LOG#069. CP(n), spheres, 1836.

Proton Electron Neutron



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})}


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(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?


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