LOG#101. Hyperstuff: the list.


Hello, world!

This short post is a list with my favourite hyperstuff related to Physmatics. It also begins a new subcategory of (generally short) posts that I have called “Lists” where I will be writing enumerative lists about some “stuff” I love…

1) Hyperpolygons ( a.k.a. polytopes).

2) Hypercubes.

3) Hyperplanes.

4) Hyperspheres.

5) Hyperlogarithms.

6) Hyperdeterminants.

7) Hypermatrices.

8) Hypergraphs.

9) Hypernumbers.

10) Hypersymmetry ( ternary and generalized n-ary supersymmetries, Clifford algebras, etc).

11) Hyperspace.

12) Hypertime.

13) Hypernuclei.

14) Hyperons.

15) Hyperdrive.

16) Hyperhydrogen atoms.

17) Hypercharge.

18) Hyperquarks.

19) Hypercolor.

20) Hyperelliptic functions and integrals.

21) Hyperlink.

Do you know some interesting hyperstuff that you would add to this list?

See you in another blog post!


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?