The Mandelbrot Set Explained The Mathemagicians Guild 2020-02-18 | We take a deep dive into the Mandelbrot Set, and try to understand what is happening under the hood. In particular we introduce the concept of orbits and their unusual behaviour. I hope by introducing such detail that you will find concepts in future videos easy to understand. This video contains many handcrafted visuals that took days code and animate, plus several hours to render them in 8k. Please subscribe!Welcome to the channel! I'm new at making mathematical explainer videos, but I hope you get something from these videos. Feel free to leave some love or constructive feedback in the comments section. Questions are also most welcome. If you loved this video then why not buy me a coffee: ko-fi.com/mathstownIn this video:00:00 Introduction1:51 Chaos & The Mandelbrot Set2:40 The Equation3:31 Looking at z squared6:48 Mandelbrot, the real part.15:24 Complex numbers18:35 Full Mandelbrot Set23:23 Mini-MandelbrotsIn this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia SetDeep Mandelbrot Zoom video (from introduction): youtube.com/watch?v=u1pwtSBTnPUPanning around the Mandelbrot: youtube.com/watch?v=BsjHdx83J9A/********************************************************************************Discord: discord.gg/q4xsmSHVTwitter: twitter.com/MathsTownPatreon: patreon.com/mathstown (Support, Downloads & Usage Rights)Website: https://www.maths.town********************************************************************************/
Polar Coordinates - Complex Analysis #3 The Mathemagicians Guild 2021-03-12 | In the 3rd complex analysis video I would like to introduce the polar form of a complex number. It may seem a little odd to bring this in so early in the series, but I think it will help me greatly when I cover multiplication and division. Multiplying functions is easier to comprehend geometrically if you think in polar coordinates. Secondly we take quick look at Euler's identity. You don't really need to understand how it is derived just yet, because we will cover the exponential function in a later video. However I feel that the exponential form needs a little explaining, otherwise it would seem to come out of nowhere.Lastly, we develop our visualization tools a little by looking making enhanced phase portraits. We can add contour lines for locations of equal magnitude and argument. Take the time to practice reading the cosine phase portrait with different constants added to it. 2D phase portraits are quite a useful way to visualise and understand complex functions. (The promised video of extra phase portraits is coming soon)In this video:00:00 Introduction00:46 Polar Coordinates03:10 How to represent the polar form.04:56 Radians (a recap)06:39 Examples08:00 Euler's Identity10:47 Enhanced Phase Portraits14:52 3D Phase Portraits.In this series:1 - youtu.be/jU7QW6AjUf4 Introduction to Complex Numbers.2 - youtu.be/nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - youtu.be/O3aJCGbyfR8 Polar Coordinates of Complex Numbers4 - [Coming Soon] Multiplication of Complex Numbers and Functions5 - [Coming Soon] Division of Complex Numbers and Functions6 - [Coming Soon] Complex Differentiation and Analytic FunctionsExtra Visuals (No Commentary):1 - youtu.be/3qEJeP6qQGA Trigonometric Functions Visualised (3D)2 - [Coming Soon] Phase Portraits of Trigonometric Functions
[Visual] Modular Form - Level 1 Weight 12 (Ramanujan Delta Function) The Mathemagicians Guild 2021-03-05 | This is a mathematical object known as a "Modular Form" visualised in 3 dimensions. Modular Forms are an area of mathematical theory that extends from complex analysis, but they are of particular interest to mathematicians studying number theory. Famously, their relation to elliptic curves was used to prove Fermat's Last Theorem (358 years after it was proposed) . This video is a collaboration with David Lowry-Duda (Institute for Computational and Experimental Research in Mathematics) who reached out to me after seeing some other visuals on the channel. Together we have been working on this little project to visualise Modular Forms. Traditionally these objects are defined algebraically, and I don't really know of any examples where they have been rendered in 3D.A Modular Form is a function defined on the upper half of the complex plane (first part of the video). The 2nd part of the video takes a closer look, so you can see the fractal like nature. Modular Forms are also commonly visualised on the Poincaré disk (as seen towards the end of the video). The colour of the function represents the complex angle (or phase), using the came colour scheme as my previous videos (red is the positive real axis). Height represents the absolute value, but we needed to scale the result. Modular forms have interesting detail close to zero, and at very very high values. To allow these features to be easily seen we have scaled it using the function atan(log(sqrt(abs(z))+1)).In this video:00:00 Upper Half Plane.01:24 A closer look02:19 Poincaré diskIf you loved this video then why not buy me a coffee: ko-fi.com/mathstownMusic from Epidemic Sound: epidemicsound.com/referral/ekoievProgrammed in Python, rendered with Blender,More Information:Detailed information on this function:lmfdb.org/ModularForm/GL2/Q/holomorphic/1/12/a/aModular Forms (Wikipedia): en.wikipedia.org/wiki/Modular_formRamanujan Tau Functionen.wikipedia.org/wiki/Ramanujan_tau_functionModularity Theoremen.wikipedia.org/wiki/Modularity_theoremDavid's Sitedavidlowryduda.com
[Visual] The Riemann Zeta Function Visualised The Mathemagicians Guild 2021-01-09 | Three different visuals exploring the Riemann Zeta function (without commentary). The 3rd visual shows shows a large part of the critical strip. These visuals are "3D phase portraits" or "modular surfaces" (not to be confused with modular functions or forms). The input is the complex plane, shown as the silver base plate. The output is the surface. The height of the surface is the absolute value of the Riemann Zeta function. The colour is the argument, or polar angle, of the Riemann Zeta function.The zeros are where the surface touches the ground plate. (sometimes there is the slightest gap, because the mesh doesn't have a vertex perfectly on the zero.) On the real axis there are the trivial zeros which are easily calculated. Next to the imaginary axis at Re(ζ) = 0.5, are the "Million Dollar Zeros". (There is a 1 million prize available if you can prove they only appear at Re(ζ) = 0.5. The Riemann Hypothesis).If you loved this video then why not buy me a coffee: ko-fi.com/mathstownIn this video:0:00 Riemann Zeta Function.1:36 Riemann Zeta Function. Height = log(1+|ζ|)3:13 The Critical Strip of the Riemann Zeta FunctionThe 2nd visual uses the log function to control the height.The 3rd visual shows only the critical strip (Re(ζ) between 0 and 1) . It is known that all the million dollar zeros are within this strip. To date, they have only be found at Re(ζ)=0.5 Sorry my camera-work is a little wonky. I'll try to improve it. I just couldn't bring myself to re-render this. Ray-tracing this took some CPU & GPU cycles! Rendered with Blender.
[Visual] Complex Trigonometric Functions Visualised The Mathemagicians Guild 2020-12-18 | In this extra video I have rendered 3D Phase Portraits (Modular Surfaces) of all six trigonometric functions: sin, cos, tan, sec, cosec & cot. Presented without any commentary, if you would like further explanation of these graphics, please see the 1st video in my Complex Analysis series. youtu.be/jU7QW6AjUf4The input to each plot is a complex number, as shown on the base plane. The output is also a complex number. The absolute value of the output is shown as height. The arguments (angle) is shown as colour. Where red it the positive real axis, and cyan is the negative real axis. The domain of the input is adjusted for presentation reasons. Where the images touches the base there is a zero (or the function is approaching zero).You will notice that only sine and cosine are analytic. The other functions all have "poles" where the value goes to infinity. You can see why they are called poles. Functions like these with isolated poles, but are otherwise analytic are known as Meromorphic functions. The functions tan and cot, both approach an absolute value of 1 in the imaginary directions (actually plus or minus i). If you were to rotate these functions 90 degrees you would get the corresponding hyperbolic functions.In this video:0:00 sin(z)0:55 cos(z)1:51 tan(z)2:42 cosec(z)3:30 cot(z)4:21 sec(z)In this series:1 - youtu.be/jU7QW6AjUf4 Introduction to Complex Numbers.2 - youtu.be/nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - [Coming Soon] Polar Coordinates of Complex Numbers4 - [Coming Soon] Multiplication of Complex Numbers and Functions5 - [Coming Soon] Division of Complex Numbers and Functions6 - [Coming Soon] Complex Differentiation and Analytic FunctionsExtra Visuals (No Commentary):1 - youtu.be/3qEJeP6qQGA Trigonometric Functions Visualised
Addition and Subtraction of Complex Numbers - Complex Analysis #2 The Mathemagicians Guild 2020-12-01 | Addition and subtraction represent translations on the complex plane. In this video we first go through the basics of adding subtracting complex numbers. A process that works as you would expect if you treat "i" as a simple constant. Then we start investigating adding simple constants to some functions. In the process we discover learn more about reading phase portraits and 3D modular surfaces. We also come across the fundamental theorem of algebra. My aim for this series is to introduce complex analysis in a visually intuitive manner, starting with the basics of complex numbers. We will start also start visualizing some simple functions very early in the series. Don't expect much in the way of rigorous proofs and definitions, there are plenty of text books for that. Instead, this series will aim to give you some visual intuition, that I hope will make any future study easier and more enjoyable.In this video:0:00 Introduction0:18 Addition of Complex Numbers1:49 Subtraction of Complex Numbers2:49 Inequalities3:26 Adding a constant to a function4:32 Phase portrait: z+1+i5:07 3D Modular Surface: z+1+i6:24 3D Modular Surface: z²7:35 Phase Portrait: z² +1 and polynomials12:41 Adding constants to cos(z)In this series:1 - youtu.be/jU7QW6AjUf4 Introduction to Complex Numbers.2 - youtu.be/nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - youtu.be/O3aJCGbyfR8 Polar Coordinates of Complex Numbers4 - [Coming Soon] Multiplication of Complex Numbers and Functions5 - [Coming Soon] Division of Complex Numbers and Functions6 - [Coming Soon] Complex Differentiation and Analytic FunctionsExtra Visuals (No Commentary):1 - youtu.be/3qEJeP6qQGA Trigonometric Functions VisualisedI don't like to interrupt my videos with ads or promos, so you won't see any mid-roll ads. You can support this channel directly by joining my Patreon page. It takes me a long time to develop these videos, so any support will mean I can spend more of my time producing animations. As a bonus you will get downloadable access to the Maths Town fractal videos. patreon.com/mathstown
Introduction to Complex Numbers - Complex Analysis #1 The Mathemagicians Guild 2020-11-11 | Introducing the complex numbers and complex analysis. This is the first video in a series covering the topic of complex analysis. We begin by introducing a complex number. Then we investigate the effects of multiplying any number by the imaginary number i. Finally, we take a look at some of the visualisation tools that we will use in later videos; phase portraits and modular surfaces. Please subscribe!My aim for this series is to introduce complex analysis in a visually intuitive manner, and we will be starting with the very basics of complex numbers. We will start visualizing some simple functions very early in the series. Don't expect much in the way of rigorous proofs and definitions, there are plenty of text books for that. Instead, this series will aim to give you some visual intuition, that I hope will make any future study easier and more enjoyable.In this video:00:00 Introduction00:30 A complex number02:17 The imaginary number "i"03:53 Visualising a complex number05:49 Multiplying a number by i06:36 Powers of i08:53 Introducing complex analysis10:43 Visualisation tools - phase portraits12:42 3D phase portraits (modular surfaces)13:42 cos(z) and cosh(z)In this series:1 - youtu.be/jU7QW6AjUf4 Introduction to Complex Numbers.2 - youtu.be/nT3WYFxvPLk Adding and Subtracting Complex Numbers 3 - youtu.be/O3aJCGbyfR8 Polar Coordinates of Complex Numbers4 - [Coming Soon] Multiplication of Complex Numbers and Functions5 - [Coming Soon] Division of Complex Numbers and Functions6 - [Coming Soon] Complex Differentiation and Analytic FunctionsExtra Visuals (No Commentary):1 - youtu.be/3qEJeP6qQGA Trigonometric Functions VisualisedI don't like to interrupt my videos with ads or promos, so you won't see any mid-roll ads. You can support this channel directly by joining my Patreon page. It takes me a long time to develop these videos, so any support will mean I can spend more of my time producing animations. As a bonus you will get downloadable access to the Maths Town fractal videos. patreon.com/mathstown Credits:Monkey Image: pixabay.com/vectors/monkey-marmoset-banana-chimpanzee-4698962
Number Sequences in the Mandelbrot Set The Mathemagicians Guild 2020-06-04 | Welcome to part 4 of our little Mandelbrot Explained series. In this video we explore the bulbs around the main cardioid, and find that they contain number sequences such as the natural numbers, Fibonacci sequence, and the rational numbers. We then investigate them in terms of their Julia Sets to try and understand visually why they are there. Finally, we look at precisely where the bulbs are attached to the cardioid. If you loved this video then why not buy me a coffee: ko-fi.com/mathstownIn this video:00:00 Introduction00:30 Period of the bulbs01:45 Signposts in the Mandelbrot Set02:42 Number Sequences04:24 Rational Numbers07:06 Julia Sets of the bulbs11:36 Location of the bulbs13:15 Why signposts?14:30 Building a Mandelbrot SetIn this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia SetMentioned in video:youtu.be/4LQvjSf6SSw Numberphile video (Mandelbrot & Fibonacci)youtu.be/TOxs1vLgQ_M Mapping a polar grid to the Mandelbrot
[Extra Visual] Period 6 orbits of a Julia Set The Mathemagicians Guild 2020-05-29 | This visual show all the orbits over a Julia Set. The location of c for this Juliet is within the period 6 bulb of the Mandelbrot, and you will notice the orbits settle down to a period 6 pattern. (These are the orbits of the Julia Set, not the Mandelbrot).In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] Period 5 orbits of a Julia Set The Mathemagicians Guild 2020-05-29 | This visual show all the orbits over a Julia Set. The location of c for this Juliet is within the period 5 bulb of the Mandelbrot, and you will notice the orbits settle down to a period 5 pattern. (These are the orbits of the Julia Set, not the Mandelbrot).In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] Period 4 orbits of a Julia Set The Mathemagicians Guild 2020-05-29 | This visual show all the orbits over a Julia Set. The location of c for this Juliet is within the period 4 bulb of the Mandelbrot, and you will notice the orbits settle down to a period 4 pattern. (These are the orbits of the Julia Set, not the Mandelbrot).In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] Period 3 orbits of a Julia Set The Mathemagicians Guild 2020-05-29 | This visual show all the orbits over a Julia Set. The location of c for this Juliet is within the period 3 bulb of the Mandelbrot, and you will notice the orbits settle down to a period 3 pattern. (These are the orbits of the Julia Set, not the Mandelbrot).In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set 4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] Period 2 orbits of a Julia Set The Mathemagicians Guild 2020-05-29 | This visual show all the orbits over a Julia Set. The location of c for this Juliet is within the period 2 circle of the Mandelbrot, and you will notice the orbits settle down to a period 2 pattern. (These are the orbits of the Julia Set, not the Mandelbrot).In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] Building a Mandelbrot Set Step-by-step The Mathemagicians Guild 2020-05-10 | This visual relates to the "How to Build a Julia Set" video. youtu.be/5T0cC6KRezo It shows the Mandelbrot forming one iteration at a time. The shape converges on the Mandelbrot Set. The shape at each iteration relates how you normally see the Mandelbrot coloured. Unlike the Julia Sets, this has little meaning as a series of transformations.Take a look at where the "8-way crosses" are found. This was the origin of the original circle. You will notice that origin maps to the bulbs in a periodic manner.The animation from one frame to the next has no real meaning, it looks cool, and makes it easier to watch, so I included it. It is simply an interpolation from one value to the next. So don't try too hard to understand the motion between frames.In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] All Period 2 orbits of the Mandelbrot Set The Mathemagicians Guild 2020-05-06 | This visual shows a series of balls located within the period 2 circle of the Mandelbrot Set. This is the area where the orbit have a period of 2. Each iteration, all these orbits bounce between 2 periodic points. This is an extra visual for the Mandelbrot Explained series of videos. If you'd like to understand what is happening a little better, please check out the related series of videos. I'm publishing these extra visuals in the hope that they may be useful for people wishing to study the Mandelbrot in more detail.In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] All Period 1 orbits of the Mandelbrot Set shown together. The Mathemagicians Guild 2020-05-06 | This visual shows a series of balls located within the main cardioid of the Mandelbrot Set. This is the area where the orbit have a period of 1. All these orbits approach a single attractive fixed point. This animation follows each orbit over 50,000 iterations to see where they finish, each near their own attractive fixed point. You'll notice that 1 or 2 of these orbits don't have time to settle down to a settle down to a single point. This is an extra visual for the Mandelbrot Explained series of videos. If you'd like to understand what is happening a little better, please check out the related series of videos. I'm publishing these extra visuals in the hope that they may be useful for people wishing to study the Mandelbrot in more detail.In this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
[Extra Visual] All orbits of the Mandelbrot Set shown together. The Mathemagicians Guild 2020-04-13 | This visual shows a series of balls on the complex plane. The equation z=z²+c is applied to every ball once per iteration. Those balls outside the Mandelbrot will escape, and head towards infinity. Those inside the Mandelbrot Set will remain bounded within the radius two circle. This video is extra footage associated with the explainer video "The Mandelbrot Set Explained". youtube.com/watch?v=7MotVcGvFMg If you look carefully you can see the period motion associated with the periods 2 and 3 bulbs. For example you will notice that the balls return to the period 3 bulb every third iteration. The colour of the balls is graduated for each ball, so that you can get a rough idea of where it started. This graphic shows the first 400 iterations.I will attempt to upload extra graphics associated with the main sequence of explainer videos, you will find them by exploring the channel playlists. Generally I'll set these videos so that subscribers aren't notified, so that subscribers aren't pestered by these uploads. Join the Maths Town Twitter feed if you would like to be notified about everything I am doing (including the Maths Town channels). twitter.com/MathsTownIn this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia Set
How to Build a Julia Set The Mathemagicians Guild 2020-04-07 | In this video we examine how to build a Julia Set by making repeated remapping of a circle on the complex plane. The technique is quite interesting because it allow us to visually understand why Julia Sets have some of their properties, such as rotational symmetry. This video only glosses over the mathematics, and don't worry if you don't exactly understand what is going on. The important part is to look at Julia Set being formed visually. I'm certainly not the first to use this technique, but there isn't much YouTube content, so I've included quite a few visuals, and everything is rendered at a super high 8k resolution. Consider playing the final visuals at slower speeds if you wish to study them. Please subscribe!If you loved this video then why not buy me a coffee: ko-fi.com/mathstownIn this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia SetThis page offered some inspiration for this video: karlsims.com/julia.html
Julia Sets, and how they relate to The Mandelbrot Set The Mathemagicians Guild 2020-04-01 | A little video introducing Julia Sets as a follow-up to the Mandelbrot Explained video. In this video we have a little look at the Julia Sets and where we find them embedded into the Mandelbrot Set. This was going to be a longer video, but I've broken it into two parts, in the next video we look into how to "build a Julia Set" using complex transformations. Please subscribe!Some terminology: In this video I refer to the set being "dust" because it is "infinitly disconnected", in more mathematical contexts you may see this referred to as "Cantor set" or "Fatou Dust".If you loved this video then why not buy me a coffee: ko-fi.com/mathstownIn this video:00:00 Introduction00:33 Julia Sets02:54 Types of Julia Sets03:57 Map of the Julia Sets05:18 Mandelbrot Set vs Julia Sets06:10 Embedded Julia Sets in the Mandelbrot Set07:18 The Mandelbrot Set Remembers11:55 A Julia Set ZoomIn this series:1 - youtu.be/7MotVcGvFMg The Mandelbrot Set Explained2 - youtu.be/dctJ7ISkU-4 Julia Sets, and how they relate to The Mandelbrot Set3 - youtu.be/5T0cC6KRezo How to Build a Julia Set4 - youtu.be/oNxPSP2tQEk Number Sequences in the Mandelbrot SetExtra Visuals (No commentary):youtu.be/JYlAPaPsoSY - All orbits of the Mandelbrotyoutu.be/-ObPTFNVByk - Period 1 orbits of the Mandelbrotyoutu.be/4y8OCqceJjo - Period 2 orbits of the Mandelbrotyoutu.be/Gmjk5G2TrSo - Building a Mandelbrot step-by-stepyoutu.be/0E9fT8hOkVQ - Period 2 orbits of a Julia Setyoutu.be/CYdKlN38VHE - Period 3 orbits of a Julia Setyoutu.be/XWboHruR6N8 - Period 4 orbits of a Julia Setyoutu.be/mt6PYJnnUHg - Period 5 orbits of a Julia Setyoutu.be/wxkuQdXcp70 - Period 6 orbits of a Julia SetIncludes content from:Music: audionautix.com11 Dimensions Video: youtube.com/watch?v=X-_LkF9V8AMAccidental Masterpiece: youtube.com/watch?v=dSA7OZHdaoA
