Everyone knows that all circles are similar. But did you know that all parabolas are similar?

The ratio of the red parabolic arc length and the blue focal parameter is
√2 + log(1+√2) = 2.29558...
for any parabola. This is the universal parabolic constant, the “π of parabolas”.

The fact that “There is only one true parabola” is shown in this @standupmaths video the last 20 seconds of which is pure nightmare fuel

The universal parabolic constant P = √2 + log(1+√2) is a transcendental number.

Proof. If P was algebraic, then so would P–√2 = log(1+√2) and by the Lindemann-Weierstrass theorem exp(P–√2) = 1+√2 would be transcendental, which it isn't.

And I haven’t even mentioned the craziest connection yet... currently rendering a video demonstrating that.. watch this space

We're randomly dropping points in a 6×6 square with uniform distribution. What's the average distance from the centre?

Corners are furthest from the centre at a distance of 3√2 ≈ 4.2426, so the average must be somewhere between 0 and 3√2. But what is it?

I got 2.33 for the average distance from the centre by dropping 1,000 random points into a 6×6 square.

What could the exact value be?