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The familiar trigonometric functions can be geometrically derived from a circle.

**But what if, instead of the circle, we used a regular polygon?**

In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.

We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) **This is why the square does not trace a straight diagonal line**, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.

Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.

More on this subject and derivations of the functions can be found in this other post

**Now you can also listen to what these waves sound like.**

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

1ucasvb

Proof of the Pythagoras Theorem.

thatscienceguy

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mockinghays

kurkcantalimadonna

*Walking into finals…*

melancholic-wallflower

mathani

impatient recursion

bigblueboo

I just finished some new code to generate **arc-length parametrizations of arbitrary curves**. To try it out, I used it on this Archimedean spiral.

Then I got curious: **exactly how does the re-parametrization redistributes the points along this curve**? In the original parametrization, the points are bunched up in the middle of the spiral, and more spaced on the outside. The arc-length parametrization makes them equally spaced along the whole path. So how do they compare?

First, I tried this with black points, but it was too confusing. Same thing for a few dots highlighted. So I decided to color them all based on the angle in the original parametrization. This is the result.

It is really interesting how the colors are bent around. It seems that the distribution is quite non-uniform, even though the spiral is rather uniform in growth.

I originally rendered this with four times as many frames, but due to the amount of colors and dimensions of the GIF, Tumblr wouldn’t accept it. It was too large. Below is the animation with twice as many frames.

**Hint:** try squinting! It blurs the colors and it looks really trippy!

1ucasvb

Drawing three 90 degree angles in a curved space. One of the reasons we know the Earth is round.

Find this interesting? Then check out this “visual demonstration of a² + b² = c².”

educational-gifs

mathani

Everybody loves gif, so I found another one. Torus.

aoozdemir

44. 4. *(line-square-cube-tesseract via folding)*

bigblueboo

Made this while working on some cyberpunx ui - transparent.

netgrind

One of the answers to the topic: *Visually stunning math concepts which are easy to explain* at **Mathematics Stack Exchange**.

I think if you look at this animation and think about it long enough, you’ll understand:

- Why circles and right-angle triangles and angles are all related
- Why sine is opposite over hypotenuse and so on
- Why cosine is simply sine but offset by pi/2 radians

scienceisbeauty

mathani

Another animation of the fractal Harriss spiral. This exploits the plastic number (1.3247…) a number whose cube is itself plus one. This number is considered by some to be the forgotten cousin of the golden ratio (whose square is itself plus one).

matthen

Visualization of the real part of a Gaussian wave packet in three dimensions. Adapted from the code of the brilliant Bernd Thaller!

quantizedconfusion

A visual demonstration of a² + b² = c² using liquid.

educational-gifs