear algebra perspective of the interesting realm of differential equations.

Why Calculus Only Works In Radians In this video, we discuss why Calculus has a bias towards radians. You see, the derivative of sin to be cos only works when theta is in radians! Corrections 0:44 This is in fact a valid identity! Thank you to @billwindsor4224 who pointed this out to me. My point in showing this graphic was that e^ipi = -1 where pi is in degrees is false, so the thumbnail is still correct. But when we have the degree symbol in the argument this identity becomes true because "180°=π" (https://math.stackexchange.com/questions/1368049/eulers-identity-in-degrees) 3:52 The right hand side should be limx_a f(x) + limx_a g(x) 4:36 The limx_0 is the first step in the proof 4:37 The limit for sinx/x is 1, and the order of multiplication should be switched For the Geometry Diagram, the whole circular sector involving theta is B Attributions: Calculus Image - https://wordsmithofbengal.wordpress.com/2021/08/02/dr-philos-the-creative-fantasy-of-differential-and-integral-calculus/ @3blue1brown Video - https://youtu.be/3d6DsjIBzJ4?si=EPTsw8CJDSsbkBYR (also inspiration) Some Latex From - http://www.deepnlp.org/blog/series-formulas-latex LaTeX Generated By - https://latexeditor.lagrida.com/ Animations are done with CapCut Diagrams and some animations were made with Desmos Chapters: 0:00 - Intro 1:00 - Proof of Euler's Identity 2:08 - Analysis of McLaurin Series 2:54 - The Derivative of sinx 3:10 - The Definition of the Derivative 3:33 - Back to sinx 4:40 - sinx/x 5:25 - The Radian and Geometry 6:08 - Conclusion 6:46 - Or is It?

In this video, I explain the proof of why there are infinitely many prime numbers using mathematical analysis.

A new (easier?) way of doing System of Linear Equations by using a new math operator that as far as I know does not have a name. This would be especially useful for teachers trying to help students learn System of Linear Equations

About infinitely nested radicals and why they're not what you think they are

Here we examine an alternate way to think about and compute derivatives.

This video talks the linear algebra topic, change of basis. It also mentions why this is important, with an important example at the end of the video. This is intended for people who are already familiar with the basics of linear algebra.

This video delves into a bit of science and mathematics history and identifies a mistake made in the Oppenheimer movie.

Presents a tidied-up version of AlphaProof's solution to problem 2 from the 2024 International Mathematical Olympiad.

A motivation for the use of a certain type of loss function in machine learning and a theorem about an surprising result of using such a loss function. Some knowledge of how neural networks are trained might be helpful.

Jenkins-Traub algorithm for polynomial zeros in a computer with a historical context and examples including fractals. Recommend some familiarity with Newton's Method or root-finding in general. There are no other videos on YouTube (or anywhere else for that matter) that explain this topic.

In this video I use the Godot game engine and Python to train a convolutional neural network from scratch which reads handwriting done in game and then uses it to fun effect. Moreover, I use this as a framing device to go over the basics of how neural networks and deep networks learn, and the rationale behind their fundamentals and the reasoning behind moving from traditional neurons to convolutional layers (and how they work). I try to keep it fun and light along the way, and have some interesting tech-demo-type outputs at the end. I believe there should be no issues as far as the AI policy, as I trained the network completely from scratch, the architecture is of my own design, and the input data is the EMNIST letters, which are fully publicly available for any use.

Discrete derivative is a good tool for finding the sum of a sequence. And it is about him and some of my discoveries about this discrete derivative that I will talk about in this video.

This video gives a full tutorial about how to create music in Desmos Graphing Calculator. It is a combination of using the tone() function, creating lists for a group of measures, and using the ticker tool for precise bpms.

This video about a tiling game which is my concept and some mathemathics related to it: pathway lengths on regular Eucledean and hyperboling tilings, and calculating the ratios of tiles in the layers in these tilings. I also add a link to downloadable contents in the description so everybody can play a bit with it: https://drive.google.com/drive/folders/1ExMJurL7j2tCT24m4qQdUt93RIq-y83P

dy/dx is a ratio and you CAN multiply and divide by dy, dx and so on like a FRACTION!!! You don't belive me? Well, I can assure you that they are numbers and it's completely rigorous to treat them as a fraction (ratio). In this video we'll see how it can be done and the other ways in which that dy/dx symbol can be interpreted (at the end all identify the same object even though they aren't the same thing!). Like d/dx as an operator (or functional) that acts on some function or as a simple notation that just means the derivative with respect to the variable x!!!

Introduction to the second part of book 10 of Euclid's Elements.

In this video I cover the problem of computing square-pyramidal numbers in-depth, leading to a rich discussion of how we can go about in general to solve a difficult math problem.

This is my entry for the Summer of Math Exposition of 2024. This video is about algorithmic complexity, the maths of problem solving. We'll solve a puzzle from the "Advent of Code", a coding challenge. The link of the repository with the code I wrote will be available soon! (when I finish to clean it ^^" ) Timestamps: - Introduction 01:04 - Problem 5 explanation 03:27 - Formalization 08:02 - Python and OOP basics 12:06 - Creating the base objects 20:34 - Part 2 naive approach 28:49 - What is Complexity? 32:01 - Complexity of our algorithm 47:03 - Another perspective 53:09 - Complexity again 1:02:04 - Conclusion & Final words 1:15:11

We may not be able to see in 4 dimensions and beyond, but we can play games in those higher dimensions. The games I discuss are Minesweeper, 5D Chess With Multiverse Time Travel, and Projective Set. I adapted these games to be playable within Minecraft, and you can play them on my public Minecraft server! Background necessary: basic arithmetic, geometry Topics covered: modular arithmetic, linear algebra, geometric analogues in higher dimensions

You get angry at the amusement park, go to a bar, talk to your inner demons, and discover a new world. A half-narrative approach for zero-knowledge functional commitment schemes, construction by (de Castro, Peikert 2023). Familiarity with linear algebra is recommended.

The video presents the math needed to expand “Tonal Space”, where a well-defined, extended set of musical intervals resides. Numerous “Rational Intervals” and “Spectral Chords” are illustrated with Manim animations and audio examples. Intervals and chords are the building blocks of musical harmony. They constitute tonality within a piece of music. If harmony is to be evolved, supplementary chords must be identified and appropriate strategies for playing have to be developed. Thus, the musical chord becomes our central entity for further research. Assuming that some “extended harmony” exists outside of our familiar cadences, the video illustrates how to identify attractive complementary chords and how to integrate them into a musical instrument for live performance. While countless timbres have been created since the early days of electronic sound design, keyboard instruments with *dynamic intonation control* have not gained much popularity on the stages of the world (with the exception of MIDI pitchbend, which affects all ringing notes by the same interval). Perhaps one reason is that more than twelve pitches per octave appear difficult to handle in a live situation. Maybe the benefits of enriched harmony have not been successfully communicated to composers and to performing musicians. Therefore, the second part of the video deals with the concept of a real instrument with enhanced intonation capabilities, designed to … • minimize dissonance and beating in polyphonic music • render an extended set of consonant musical chords • freely modulate through 12 chromatic keys • implement innovative strategies for changing the pitch of multiple notes (individually and simultaneously) over a wide range - continuously or in discrete steps • maintain the best possible compatibility with twelve tone equal temperament (12tet) • be suitable for live performance A piano keyboard (that can simultaneously address up to twelve pitches per octave) is a good starting point for explaining a process we call “dynamic pitch mapping”, which describes the assignment of a given “sequence of intervals” to successive keys at performance time. For best tonal compatibility with instruments tuned in 12tet, the twelve equally tempered pitches per octave (forming a stack of “irrational intervals”) are an integral part of the instrument's design. It's up to the player to assign one (at a time) of the 12tet-pitches to the appropriate key during live performance. While such an assignment slightly retunes all pitches, the actual intervals between the keys remain unaffected. The subject of this video is at the intersection of art, math and engineering. It would be awesome to take the ideas that have been prototyped so far and turn them into an instrument available to anyone interested.   *Video Chapters* 00:00 Introduction 00:50 Should I watch? 02:25 Frequency Ratio and Musical Interval 03:39 Expanding a Plane of Tonal Space 05:16 Definition of Terms 07:02 Triads in Tonal Space 09:10 3D-Tonal Space 09:45 Balance - A Property of Spectral Chords 10:22 Added Notes and Extended Chords 11:38 Hyperbolic Stretching of Intervals 12:09 Hyperbolic Stretching of Chords 13:08 A Real Musical Instrument 14:58 Dynamic Mapping of Intervals 17:57 Musical Score 18:28 Proof of Concept 19:28 Polar Art 20:30 Intermediary Chords in Standard Chord Progressions 21:43 Coda (until 22:36) *Acknowledgements* Two decades ago, I was lucky enough to find Tim Thompson's “Keykit” on the Internet, a programming environment for experimental real-time processing and creation of MIDI messages. The Keykit language features an ingenious generic “phrase” data type, that stores a tempo-related sequence of MIDI events. The software of the instrument described here was written entirely in Keykit. Keykit is still available on GitHub. Stephan Schmitt, co-founder of Native Instruments and director of Nonlinear Labs in Berlin encouraged me to put my ideas on paper and produce the first sound examples. Stephan is a brilliant motivator. Grant Sanderson's Manim library was a major inspiration for the dynamic visualization of tonal relationships. In a last-minute move, Joachim Schmidt, Hamburg, mastered the audio track of the video. Thanks to all of you, to the 3b1b team and of course to the Manim CE community! *Resources* Sethares, William A. Tuning Timbre Spectrum Scale. London: Springer Verlag, 1999. Helmholtz, Hermann. On the Sensations of Tone. New York: Dover Publications, Inc., 1954. Eskelin, Gerald. Lies my music teacher told me. Woodland Hills CA, USA: Stage 3 Publishing, 1997 Numerous contributors of royalty-free photography and audio at pixabay.com – thank you so much for sharing your art. This video was created by Holger Stoltenberg (Hamburg, DE) for the community-based Summer of Math Exposition (SoMEpi), 2024. You can contact the author at [email protected]

This is a tutorial with a fun philosophical background story on how to use Conway's criteria to create a unique tile to tile the plane. I am demonstrating the criteria as well as the tiling I came up with, so viewers can use the example to create their own.

This video tries to show how a flag as complex, as that of Nepal, can be constructed with constructions as simple as bisectors and straight lines. There is literally no measurement needed. The steps to construct the flag, of Nepal as shown in the video, are clearly and remarkably laid into the Constitution of Nepal. I am trying to learn better ways to make math content, something that is truly unique, and this is just (hopefully) one of the many videos along that learning curve :))

------------------------------ TIMESTAMPS 0:00 Intro 0:12 Finding Pi With a Rope and the Unit Circle 0:37 Extra Facts About Pi 1:11 Circumference Intro 1:40 Finding the Circumference 3:57 Tying it all Together (there is a pun there somewhere) 4:23 Conclusion 4:56 Outro ------------------------------ CREDIT The Easiest Way to Calculate Pi" By The Tipping Point Math https://youtu.be/bCiQOwP4LrY?si=NdRq6Se3FxNpDZOm Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/playlist/3zNK20qC96mVSww60lVi1k Song : Barbershop - Vishmak & Brainshaw Stream / Download : https://streamlink.to/Bbsh Music promoted by BEATS MUSIC Video link : • Vishmak & Brainshaw - Barbershop (No ...

In this video, we explore how we can use the basic rules of probability to calculate the probability of a given card being prized in the Pokémon TCG. Full list of prizing probabilities: https://lastlegume.github.io/blog/prize_probability#correct-probabilities I'd also like to credit this article, which I read after I finished all of my calculations and taught me that the order of hands and prizes does not matter in the naive case. https://www.pojo.com/Features/X-Act/2005/Odds%20In%20Pokemon%202.htm

The Erdos-Rado theorem is a theorem about coloring infinite sets and finding monochromatic subsets. After becoming stumped by the proof of this theorem, I created this video to make sure I understood why the theorem is true. A basic understanding of cardinals and their operations is needed. You also need to be comfortable taking infinite unions of sets.

This video describes the method for computing pi developed by Francois Viete in the 16th century. It uses spreadsheets to see how you could do the same calculation today. It also introduces the Richardson extrapolation technique to improve convergence.

In this video, we explore some basic concepts of Metric. Using the understanding of Metric we understand the mathematical meaning of a Circle. We take a look at the Manhattan or Taxi-Cab Metric or L 1 Metric and see how a square with its diagonals aligned to the co-ordinate axes, is a circle for L-1 Metric.

Hi, I am Rudransh Bhardwaj, a young highschooler , shared some details about my project of genetic algorithm and explains how it works....

I explain an algorithm designed for my mother to help her find which grades of my younger brother were duplicated by his teachers.

This video talks about number sets, the Zn groups, pairs of numbers, sets of numbers, positional number systems, word-based number systems, arithmetic and the concept of mapping all the elements of one set to all the elements of another set. While in the process of making the video, I used https://www.pexels.com/ images for decoration purposes. In it, I used the song Adventure (Remaster). On the 18th of August, this video will be made public. The Pexels.com image policy: (https://www.pexels.com/license/) The song I used: https://youtu.be/-6MFPOJEQj4?si=oWlj8ZiMRfn-i0Yw The song's licence: https://creativecommons.org/licenses/by/4.0/deed.en

Have you ever wondered what makes our device secure? What stops hackers from getting our data? Well, it's cryptography. Elliptic Curve Cryptography or ECC for short is one of the ways to encrypt data, but how does it compare to the Rivest-Shamir-Adleman algorithm? Today, we'll find out. How the animation was done: https://github.com/DavidRoloham/Elliptic-Curve-Cryptography-Manim The video will start with some exposition on cryptography, go over what an elliptic curve is and how addition is done on it. Then will briefly talk about how it's hard to reverse the keys due to ECDLP, and show a simplified example of how key generation is done. Then, it will compare ECC and RSA (briefly), and back to exposition on ECC to link back to the start.

Two novel proofs of the pythagorean theorem.

In the previous two episodes of the series, we performed mathematics on a plane containing two entities and the core of the Earth. In this episode, we find the equation of this plane.

Cellular automata can reach complex structures, in this video we try to understand if a phenomenon such as life can occur in it and under which conditions.

In this video, we're exploring Cannon's distributed memory matrix multiplication algorithms. These algorithms are key to understanding how large-scale computations are performed in distributed systems. The animations in this video were created using Manim, an open-source mathematical animation framework. For those interested in the technical details, the source code for these animations is available on my GitHub repository. GitHub Repository: https://github.com/bhavik-goplani/Matrix_mult_manim Enjoy the video, and happy learning!

Seven prisoners are given the chance to be set free tomorrow. An executioner will put a hat on each prisoner's head. Each hat can be one of the seven colors of the rainbow and the hat colors are assigned completely at the executioner's discretion. Every prisoner can see the hat colors of the other six prisoners, but not his own. They cannot communicate with others in any form, or else they are immediately executed. Then each prisoner writes down his guess of his own hat color. If at least one prisoner correctly guesses the color of his hat, they all will be set free immediately; otherwise they will be executed. They are given the night to come up with a strategy. Is there a strategy that they can guarantee that they will be set free?

A short video about my viewing of an edition of Euclid's elements with pop-up diagrams. If you'd like to link to the video, please use this link: https://youtu.be/E-lj1b9R-wg (Only one difference: I had to change the logo on request from the libary where I saw the book)

How much would you spend to buy Minecraft? Java edition at most comes to $39.99 USD. Would you spend $50 for Minecraft, $100? Some break Mojang's terms of service and buy Minecraft accounts for thousands of dollars! Why would someone do this? In this video, we make a brief overview of this online black market. Then, using linear regression, try to correlate something with account prices. Within is embedded a multivariable calculus lesson, and its application in least squares regression.

An introductory video in Group Theory, and its applications in the RSA cryptosystem

A way to see the basic trigonometric identities geometrically. i.e. what do these identities really mean geometrically ? sin(2X) = 2sin(X)cos(X) cos(2X) = cos(X)^2 - sin(X)^2 While it is easy to derive the identities by applying e^(iX) = cos X + i sinX , I was curious to see it more geometrically, and helpfully, I received some insights and was able to see how this lines up. [ Brown paper based explainer is obviously and totally inspired by and in homage to the @numberphile channel. :) ]

In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to random variables having a standard uniform distribution. (wikipedia)

This is a video about division by 0, how to do it, and wheel theory. This also lets people understand that there's more to math that might initially seem, by exploring non-standard ways of using it.