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

Proton Electron Neutron


    \[ V(CP^n)=\dfrac{\pi^n}{n!}\]

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

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 l \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 we can write 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).

9) 6! \cdot V(CP^5), 6! times the volume of the 5D complex projective space CP^5.

Addendum: Nice sphere volumes are

(1)   \begin{equation*}V(S^0)=2 R\end{equation*}

(2)   \begin{equation*}V(S^1)=\pi R^2\approx 3.14159 R^2\end{equation*}

(3)   \begin{equation*}V(S^2)=\dfrac{4}{3}\pi R^3\approx 4.11879\end{equation*}

(4)   \begin{equation*}V(S^3)=\dfrac{\pi^2}{2} R^4\approx 4.9348R^4\end{equation*}

(5)   \begin{equation*}V(S^4)=\dfrac{8\pi^2}{15} R^5\approx 5.26379R^5\end{equation*}

(6)   \begin{equation*}V(S^5)=\dfrac{\pi^3}{6} R^6\approx 5.16771R^6\end{equation*}

(7)   \begin{equation*}V(S^6)=\dfrac{16\pi^3}{105} R^7\approx 4.72477R^7\end{equation*}

(8)   \begin{equation*}V(S^7)=\dfrac{\pi^4}{24} R^8\approx 4.05871R^8\end{equation*}

(9)   \begin{equation*}V(S^8)=\dfrac{32\pi^4}{945} R^9\approx 3.29851R^9\end{equation*}

(10)   \begin{equation*}V(S^9)=\dfrac{\pi^5}{120} R^{10}\approx 2.55016R^{10}\end{equation*}

(11)   \begin{equation*}V(S^{10})=\dfrac{64\pi^5}{10395} R^{11}\approx 1.8841R^{11}\end{equation*}

(12)   \begin{equation*}V(S^{11})=\dfrac{\pi^6}{720} R^{12}\approx 1.33526R^{12}\end{equation*}

(13)   \begin{equation*}V(S^{12})=\dfrac{128\pi^6}{135135} R^{13}\approx 0.919629R^{13}\end{equation*}

(14)   \begin{equation*}V(S^{13})=\dfrac{\pi^7}{5040}R^{14}\approx 0.599265R^{14}\end{equation*}

(15)   \begin{equation*}V(S^{14})=\dfrac{256\pi^7}{2027025} R^{15}\approx 0.381443R^{15}\end{equation*}

(16)   \begin{equation*}V(S^{15})=\dfrac{\pi^8}{40320} R^{16}\approx 0.235331R^{16}\end{equation*}

(17)   \begin{equation*}V(S^{16})=\dfrac{512\pi^8}{34459425} R^{17}\approx 0.140981R^{17}\end{equation*}

(18)   \begin{equation*}V(S^{23})=\dfrac{\pi^{12}}{479001600} R^{24}\approx 0.00192957R^{24}\end{equation*}

(19)   \begin{equation*}V(S^{24})=\dfrac{8192\pi^{12}}{7905853580625} R^{25}\approx 0.000957722 R^{25}\end{equation*}

(20)   \begin{equation*}V(S^{25})=\dfrac{\pi^{13}}{6227020800} R^{26}\approx 0.000466303R^{26}\end{equation*}

(21)   \begin{equation*}V(S^{26})=\dfrac{16384\pi^{13}}{213458046676875} R^{27}\approx 0.000222872R^{27}\end{equation*}

(22)   \begin{equation*}V(S^{91})=\begin{cases}\dfrac{\pi^{46}R^{92}} {5502622159812088949850305428800254892961651752960000000000} \\ \; \\ \approx 1.34377\cdot 10^{-35} R^{92}\end{cases}\end{equation*}

And finally, two more…The 4096-dimensional sphere

(23)   \begin{equation*}V(S^{4095})\approx 8.70008138919055\times 10^{-4877} R^{4096}\end{equation*}

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)   \begin{equation*}V(S^\infty)=0\end{equation*}

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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