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 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.
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?
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?
What could the exact value be?
Pick uniformly distributed random points in a 6×6 square. The average distance from the centre is
√2 + log(1+√2) = 2.29558...
It's the universal parabolic constant! How crazy is that‽ Where is the parabola in this problem about random points in a square???
√2 + log(1+√2) = 2.29558...
It's the universal parabolic constant! How crazy is that‽ Where is the parabola in this problem about random points in a square???
It could also be a coincidence. Either way, I think it's a nice fact.
Here is another way of looking at this:
Can you formulate the probability density function of distances of randomly selected points to the centre?
What is the probability P(d) of a point being at distance d from the centre?
Can you formulate the probability density function of distances of randomly selected points to the centre?
What is the probability P(d) of a point being at distance d from the centre?
If you were traumatized by the Matt Parker video above, you should watch @TobyHendy's Bob Ross style introduction to parabolas. It's very soothing 🤗
@TobyHendy Returning to the probability question:
The probability of being at distance d from the centre is proportional to the length of arc(s) of radius d that's inside the square. To get probabilities, one needs to normalize by 36, the area of the square.
The probability of being at distance d from the centre is proportional to the length of arc(s) of radius d that's inside the square. To get probabilities, one needs to normalize by 36, the area of the square.
Now, to get the expected value, all you need to do is calculate the definite integral of d×P(d) over the interval [0,3√2].
The results is, of course, the universal parabolic constant.
The results is, of course, the universal parabolic constant.
A final surprise! We can connect e, π, and the universal parabolic constant.
Plot the function y=exp(-x) over the positive x-axis and rotate the graph about the x-axis. The surface you get has area π × the parabolic constant.
Plot the function y=exp(-x) over the positive x-axis and rotate the graph about the x-axis. The surface you get has area π × the parabolic constant.
A nice animation of the similarity of parabolas.
