# LOG#069. CP(n), spheres, 1836.

Data:

$V(CP^n)=\dfrac{\pi^n}{n!}$

$V(S^{d-1})=\dfrac{2\pi^{d/2}}{\Gamma (d/2)}$

where we take the radius of the sphere equal to 1 without loss of generality.

Thus, $6\pi^5$ is 6!=720 times the volume of the complex projective space $CP^5$. 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 ($d'=d+n$ and $d>d'$) is equal to:

$\dfrac{V(S^{d-1})}{V(S^{d'-1})}=\dfrac{\pi^{(d-d')/2}\Gamma (d'/2)}{\Gamma (d/2)}=\dfrac{\pi^{n/2}\Gamma ((d-n)/2)}{\Gamma (d/2)}=C(d,n) \pi^{n/2}$

We want $\pi^5$, so we fix $n=10$ there:

$\dfrac{\pi^{10/2}\Gamma ((d-10)/2)}{\Gamma (d/2)}=C(d,10) \pi^{5}\equiv k(d)\pi^5$

and where we have defined the dimensional dependent coefficients (note that $d=d'+10$, and $d-1=d'+9$, or $d'-1=d-11$)

$k(d)=\dfrac{\Gamma ((d-10)/2)}{\Gamma (d/2)}=\dfrac{32}{(d-10)(d-8)(d-6)(d-4)(d-2)}=\dfrac{V(S^{d-1})}{V(S^{d'-1})}$

$k(d)=\dfrac{V(S^{d-1})}{V(S^{d-11})}$

I can obtain some numbers very easily:

$k(11)=\dfrac{V(S^{10})}{V(S^{0})}=\dfrac{32}{1\cdot 3\cdot 5\cdot 7 \cdot 9}$

$k(12)=\dfrac{V(S^{11})}{V(S^{1})}=\dfrac{1}{12^2}$

$k(13)=\dfrac{V(S^{12})}{V(S^{2})}=\dfrac{32}{3\cdot 5\cdot 7 \cdot 9\cdot 11}$

$k(14)=\dfrac{V(S^{13})}{V(S^{3})}=\dfrac{1}{6\cdot 12\cdot 14 }$

$k(15)=\dfrac{V(S^{14})}{V(S^{4})}=\dfrac{32}{5\cdot 7\cdot 9 \cdot 11\cdot 13}$

$k(16)=\dfrac{V(S^{15})}{V(S^{5})}=\dfrac{1}{12\cdot 14\cdot 15}$

$k(17)=\dfrac{V(S^{16})}{V(S^{6})}=\dfrac{32}{7\cdot 9\cdot 11 \cdot 13\cdot 15}$

$k(18)=\dfrac{V(S^{17})}{V(S^{7})}=\dfrac{1}{ 4\cdot 10 \cdot 12\cdot 14}$

I stop here since $S^7$ is the last parallelizable sphere. We get

$6\pi^5\approx 1836\approx \dfrac{m_p}{m_e}=\dfrac{\mbox{Proton (rest) mass}}{\mbox{Electron (rest) mass}}$

as the ratio between the volumes of the following spheres:

1) $\dfrac{6\cdot 945}{32}$ times the ratio of the 10-sphere and the 0-sphere volumes, k(11).

2) $6\cdot 12^2$  times the ratio of 11-sphere and the 1-sphere (the circle) volumes, k(12).

3) $\dfrac{6\cdot 10395}{32}$  times the ratio of 12-sphere and the 2-sphere  volumes, k(13).

4) $6^2\cdot 12\cdot 14$ times the ratio of 13-sphere and the 3-sphere volumes, k(14).

5) $\dfrac{6\cdot 45045}{32}$  times the ratio of 14-sphere and the 4-sphere  volumes, k(15).

6) $6\cdot 2520$  times the ratio of 15-sphere and the 5-sphere  volumes, k(16).

7) $\dfrac{6\cdot 135135 }{32}$  times the ratio of 16-sphere and the 6-sphere volumes, k(17).

8) $6\cdot 6720$  times the ratio of 17-sphere and the 7-sphere volumes, k(18).

PS: Made by hand and the only use of my brains and head. No calculators, no computers assisted me in the calculations. I am obsolete, amn’t I?