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

I got sidetracked again by a puzzling little mathematical miracle. And, as usual, I could not help myself and just had to figure it out. Here is the result of my efforts.

00:00 Intro
08:45 The Coin rotation paradox
16:00 Mystery number explanation
18:38 Challenges and the new book
19:53 One-minute animation on how to figure the sum of the angles in a star
21:04 Thank you :)

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:

Some fun and helpful links.

The animation in geogebra (by Juan Carlos Ponce Campuzano):

The new book by Marty and me:

New short videos on Mathologer 2:

New Mathologer instagram account:

Hypotrochoids:

Spirograph:
(a nice app)

Coin rotation paradox

Tusi couple:

Funfair Twister ride:

Today's music is Altitude by Muted.

Enjoy!

Burkard

[Mathologer]

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:

Here is the complete list in the order that received them :)

Other submissions without online implementations by
Stefan Muntean, Qubix (Tim), Daniel Wilckens

[Cryguy]

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.

[freon]

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.

[DanBurgaud]

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

[donaldasayers]

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.

[lagomoof]

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.

[jimm]

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.

[Tara_Li]

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

[yekhantlu]

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

[hinsxd]

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

[Mr.-Good]

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

[bobjeaniejoey]

4:30 to 4:45 - Mathologer demonstrating his acceptance of less than precision.
How human!
I love watching and listening to you, Mathologer!

P.S. - That animation jumps out to me in 3D when the shapes containing more than 3 points are being shown!

[wintrsmith]

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

[Saka_Mulia]

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

[SKYGUY]

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.

[rpmendez]

The discussion around the 2:37 mark sets up a lovely proof that the vertex-to-vertex segments radiating from a single vertex in a regular polygon divide the associated interior angle evenly.

[marsegan]

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.

[bot]

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°

[avoirdupois]

I really enjoy watching your enthusiastic explantation of these concepts. The animations are beautiful.

[nealmcb]

I love the beautiful sequence of adding elements starting at @6:40