The Simplex of the Center

Calculus Level 3

If $n+2$ points are placed on the surface of a hypersphere, or $n$ -sphere, what equation expresses the odds that the $n$ -simplex formed by the points contains the center of the hypersphere?

f(x)=\frac{1}{2^{n+1}}

f(x)=\frac{π}{e^{n+1}}

f(x)=\frac{π}{2^{n+1}}

f(x)=\frac{π}{e^{n}}

f(x)=\frac{1}{e^{n}}

1 solution

Tyler Skywalker
Dec 9, 2017

This equation can be best visualized as 3 points on 1-sphere, or circle. Imagine that there are two diameters inside the circle. Now imagine that there is a random point, point 1, in one of the four sections of the circle, which were created with the imagination of the diameters. As for the other two points, they can each be visualized as an endpoint of one of the diameters. Since for each diameter there are 2 endpoints, there are $2^2$ , or $2^{n+1}$ , combinations of endpoints. For only one of these combinations, the triangle formed between the endpoints and point 1 will contain the center of the circle. That means that the probability of containing the center of the n-sphere is, in this case $\frac{1}{2^{2}}$ , but will always be $\frac{1}{2^{n+1}}$ .

Q.E.D.

Did you get this from 3Blue1Brown?

kb e - 3 years, 5 months ago

Yes, but I morphed the question a bit.

Tyler Skywalker - 3 years, 5 months ago

@Tyler Skywalker , I like your problem "Integral of a hot dog". It is very good :)

A Former Brilliant Member - 3 years, 5 months ago

Thank you!

Tyler Skywalker - 3 years, 5 months ago

The Simplex of the Center

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