The Perfect Road for a Square Wheel and How to Design It Morphocular 2022-01-05 | How do you design a road that a square wheel will roll smoothly over? And what about other wheel shapes? How do you even approach such a problem?=Chapters=0:00 - Intro1:36 - The Dynamics of Rolling4:05 - Vertical Alignment Property7:16 - Stationary Rim Property8:29 - Describing the Road and Wheel13:04 - The Road-Wheel Equations17:02 - The Perfect Road for a Square Wheel22:40 - Building the Road Visually25:54 - Wrap Up===============================Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive (or if you want spoilers for the next video), I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:https://web.mst.edu/~lmhall/Personal/RoadsWheels/RoadsWheels.pdf===============================CREDITS► The song at the beginning of this video is "Rubix Cube" and comes from Audionautix.com► Thumbtack icon comes from Mister Pixel of the Noun Project.===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
What Gear Shape Meshes With a Square? Morphocular 2024-04-13 | Stay informed and get the full picture on every story by subscribing through the link https://ground.news/morphocular to get 40% off unlimited access with the Vantage subscription which is only $5/month.How do you design the perfect gear to partner with a given shape? It's tempting to think the way to do it is to treat both gears as if they're rolling on each other without slipping, but it turns out most gears by their very nature must slip as they spin. Why is that?Playlist of Weird Wheel videos: youtube.com/playlist?list=PLBVP28tRh1pXjZxJzZ2poJf5OhETqSksI=Chapters=0:00 - Wheels are not gears!2:03 - What's wrong with wheels?5:32 - Ground News ad7:21 - How to design actual gears12:07 - Envelopes18:50 - Parametrizing an orbiting gear22:04 - Computing the envelope25:22 - Example gear pairs29:05 - Resolving road-wheel clipping30:39 - Outro===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend, Carlos Herrado, James SpearIf you want to support the channel, you can become a patron atpatreon.com/morphocularThanks for your support!===============================CREDITSThe music tracks used in this video are (in order of first appearance): Rubix Cube, Checkmate, Ascending, Orient, Falling SnowThe track "Rubix Cube" comes courtesy of Audionautix.comThe animation of the moving point of contact between two gears comes from Claudio Rocchini. Original source: commons.wikimedia.org/wiki/File:Involute_wheel.gif===============================The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
I Get Interrogated For Reaching 100,000 Subscribers Morphocular 2024-04-12 | I was cursed with 100,000 subscribers and now I have to answer for it :(=Chapters=0:00 - Intro0:18 - When did you start liking math?1:10 - How do you get your video ideas?2:22 - Which video was hardest to make?3:03 - What are your favorite math topics?3:42 - How do you make your animations?4:59 - Tips for animating math explainers7:10 - Lightning Round!8:07 - Thanks everyone!!!=Links!=Manim Homepage: https://www.manim.communityManim Subreddit: reddit.com/r/manimManim Discord: https://manim.community/discordMorpho: github.com/morpho-matters/morpholibMotion Canvas: motioncanvas.ioMathemaniac: youtube.com/@mathemaniacMorphocular Patreon: patreon.com/morphocular===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend, Carlos Herrado===============================CREDITSThe music tracks used in this video are (in order of first appearance): Morpho Thinker, Bug Catching, Frozen in Love
The Subtle Reason Taylor Series Work | Smooth vs. Analytic Functions Morphocular 2023-12-22 | Get Surfshark VPN at https://surfshark.deals/MORPHOCULAR and enter promo codeMORPHOCULAR for a Holiday Special offer of 5 extra months for free with the Surfshark Onepackage.Taylor series are an incredibly powerful tool for representing, analyzing, and computing many important mathematical functions like sine, cosine, exponentials, and so on, but in many ways, Taylor series really shouldn't work as well as they do, and there are functions out there that can't be represented with them. What are these functions? And what's so special about so many of our familiar functions that we can compute them with Taylor series?=Chapters=0:00 - How to calculate e^x4:16 - Surfshark ad5:15 - Why Taylor series shouldn't work6:54 - A pathological function8:25 - Taylor's Theorem10:48 - Analytic functions vs. smooth functions12:53 - The simplicity of complex functions14:10 - The uses of non-analytic smooth functions14:53 - See you next time!===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend.===============================CREDITSThe music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Rubix Cube, OrientThe track "Rubix Cube" comes courtesy of Audionautix.com===============================The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Finding Velocity On a Sphere Using a 3D Eulers Formula Morphocular 2023-11-22 | Using a generalized version of Euler's Formula, exponential functions can be used to algebraically represent rotations in any dimension. But what is this generalized formula, and what can we use this representation for?Previous episode: youtu.be/Y1gOYtQYRXoPatreon: patreon.com/morphocular=Chapters=0:00 - Intro2:41 - Tilt Product Powers5:13 - Generalized Euler's Formula Derived6:57 - Classic From General9:35 - Polar and Spherical Coordinates13:20 - Trouble With the Matrix Exponential15:56 - Computing Velocity of a Sphere Point22:02 - Interpreting the Result23:03 - Wrap Up and a Look Ahead===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend.===============================CREDITSThe music tracks used in this video are (in order of first appearance): Rubix Cube, Checkmate, Ascending, Orient, Falling Snow.The track "Rubix Cube" comes courtesy of Audionautix.com===============================The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
The Concept So Much of Modern Math is Built On | Compactness Morphocular 2023-08-18 | Go to brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription.Compactness is one of the most important concepts in Topology and Analysis, but it can feel a little mysterious and also contrived when you first learn about it. So what is compactness, intuitively? And why is it so fundamental to so much of modern math?=Chapters=0:00 - Intro2:26 - Formal Definition3:03 - Topology Review4:03 - Unpacking the Definition6:33 - What Do Compact Sets Look Like?8:02 - Sequential Compactness10:03 - Making a Set Sequentially Compact14:46 - What is Compactness Good For?18:45 - Wrap Up19:22 - Brilliant Ad===============================The quote about compactness being a "gate-keeper" topic to math students comes from the paper "A pedagogical history of compactness" by Manya Raman-Sundstrom which provides, well, exactly what it says. You can find it here:arxiv.org/abs/1006.4131===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend.To support future videos, become a patron at patreon.com/morphocularThank you for your support!===============================CREDITSThe music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Rubix Cube, Falling SnowThe track "Rubix Cube" comes courtesy of Audionautix.com===============================The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
When CANT Math Be Generalized? | The Limits of Analytic Continuation Morphocular 2023-07-01 | There's often a lot of emphasis in math on generalizing concepts beyond the domains where they were originally defined, but what are the limits of this process? Let's take a look at a small example from complex analysis where we actually have the tools to predict when this is impossible.This video is a participant in the third Summer of Math Exposition (#SoME3) hosted by 3Blue1Brown to encourage more math content online. To learn more, see this:3blue1brown.substack.com/p/some3-begins=Chapters=0:00 - Intro1:15 - Extending a Geometric Series3:39 - Complex Power Series6:23 - Analytic Continuation8:30 - Analyzing the Gap Series11:51 - Visualizing the Gap Series19:21 - Gap Theorems===============================This video was generously supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor, Mfriend.To support future videos, become a patron at patreon.com/morphocularThank you for your support!===============================CREDITSThe music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Orient, Faultlines===============================The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Eulers Formula Beyond Complex Numbers Morphocular 2023-05-13 | Go to brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription.The famous Euler's Formula for complex numbers provides an elegant way to describe 2D rotation, but is there a way to make it work for 3D or higher dimensions?Previous video about complex numbers: youtu.be/4KlvI_uK9zs3Blue1Brown's video on the matrix exponential: youtu.be/O85OWBJ2ayo=Chapters=0:00 - Intro3:03 - 3D vs 2D5:07 - A Brief Overview of Matrices8:47 - How To Exponentiate a Matrix12:06 - 3D Rotations via Matrix Exponentials17:54 - How to Build Rotation Generators24:54 - 3D Euler27:10 - A Look Ahead28:25 - Brilliant ad===============================CREDITSThe music tracks used in this video are (in order of first appearance): "Checkmate", "Dream Escape", "Ascending", "Rubix Cube", "Orient", "Falling Snow"The track "Rubix Cube" comes courtesy of Audionautix.com===============================This video was supported in part by these patrons on Patreon:Marshall Harrison, Michael OConnor.To support future videos, become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Complex Numbers Have More Uses Than You Think Morphocular 2023-03-04 | Complex numbers are often seen as a mysterious or "advanced" number system mainly used for solving similarly mysterious or "advanced" problems. But really, once you get used to them, they're really an elegant and (ironically) simple mathematical tool with application to more down-to-earth problems besides Quantum Mechanics or advanced Differential Equations or something. Let's see what these numbers can do!Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!=Chapters=0:00 - Intro1:45 - Complex number basics3:31 - Interpreting complex number multiplication7:55 - Angular velocity10:42 - Calculating angular velocity using complex numbers15:18 - Interpreting the formula17:59 - More uses of complex numbers19:48 - Special announcement!===============================CREDITSThe music tracks used in this video are (in order of first appearance): "Dream Escape", "Checkmate", "Orient", "Rubix Cube", "Frozen in Love"The track "Rubix Cube" comes courtesy of Audionautix.com===============================The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
How to Design a Wheel That Rolls Smoothly Around Any Given Shape Morphocular 2022-12-20 | Go to brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription.In previous videos, we looked at how to find the ideal road for any given wheel shape and vice-versa, but what about getting two wheels to roll smoothly around each other? Would two such wheels work as gears?Episode 1: youtu.be/xGxSTzaID3kEpisode 2: youtu.be/Y0aOxj5lrKY=Chapters=0:00 - Intro1:23 - Defining smooth rolling2:30 - Sidenote about gears3:16 - The Wheel-Coupling Equations7:34 - Sanity check9:24 - The partner for an ellipse12:24 - The connection between ellipses and parabolas13:23 - Finding self-coupling wheels16:35 - The partner for a square19:21 - A look back20:20 - A fractal wheel??20:47 - Brilliant ad===============================I would also like to thank the user @BeekersSqueakers whose comment I think it was that taught me that a partner wheel can be generated by first generating a road and then generating a wheel on its underside. This comment was directly responsible for inspiring the technique shown in this video to easily generate self-coupling wheels, and dramatically simplified the second half of this video! So a seriously genuine thank you to @BeekersSqueakers and to all those who actually took up the call to answer my challenge problems in a comment! They can have a surprisingly big impact sometimes!===============================For a deeper dive into the concepts explored in these videos, take a look at the paper "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). You can find it here:https://web.mst.edu/~lmhall/Personal/RoadsWheels/RoadsWheels.pdf===============================CREDITSThe music tracks used in this video are (in order of first appearance): "Rubix Cube", "Checkmate", "Ascending", "Orient", "Falling Snow"The track "Rubix Cube" comes courtesy of Audionautix.com===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Is this one connected curve, or two? Bet you cant explain why... Morphocular 2022-10-18 | Go to brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription.One of the most fundamental properties of geometry is connectedness: when a shape is a single continuous entity. But how do you define this idea precisely so that you can apply it even to extremely bizarre shapes in very strange spaces?=Chapters=0:00 - The Topologist's Sine Curve1:54 - The goal of this video3:06 - Path-Connectedness6:56 - A new definition?7:37 - Topology basics10:27 - A snag12:04 - Connectedness 2.0 and the Topologist's Sine Curve15:48 - Who's right?17:38 - The Ordered Square19:32 - The connection between the definitions21:18 - The underappreciated art of crafting definitions22:16 - Brilliant ad===============================CREDITSThe songs used in this video are (in order of first appearance) "Dream Escape", "Checkmate", "Rubix Cube", and "Twinkle in the Night".The song "Rubix Cube" comes courtesy of Audionautix.com===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
What Lies Between a Function and Its Derivative? | Fractional Calculus Morphocular 2022-08-11 | Can you take a derivative only partway? Is there any meaning to a "half-derivative"? Does such a concept even make sense? And if so, what do these fractional derivatives look like?Previous video about Cauchy's Formula for Repeated Integration:youtu.be/jNpKKDekS6kA really nice video that derives the gamma function from scratch:youtu.be/v_HeaeUUOnc=Chapters=0:00 - Interpolating between polynomials1:16 - What should half derivatives mean?3:56 - Deriving fractional integrals8:22 - Playing with fractional integrals9:12 - Deriving fractional derivatives13:53 - Fractional derivatives in action16:19 - Nonlocality17:54 - Interpreting fractional derivatives18:51 - Visualizing fractional integrals22:10 - My thoughts on fractional calculus23:10 - Derivative zoo===============================MAIN SOURCES USED FOR THIS VIDEOPodlubny, Igor. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, 1999Podlubny, I.: "Geometric and physical interpretation of fractional integration and fractional differentiation." Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367--386.- (for the visualization trick for fractional integrals)Edmundo Capelas de Oliveira, José António Tenreiro Machado, "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459, 6 pages, 2014. doi.org/10.1155/2014/238459- (for the zoo of alternative fractional derivatives)===============================Minor correction: The footnote at 7:34 should say the trig substitution produces another *whole* factor of pi (not a root pi) in the numerator which then cancels the *two* root(pi)'s that appear in the denominator from applying the half integral formula twice.===============================CREDITSThis video uses the song "Rubix Cube" coming courtesy of Audionautix.com===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:github.com/morpho-matters/morpholib===============================This video is part of the 3Blue1Brown Summer of Math Exposition 2 (#SoME2). You can find out more about it here:summerofmathexposition.substack.com/p/the-summer-of-math-exposition-is
How to do two (or more) integrals with just one Morphocular 2022-06-16 | Is there a way to turn multiple, repeated integrals into just a single integral? Meaning, if you, say, wanted to find the second antiderivative of 6x, is there a way to compute it all in one step just using a single integral? Turns out there is! In fact, any number of repeated antiderivatives can be compressed into just a single integral expression. How is that possible? And what does that single integral expression look like?My old video about Double Integrals: youtu.be/gifQWtTWqEYThe sequel to this video about fractional calculus: youtu.be/2dwQUUDt5IsA really nice video that derives the gamma function from scratch:youtu.be/v_HeaeUUOnc=Chapters=0:00 - Intro0:51 - Why Compress Integrals?2:29 - Analyzing the Problem3:46 - Visualizing a 2-Fold Integral5:25 - Deriving the Formula10:56 - Testing the Formula12:14 - How Is This Not Impossible?13:49 - Higher-Order Integrals15:22 - Application to Numerical Integrals16:25 - The Gamma Function===============================For more on applying Cauchy's Formula to numerical integration, see this paper:Tvrdá, Katarína & Minárová, Mária. (2018). "Computation of Definite Integral Over Repeated Integral." Tatra Mountains Mathematical Publications. 72. 141-154. 10.2478/tmmp-2018-0026. researchgate.net/publication/331705202_Computation_of_Definite_Integral_Over_Repeated_Integral===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
How to Design the Perfect Shaped Wheel for Any Given Road Morphocular 2022-04-16 | Last video, we looked at finding the ideal road for a square wheel to roll smoothly on, but what about other wheel shapes like polygons and ellipses? And what about the inverse problem: finding the ideal wheel to roll on any given road, such as a triangle wave?Previous episode: youtube.com/watch?v=xGxSTzaID3k=Chapters=0:00 - Intro & Review1:48 - Polygon Wheels3:49 - Elliptical Wheel5:30 - Focus-centered Ellipse8:50 - Wheels From Roads11:24 - How to Get a Closed Wheel14:10 - The Many Wheels for a Sinewave16:24 - The Wheel for a Triangle Wave Road19:16 - The Wheel(s) for a Cycloid Road20:24 - The Wheel for a Parabolic Road20:58 - A Look Ahead and a Challenge===============================Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive, I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:https://web.mst.edu/~lmhall/Personal/RoadsWheels/RoadsWheels.pdf===============================CREDITS► The song at the beginning of this video is "Rubix Cube" and comes from Audionautix.com===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Can an Uncountable Sum Ever Be Finite-Valued? | Why Measure Infinity? Morphocular 2021-11-07 | Traditional infinite sums deal with only COUNTABLY infinitely many terms. But is it ever possible to add up UNCOUNTABLY many terms and get a finite sum? And if so, can it give us a way to extend the dot product from finite-dimensional vectors to functions?=Chapters=0:00 - Intro1:23 - Functions as vectors3:21 - Uncountable sums6:45 - Analyzing an uncountable sum10:52 - Resolution===============================A few sidenotes on the video:► What I've been calling a "dot product" on functions and sequences is known more formally as an "inner product". I believe the term "dot product" is usually reserved for dealing with traditional finite-dimensional vectors.► I described the "components" of a function as coming from each real number input you can plug in, but that was mainly to supply a hypothetical train of thought that would motivate the inquiry that followed. I think those who work in Functional Analysis usually think of function "components" in a somewhat different way (e.g. a Fourier Series decomposition).===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Navigating an Infinitely Dense Minefield | Why Measure Infinity? Morphocular 2021-09-13 | If you're in to math at all, there's a good chance you've encountered the idea that infinity can come in different sizes. And while that's cool, and keeps pure mathematicians off the streets, is there any practical use for it? Can you solve any problems with it? And does it matter at all to broader mathematics? To find out, we'll have to find a path thru an infinitely dense minefield.=Chapters=0:00 - Who cares about infinity?2:15 - How to measure infinity8:00 - The infinite minefield10:45 - How many mines are there?12:25 - Finding a way out15:53 - Why it all matters===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
Can you change a sum by rearranging its numbers? The Riemann Series Theorem Morphocular 2021-07-25 | Normally when you add up numbers, the order you do so doesn't matter and you get the same sum regardless. And, of course, the same holds true even if you add up infinitely many numbers.....Right?=Chapters=0:00 - Let's rearrange a sum!1:48 - Investigation6:32 - Riemann Series Theorem explained visually13:58 - Resolving objections18:52 - A step further and a challenge20:07 - Significance of the Riemann Series Theorem21:47 - Final thoughtsThis video is a participant in the 3Blue1Brown First Summer of Math Exposition (SoME1). You can find out more about it here:3blue1brown.com/blog/some1#SoME1===============================Want to support future videos? Become a patron at patreon.com/morphocularThank you for your support!===============================The animations in this video were mostly made with a homemade Python library called "Morpho".If you want to play with it, you can find it here:github.com/morpho-matters/morpholib
