Look Who it Was Random video

Running Circles Around Circles: Part 4: Irrational Hypocycloids

614 views · Published 19 January 2017 · 16:04 · Indexed 22 September 2026

Channel: Math Easy Solutions · 2017 · Education

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In this video I go over Part 4 of the Laboratory Project titled Running Circles Around Circles and in this part I deal with hypocycloids in which the outer circle has a radius that is irrational. I test out how the curves change for the radius being equal to the irrational number 2^(1/2) and e - 2. I use the amazing Desmos graphing calculator again to plot out these parametric equations, and play around with the curves for larger and larger values of the parameter θ. For irrational radii less than 1, we obtain hypocycloids with cusps that are on the "inside ring" of the outer circle. For greater than 1 we get hypocycloids with cusps on the "outside ring".

Also, I show how the curves change when we have simple, rational, radii such as 3 but when as we increase the number of decimal place values, such as 3.1, 3.12, 3.123, etc., we get increasingly more complex hypocycloids. But when ever there is a pattern, such as 3.121212, the curve also obtains a pattern that repeats itself. But for irrational numbers, as we get larger and larger ranges of values for θ we eventually will obtain a parametric curve that has points arbitrarily close to all other points, thus looking like a solid ring or washer.

This is one of the coolest and most amazing videos on the power of parametric equations and curves to graph not only amazing shapes and curves, but also as a way to graph seemingly solid filled shapes. So make sure to watch this  video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuFxSw6viMdEkL657g

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/running-circles-around-circles-part-4-irrational-hypocycloids

Related Videos: 

Laboratory Project: Running Circles Around Circles: Part 3: https://youtu.be/uCdvK29T5Z0
Laboratory Project: Running Circles Around Circles: Part 2: https://youtu.be/-hCw2HJF0YQ
Laboratory Project: Running Circles Around Circles: Part 1: https://youtu.be/MiCqDNQN7_E
Parametric Curves: Superellipses: https://youtu.be/JqkjQjtenPQ
Parametric Curves: Example 11: Conchoids of Nicomedes: https://youtu.be/Ir1tVdwGWec
Parametric Curves: Example 7: The Cycloid: Proof Part 1: https://youtu.be/XmHpIZqeRKk
Parametric Curves: Example 6: Graphing Devices: https://youtu.be/cwpEasc9mhQ
Parametric Curves: Example 5: Lissajous Figure: https://youtu.be/0OT1OnnAQ9I
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers .

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