Have you ever heard that the unit n-ball has maximal volume at dimension n = 5? This raises a natural question: what makes the fifth dimension so special? Why does the volume of the n-ball grow with radius until the fifth dimension, and then decrease to zero?

This is because the radius is NOT the natural way to describe an n-sphere. The diameter is. No, this isn't a pithy tau vs. pi debate, it's a very real issue. The n-sphere is contained within an n-cube for any dimension n. The volume of the unit n-cube is unitary at all dimensions.

Imagine a circle in a square. There are four corners of empty space. A sphere inside a cube has 8 corners of empty space. And so on. This tells us the n-sphere, as n grows, takes up less and less space within the cube (in other words, there is more and more empty space in the n-cube). This implies for arbitrarily large n, the volume of the n-sphere goes to zero.

However it doesn't explain the spooky fifth dimension maximal volume for unit radius. The real problem is the argument that the volume of the n-sphere GROWS up until the fifth dimension (or even for non-unitary radius, the volume of the n-ball grows until it hits a peak, and then decreases).

The formula for the n-ball should be written with the diameter. This is because the n-sphere is contained within the n-cube, and it is the diameter of the n-sphere that connects the edges of the n-sphere to the n-cube edges.

When you consider a "unit-sphere" as one of unit diameter (to match the unit-length of the n-cube). you find the that the volume of the n-sphere is monotone decreasing as expected since it is contained within the cube (with the arguments provided earlier).

tl;dr: radius is unnatural descriptor for a sphere; the diameter is the natural measurement for it. A unit n-ball is one with diameter = 1.

>>16722760 (OP)
>what makes the fifth dimension so special
There are a gazillion bazillion special dimensions. Let’s stick to real vector spaces.
>1
the only dimension where puncturing it yields a disconnected space
>2
The only dimension where the special orthogonal group is abelian. The only dimension where a punctured space has a non-trivial fundamental group.
>3
The only dimension with a well-behaved cross product of vectors
>4
The only dimension in uncountably many smooth atlases on it
>7
Has a weird cross product like structure governed by the G2 exceptional Lie group
>10 dimensions
the special orthogonal group in 10 dims is very “special” in that the Dynkin diagram has a ton of symmetry, which yields particularly nice representations

>>16722760 (OP)
well said, i suppose we should rename it to the unimeter

>>16722784
>There are a gazillion bazillion special dimensions. Let’s stick to real vector spaces.
not until the string theorists themselves put away their own multi verse theories but simultaneously disregard maxwells own gripes on the matter

>>16722760 (OP)
Appreciate a thread that is not about quackery for once. Do you mean the diameter as the largest distance between points on the sphere?