#mathmatrivia

[mathmatrivia]

So how big does the inner circle need to be in order to put those four points on one straight line?

It turns out to be about 38%. But this is math. We want to know exactly how much and we want to know how to find the answer!

Let's consider the bigger circle a unit circle, so R = 1. Then consider this triangle.

Ten equal sectors define the entire 360°, so they're 36° each. The center-touching angle of this triangle encompasses 2 of those sectors. Therefore, 72°.

Unfortunately, I have to use cosines now. Those who fear trigonometry may want to look away.

Cosine equals adjacent over hypotenuse. We defined the hypotenuse to be 1, because it's a unit circle. So the cosine of 72° is the height of that triangle divided by one. Or just the height of that triangle. h = cos 72°

Now lets look at this slightly different triangle.

That's only 1 out of 10 equal sectors that make up the entire circle. That's a 36° angle. And the cosine of that one is the height of the triangle (the same as the other triangle) divided by the radius of the inner circle — exactly we're trying to find! cos 36° = h / r But we know what h is. cos 36° = cos 72° / r Now we solve for r. r⋅ cos 36° = cos 72° r = cos 72° / cos 36°

That's the exact answer: cos 72° / cos 36°

If you throw it into a calculator, you'll get something like ≈ 0.3819660112510519363...

Like I said, about 38%. Or 38.197%, if you want to be pedantic. Which I hope you do.

And if you don't want your stars straight... that's fine, too. Every star is perfect for someone. ❤️🧡💛💚💙💜

[mathmatrivia]

This divide symbol ÷ is called an obelus. It was first used for mathematical division by Swiss Mathematician Johann Rahn in his 1659 book Teutsche Algebra (meaning "German Algebra"). Before that, it was a symbol used in the hand-copying of documents to indicate that some passage might need to be removed; the process was called obelism. The obelus is a common division sign in English, but can be unknown or mean different things in cultures that don't speak English.

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More information about:

Swiss Mathematician Johann Rahn https://mathshistory.st-andrews.ac.uk/Biographies/Rahn/

Hand-copying ancient texts: https://en.wikipedia.org/wiki/Obelism

[mathmatrivia]

How to do square roots on paper.

I also made an animation with pause and play buttons here. Reload that page to see it calculate the square root of a different pseudorandom number.

I had planned to make a slideshow that walked the audience through the process step by step, but it was over 20 slides, too much text, and just... a lot. If you want to see that, it's here.

[mathmatrivia]

A degenerate triangle occurs when all three vertices of a triangle lie on the same line. It is as if you took a plain old triangle and flattened it out.

When the sum of the two smaller side lengths is equal to the longest side length, it forms a degenerate triangle with angles 0°, 0°, and 180°.

But are they really triangles?

They have three sides. Check They have three angles. Check Do they really have three vertices? When one angle is 180°, does that really count as a vertex? They have an area of zero. That's better than undefined or indeterminate, but it's not a positive number. Does that count? The three angles sum to 180°. Check They fit on a plane. Check Triangles are supposed to be two dimensional shapes. Degenerate triangles only have one dimension.

When triangles have two equal angles, they also have two equal side lengths. That's what makes an isosceles triangle. But degenerate always have two angles of 0° whether they have two equal side lengths or not. They could even have two equal side lengths that are not opposite the two equal angles. That's not how isosceles triangles are supposed to work!