# LOG#069. CP(n), spheres, 1836. Data:  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

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) And finally, two more…The 4096-dimensional sphere

(23) 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:

(24) 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?

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