The 3-4-7 miracle. Why is this one not super famous?

The winner of Marty and my book Putting Two and Two together is Alexander Svorre Jordan. Congratulations. :) Thank you again to everybody who submitted an implementation of the dance. Here are five particularly noteworthy submissions: (Kieran Clancy) kieranclancy.github.io/star-animation/ (this was the very first submission submitted in record time :) (Liam Applebe) tiusic.com/magic_star_anim.html (an early submission that automatically does the whole dance for any choice of parameters) (Pierre Lancien) lab.toxicode.fr/spirograph/ (with geared circles) (Christopher Gallegos) gallegosaudio.com/MathologerStars (very slick interface) (Matthew Arcus) www.shadertoy.com/view/7tKXWy (implements the fact that BOTH types of rotating polygons are parts of circles rolling around DIFFERENT large circles) Here is the complete list in the order that received them :) (Kieran Clancy) kieranclancy.github.io/star-animation/ (Richard Copley) bustercopley.github.io/star/ (Liam Applebe) tiusic.com/magic_star_anim.html (Nan Ma) observablehq.com/@nanma80/rotating-dots-along-star… (Owen Bechtel) owenbechtel.com/mathologer-animation.html (Morten Barklund) spirograph.live/ (Chris du Plessis) www.desmos.com/calculator/1pf7ftgo8i (r57shell) www.desmos.com/calculator/h01p97im0o (Adam Zimny) github.com/AdamZimny349/StarAnimationManim (Gary Au) www.desmos.com/calculator/wcdzino9pa/ (Tamir Daniely) jsfiddle.net/tdaniely/ojcgydze/595 (Alexander Svorre Jordan) alex6480.github.io/MathologerCircles/ (Niyoko Yuliawan) 347-challenge.niyoko.id/ (Gijs Schröder) www.shadertoy.com/view/slKXRy (Azai) www.shadertoy.com/view/NtKXR3 (Hall Holden) editor.p5js.org/pi3point14/full/1aRnJUX9S (Eric Kaiser) misc.eric-kaiser.net/starograph (Nahuel Fouilleul) nahuelfouilleul.free.fr/programmation%20jeux/hypot… (Pierre Lancien) lab.toxicode.fr/spirograph/ (Anton Älgmyr) www.algmyr.se/347/ (Lars Christensen) larsch.github.io/dancing-polygons/ (Ivan Sorokin) sorokin.github.io/littlemiracle/littlemiracle.html (Robin Lasne) twelveothirteen.itch.io/mathologer-spirograph-chal… (Christopher Gallegos) gallegosaudio.com/MathologerStars (Matthew Arcus) www.shadertoy.com/view/7tKXWy Other submissions without online implementations by (Joseph McGowan) github.com/Joetrahedron/star-animation (Octavia Togami) github.com/octylFractal/StellatedRoller (Jérémie Marquès) github.com/Joetrahedron/star-animation (Emmanuel Pinto) github.com/EmmanuelPinto/manim-3-4-7 Stefan Muntean, Qubix (Tim), Daniel Wilckens

There is a special place in heaven for people like you who make math accessible and fun, who put the intermediate logical steps in their demonstrations, thank you.

If I had a math teacher like you instead of the monstrous bully in Gr 9 I'd probably be a retired engineer today having worked on super-projects. You're an inspiration.

My solution to the coin paradox: imagine the two coins are like gears with the centers fixed in space. Both turn once every time the other one does. Then imagine the camera rotates with one of them, so it looks fixed. This feels more intuitive since nothing "rolls away" or gets wound around.

Once you add both the triangles and squares, it looks like a 4D projection.

The off-by-one counter-intuitive logic reminded me of (and is clearly related to) how there are only 11 places on the face of a 12-hour clock where the hands point the same direction. One of my favourite weird facts is that if either one of the two hands on that clock went backwards (what a strange clock that would be!) the hands would point in the same direction in 13 places instead.

I started playing with Meccano gears when I was 7 and rapidly discovered the anomalies that seem to happen when one gear rotates around another. It really bugged me til I got it figured out. Later, during my engineering degree, I couldn't understand why the lecturer made such a meal of it all, with complicated formulas, when to me it was (by now) all so obvious but then he lacked your clear insight and pretty animations. Many years ago Meccano issued a striking clock kit. A striking mechanism requires a wheel that goes round once in 12 hours in 1+2+3...+12=78 steps, one for each hour strike. Awkwardly 78=6*13. They had to use a 12:1 geared ratio with 13 given by mounting the 12:1 gears to go round with the big wheel epicyclically. For more puzzlement tie a loop of string into a trefoil knot and drop it over a post and discover that the string only goes around the post twice despite the three loops. Or count the number of times on a clock, the minute hand crosses the hour hand in 12 hours.

A Parker heptagon!

8:07 If you can make a 12 pointed star with this, you can design a CLOCK with this fancy movement!

The amount of effort you put into making these animations are truly astonishing.

Simple solution for the sum of 7-star angles: 1. Without loss of generality, assume the 7 points lies on a circle 2. Those 7 angles are produced by 7 unique arcs that form a whole circle 3. For a certain arc, the angle at centre produce by that arc is always twice of the angle at circumference (Theorem "Angle at centre is twice angle at circumference") 4. Therefore, twice of the sum of all angles is 360 degrees. 5. Sum of required angles = 180 degrees

3,4,7 is actually really useful in music. Intervals of 3 and 4 are periodic complements in mod 12, because they also function as the smallest interval in basic trichords, intervals of 7 are the most common. I often visualize the structure of chords in a way similar to the graphic representation used in this video. The triangles and squares form a continuum that you can rotate along to find relatively valent harmonic structures in a set of closely related keys.

This is why I love your channel, a machine gun of “Aha!” moments makes me feel like Baldrick with a cunning idea. Love it!

Fascinating and entertaining... and it completely took my mind off of the troubles in the world for the duration. That fact in itself made it worth the 23 minutes spent watching. thank you.

6:32 The beauty of the movement looks like a 4-dimensional hyper cube / Tesseract. So many great videos on this channel. Thank You!

Your video is a treasure. I hope it is incorporated in early STEM learning. My experience watching, caused me dynamic insights in three separate domains. In my opinion, Mathologer is a "miracle" math educator.

this animation looks like a 4d shape of some kind i know it probably isn't one but it's still cool

The sum of angles: Imagine a point with a nose walking along the edges of the star. In every corner, it will rotate 180°-a, where a is any angle of the star. If you carefully watch its nosr, it does three rotations, so 360°×3 or 180°×6. We get (180°-a)×7=180°×6, so the sum of angles is equal to 7a, which from the equation is 180°

God, those spirograph animations are incredibly beautiful, the dancing cubes makes me think of hypercubes and the potential of seeing higher dimensions in this kids toy.

Coin paradox blew my mind. Great video, great visual and explanations.