Hypersport
Perhaps rather more attention is given to polychora than they deserve. On
account of the fascinating networks of them which can be constructed. No
doubt they possess a symmetry which might startle the horses; but, for sheer
simplicity, and unbelievable *roundness* - and mystique -give me the
unbeatable glome.
For unit radius, the surface of a circle is 2 pi or 6.28. Rising to 4 pi or 12.57
for a sphere, 2 pi^2 or 19.74 for a glome, and a maximum of 16 pi^3/15 or
33.07 in the 7th dimension or heptaspace.
After which it actually decreases for successive dimensions. The content of
the ball performs similarly; rising to a maximum of 8 pi^2/15 or 5.26 units in
5 dimensions, then decreasing. All of which is well-known.
What is more obvious but less well-known, is the increasing corner-wastage
of these balls as a result of their progressive roundness.
For a circle in a square, this is 21.46%; for a sphere in a cube, 47.64%; for a
glome in a tetracube, 69.16%; a pentaglome in a pentacube, 83.55%.
And by the 11th dimension, a hendecaglome in a hendecacube, 99.91%.
This means that if an 11-ball of diameter 1 is exactly fitted in the smallest
possible hypercubical box, the hypervolume of which is L^11 or 1 unit for
unit side, touching all its 22 decacube sides at their exact centres, its content
will be 64 pi^5/10395. And although being the largest ball which can be
fitted into the box, it will occupy less than a thousandth of it.
The moral would seem to be: if travelling in the 11th dimension, think
carefully before posting home a basket-ball. Because the corners of your
parcel (all 2048 of them) are going to be awfully empty. Not to mention the
colossal amount of red tape involved.
Simeon