where we take the radius of the sphere equal to 1 without loss of generality.
Thus, is 6!=720 times the volume of the complex projective space . I have not found any other (simpler) way to remember that number ( it is interesting since it is really close to the proton to electron mass ratio).
We can also find that number with ratios between (hyper)-spherical volumes. There are many (indeed infinite) solutions. The ratio between the volume of a d-sphere and a d’-sphere ( and ) is equal to:
We want , so we fix there:
and where we have defined the dimensional dependent coefficients (note that we can write , and , or )
I can obtain some numbers very easily:
I stop here since is the last parallelizable sphere. We get
as the ratio between the volumes of the following spheres:
1) times the ratio of the 10-sphere and the 0-sphere volumes, k(11).
2) times the ratio of 11-sphere and the 1-sphere (the circle) volumes, k(12).
3) times the ratio of 12-sphere and the 2-sphere volumes, k(13).
4) times the ratio of 13-sphere and the 3-sphere volumes, k(14).
5) times the ratio of 14-sphere and the 4-sphere volumes, k(15).
6) times the ratio of 15-sphere and the 5-sphere volumes, k(16).
7) times the ratio of 16-sphere and the 6-sphere volumes, k(17).
8) times the ratio of 17-sphere and the 7-sphere volumes, k(18).
9) , 6! times the volume of the 5D complex projective space .
Addendum: Nice sphere volumes are
And finally, two more…The 4096-dimensional sphere
with a fantastic fraction that can not be written in the margin or space of this page easily. The final one, surprisingly, the infinite-dimensional sphere volume is zero:
The amazing vanishing sphere volume with increasing dimensions!!!!!!!!!!
PS: Made by hand and the only use of my brains and head. No common calculators but only software helped me. I am obsolete, amn’t I?