Calculus in hyperdimensional geometry

Here's an intuitive illustration of why the volume of the n-hypersphere of radius $1$ approaches value of $0$ as $n$ increases. Let's say $n=100$ . Then for a 100-hypersphere of radius, all coordinates inside the sphere satisfy the relationship

$\displaystyle \sum _{ k=1 }^{ 100 }{ { a }_{ k }^{ 2 } } \le 1$

where $\left| { a }_{ k } \right| \le 1$ . To make this a little easier to see, let $\left| { a }_{ k } \right| \le 100$ be integers, so that we have

$\displaystyle \sum _{ k=1 }^{ 100 }{ {\left(\dfrac {{ a }_{ k }}{100} \right) }^{ 2 } } \le 1$

or, after multiplying both sides by $100$

$\displaystyle \sum _{ k=1 }^{ 100 }{ { a }_{ k }^{ 2 } } \le { 100 }^{ 2 }$

which means on average

${a_k}^{2} \le 100$ , or $\left| { a }_{ k } \right| \le 10$ , out of a possible range $\left| { a }_{ k } \right| \le100$

As $n$ increases, the range for ${a_k}$ as a fraction of the whole drops down to $0$