A circle inscribed in a square covers 79% of the square.
A ball inscribed in a cube fills 52% of the cube.
A 4-ball inscribed in a hypercube fills 31% of the hypercube.
A 5-ball inscribed in a 5-cube fills 16% of the 5-cube...
A 9-ball inscribed in a 9-cube fills it up less than 1% of the way. Yet there's no room to fit a second same sized ball without intersecting the first - which contradicts my intuition of balls as especially compact shapes.
In dimensions 23 and up you can fit a little cube in the corner of the diagram that has a larger volume than the inscribed ball! Which is especially disconcerting, because the 23-cube has 8,388,608 corners. And even if you inscribe the little cubes in all eight million corners, your big cube will still be 99.5% empty space.
batshit-birds
