Roger Bagula
2008-11-02 18:47:08 UTC
The first model is based on the idea that
tilings in an hyperbolic space behave like Gamma 1 modular groups
as triangulations ( polygonal tilings in hyperbolic spaces):
This model is based on an Escher like Gamma modular form
hyperbolic tiling of E_11 as an extended Platonic solid.
The model is a fractal scaling one in which the vertex number
of the hyperbolic polygons are used to both radially scale
and number the particles that time a limited quantum mass state.
It is a lattice in 2d space that can be mapped to a 3d manifold.
4, 6, 8, 12,20 are the vertex numbers of the platonic solids:
my v[n] function is a generalization of that.
I use a radius ratio function of powers of vertex minus one time Pi.
The quantum radius is taken as:
r=2*Pi*hbar/(m*c)
The temperature is the measured background temperature:
and the Hawking entropy formula is used to determine a black hole
radiator mass
at that temperature.
The number of particles in the model are the count of
hyperbolic tiles at different levels of the fractal scaling.
The finding is that the level n=8 accounts for an E_11
model hyperbolic big bang with the temperature
that is observed.
The level n=8 particle number is;
1239964645920142118597441
or about 10^24
and the particle mass at that level is;
1.083432147883765*10^12/1239964645920142118597441=8.737605152280621*10^-13
gm
which is less than the classical 10^(-5) Planck mass.
Mathematica:
Clear[v, r, n, num, m0]
(* Vertices exceptional groups*)
v[n_] := 2*(If[n == 0, 0, 2^(n - 1)] + 2)
Table[v[n], {n, 0, 30}]
{4, 6, 8, 12,20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388,
32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308,
8388612, 16777220, 33554436, 67108868,
134217732, 268435460, 536870916, 1073741828}
(*Hyperbolic tiling numbers for E - 11*)
Table[Sum[If [n == 0, 1, 1027^(n - 1)*1028], {n, 0, m}], {m, 0, 10}]
{1, 1029, 1056785, 1085318197, 1114621788321, 1144716576605669,
1175623924174022065, 1207365770126720660757, 1239964645920142118597441,
1273443691359985955799571909, 1307826671026705576606160350545}
num[m_] := Sum[If [n == 0, 1, 1027^(n - 1)*1028], {n, 0, m}]
(* relative E_11 fractal/ tiling radius*)
r[n_] := 1/(Pi*(v[10] - 1)^n)
(* comic back ground radiation temperature as Hawking radiation*)
T = 2.725;
c = 2.997925*10^10;
hbar = 1.0545919*10^(-27);
G = 6.67328*10^(-8);
kb = 1.380622810^(-16);
(* Hawking black hole mass by entropy temperature*)
m0 = c^3*hbar/(8*G*kb*Pi*T)
1.083432147883765`*^12
(* mass at each tiling level for numbers of particles*)
m[k_] = m0/num[k];
Table[m[n]*r[n]/(2*Pi*hbar/c), {n, 0, 10}]
{1.5603007215725785`*^48, 1.476462737924984`*^42,
1.3998476799928784`*^36, 1.3272107597432808`*^30,
1.2583429105873632`*^24, 1.1930485561574218`*^18, 1.1311422708178325`*^12,
1.0724482505153765`*^6, 1.0167998135211762`, 9.64038927080962`*^-7,
9.140157586270618`*^-13}
p = m[8]*r[8]/(2*Pi*hbar/c)
1.0167998135211762`
(* alpha residue of first order moment *)
(p - 1)*137.03608
2.302180589672985`
The second model uses a Tetrahedron as the basic figure:
This model using a four vertex hyperbolic uses
50 levels instead of 8...
(Nature usually shows 7 levels above one.).
Gauge mass is about the same
at
7.545864220675408`*^-13 gm
but there is a 3/2
factor necessary to get the quantum level near one.
The tetrahedrons are
actually scale slower than the larger symmetry of E_11.
Mathematica;
In[49]:=
Clear[v,r,n,num,m0]
(* Vertices exceptional groups*)
v[n_]:=2*(If[n==0,0,2^(n-1)]+2)
Table[v[n],{n,0,30}]
(*Hyperbolic tiling numbers for E_6*)
Table[Sum[If [n==0,1,3^(n-1)*4],{n,0,m}],{m,0,50}]
num[m_]:=Sum[If [n==0,1,3^(n-1)*4],{n,0,m}]
(* relative E_11 fractal/ tiling radius*)
r[n_]:=1/(Pi*(v[0]-1)^n)
(* comic back ground radiation temperature as Hawking radiation*)
T=2.725;
c=2.997925*10^10;
hbar=1.0545919*10^(-27);
G=6.67328*10^(-8);
kb=1.380622810^(-16);
(* Hawking black hole mass by entropy temperature*)
m0=c^3*hbar/(8*G*kb*Pi*T)
(* mass at each tiling level for numbers of particles*)
m[k_]=m0/num[k];
Out[51]=
{4,6,8,12,20,36,68,132,260,516,1028,2052,4100,8196,16388,32772,65540,131076,\
262148,524292,1048580,2097156,4194308,8388612,16777220,33554436,67108868,\
134217732,268435460,536870916,1073741828}
Out[52]=
{1,5,17,53,161,485,1457,4373,13121,39365,118097,354293,1062881,3188645,\
9565937,28697813,86093441,258280325,774840977,2324522933,6973568801,\
20920706405,62762119217,188286357653,564859072961,1694577218885,5083731656657,\
15251194969973,45753584909921,137260754729765,411782264189297,\
1235346792567893,3706040377703681,11118121133111045,33354363399333137,\
100063090197999413,300189270593998241,900567811781994725,2701703435345984177,\
8105110306037952533,24315330918113857601,72945992754341572805,\
218837978263024718417,656513934789074155253,1969541804367222465761,\
5908625413101667397285,17725876239305002191857,53177628717915006575573,\
159532886153745019726721,478598658461235059180165,1435795975383705177540497}
Out[60]=
\!\(1.083432147883765`*^12\)
In[34]:=
Table[m[n]*r[n]/(2*Pi*hbar/c),{n,0,55}]
Out[34]=
\!\({1.5603007215725785`*^48, 1.0402004810483857`*^47, \
1.0198043931846919`*^46, 1.0903568983735697`*^45, 1.196457880202882`*^44, \
1.3239155925269006`*^43, 1.468998083677755`*^42, 1.6314735939618026`*^41, \
1.8124721251924304`*^40, 2.013755599699384`*^39, 2.237468329203007`*^38, \
2.4860618873257212`*^37, 2.762285788174756`*^36, 3.0692045062202207`*^35, \
3.4102265161397066`*^34, 3.789140309416832`*^33, 4.2101558015477`*^32, \
4.677950854384729`*^31, 5.197723158122356`*^30, 5.775247948500309`*^29, \
6.416942163159983`*^28, 7.12993573616281`*^27, 7.922150817706228`*^26, \
8.802389797357865`*^25, 9.780433108140776`*^24, 1.0867147897921367`*^24, \
1.2074608775463436`*^23, 1.3416231972735392`*^22, 1.4906924414149782`*^21, \
1.6563249349055077`*^20, 1.8403610387838884`*^19, 2.0448455986487613`*^18, \
2.272050665165289`*^17, 2.5245007390725432`*^16, 2.8050008211917145`*^15, \
3.1166675791019056`*^14, 3.4629639767798938`*^13, 3.84773775197766`*^12, \
4.2752641688640674`*^11, 4.750293520960074`*^10, 5.278103912177859`*^9, \
5.864559902419845`*^8, 6.516177669355383`*^7, 7.240197410394871`*^6, \
804466.3789327634`, 89385.15321475148`, 9931.683690527943`, \
1103.5204100586604`, 122.6133788954067`, 13.6237087661563`, \
1.5137454184618115`, 0.16819393538464567`, 0.018688215042738408`, \
0.002076468338082045`, 0.00023071870423133835`, 0.000025635411581259828`}\)
In[47]:=
p=m[50]*r[50]/((3/2)*2*Pi*hbar/c)
Out[47]=
1.00916
(p-1)*137.03608
Out[48]=
1.25575
In[62]:=
1.083432147883765*10^12/1435795975383705177540497
Out[62]=
\!\(7.545864220675408`*^-13\)
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net
tilings in an hyperbolic space behave like Gamma 1 modular groups
as triangulations ( polygonal tilings in hyperbolic spaces):
This model is based on an Escher like Gamma modular form
hyperbolic tiling of E_11 as an extended Platonic solid.
The model is a fractal scaling one in which the vertex number
of the hyperbolic polygons are used to both radially scale
and number the particles that time a limited quantum mass state.
It is a lattice in 2d space that can be mapped to a 3d manifold.
4, 6, 8, 12,20 are the vertex numbers of the platonic solids:
my v[n] function is a generalization of that.
I use a radius ratio function of powers of vertex minus one time Pi.
The quantum radius is taken as:
r=2*Pi*hbar/(m*c)
The temperature is the measured background temperature:
and the Hawking entropy formula is used to determine a black hole
radiator mass
at that temperature.
The number of particles in the model are the count of
hyperbolic tiles at different levels of the fractal scaling.
The finding is that the level n=8 accounts for an E_11
model hyperbolic big bang with the temperature
that is observed.
The level n=8 particle number is;
1239964645920142118597441
or about 10^24
and the particle mass at that level is;
1.083432147883765*10^12/1239964645920142118597441=8.737605152280621*10^-13
gm
which is less than the classical 10^(-5) Planck mass.
Mathematica:
Clear[v, r, n, num, m0]
(* Vertices exceptional groups*)
v[n_] := 2*(If[n == 0, 0, 2^(n - 1)] + 2)
Table[v[n], {n, 0, 30}]
{4, 6, 8, 12,20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388,
32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308,
8388612, 16777220, 33554436, 67108868,
134217732, 268435460, 536870916, 1073741828}
(*Hyperbolic tiling numbers for E - 11*)
Table[Sum[If [n == 0, 1, 1027^(n - 1)*1028], {n, 0, m}], {m, 0, 10}]
{1, 1029, 1056785, 1085318197, 1114621788321, 1144716576605669,
1175623924174022065, 1207365770126720660757, 1239964645920142118597441,
1273443691359985955799571909, 1307826671026705576606160350545}
num[m_] := Sum[If [n == 0, 1, 1027^(n - 1)*1028], {n, 0, m}]
(* relative E_11 fractal/ tiling radius*)
r[n_] := 1/(Pi*(v[10] - 1)^n)
(* comic back ground radiation temperature as Hawking radiation*)
T = 2.725;
c = 2.997925*10^10;
hbar = 1.0545919*10^(-27);
G = 6.67328*10^(-8);
kb = 1.380622810^(-16);
(* Hawking black hole mass by entropy temperature*)
m0 = c^3*hbar/(8*G*kb*Pi*T)
1.083432147883765`*^12
(* mass at each tiling level for numbers of particles*)
m[k_] = m0/num[k];
Table[m[n]*r[n]/(2*Pi*hbar/c), {n, 0, 10}]
{1.5603007215725785`*^48, 1.476462737924984`*^42,
1.3998476799928784`*^36, 1.3272107597432808`*^30,
1.2583429105873632`*^24, 1.1930485561574218`*^18, 1.1311422708178325`*^12,
1.0724482505153765`*^6, 1.0167998135211762`, 9.64038927080962`*^-7,
9.140157586270618`*^-13}
p = m[8]*r[8]/(2*Pi*hbar/c)
1.0167998135211762`
(* alpha residue of first order moment *)
(p - 1)*137.03608
2.302180589672985`
The second model uses a Tetrahedron as the basic figure:
This model using a four vertex hyperbolic uses
50 levels instead of 8...
(Nature usually shows 7 levels above one.).
Gauge mass is about the same
at
7.545864220675408`*^-13 gm
but there is a 3/2
factor necessary to get the quantum level near one.
The tetrahedrons are
actually scale slower than the larger symmetry of E_11.
Mathematica;
In[49]:=
Clear[v,r,n,num,m0]
(* Vertices exceptional groups*)
v[n_]:=2*(If[n==0,0,2^(n-1)]+2)
Table[v[n],{n,0,30}]
(*Hyperbolic tiling numbers for E_6*)
Table[Sum[If [n==0,1,3^(n-1)*4],{n,0,m}],{m,0,50}]
num[m_]:=Sum[If [n==0,1,3^(n-1)*4],{n,0,m}]
(* relative E_11 fractal/ tiling radius*)
r[n_]:=1/(Pi*(v[0]-1)^n)
(* comic back ground radiation temperature as Hawking radiation*)
T=2.725;
c=2.997925*10^10;
hbar=1.0545919*10^(-27);
G=6.67328*10^(-8);
kb=1.380622810^(-16);
(* Hawking black hole mass by entropy temperature*)
m0=c^3*hbar/(8*G*kb*Pi*T)
(* mass at each tiling level for numbers of particles*)
m[k_]=m0/num[k];
Out[51]=
{4,6,8,12,20,36,68,132,260,516,1028,2052,4100,8196,16388,32772,65540,131076,\
262148,524292,1048580,2097156,4194308,8388612,16777220,33554436,67108868,\
134217732,268435460,536870916,1073741828}
Out[52]=
{1,5,17,53,161,485,1457,4373,13121,39365,118097,354293,1062881,3188645,\
9565937,28697813,86093441,258280325,774840977,2324522933,6973568801,\
20920706405,62762119217,188286357653,564859072961,1694577218885,5083731656657,\
15251194969973,45753584909921,137260754729765,411782264189297,\
1235346792567893,3706040377703681,11118121133111045,33354363399333137,\
100063090197999413,300189270593998241,900567811781994725,2701703435345984177,\
8105110306037952533,24315330918113857601,72945992754341572805,\
218837978263024718417,656513934789074155253,1969541804367222465761,\
5908625413101667397285,17725876239305002191857,53177628717915006575573,\
159532886153745019726721,478598658461235059180165,1435795975383705177540497}
Out[60]=
\!\(1.083432147883765`*^12\)
In[34]:=
Table[m[n]*r[n]/(2*Pi*hbar/c),{n,0,55}]
Out[34]=
\!\({1.5603007215725785`*^48, 1.0402004810483857`*^47, \
1.0198043931846919`*^46, 1.0903568983735697`*^45, 1.196457880202882`*^44, \
1.3239155925269006`*^43, 1.468998083677755`*^42, 1.6314735939618026`*^41, \
1.8124721251924304`*^40, 2.013755599699384`*^39, 2.237468329203007`*^38, \
2.4860618873257212`*^37, 2.762285788174756`*^36, 3.0692045062202207`*^35, \
3.4102265161397066`*^34, 3.789140309416832`*^33, 4.2101558015477`*^32, \
4.677950854384729`*^31, 5.197723158122356`*^30, 5.775247948500309`*^29, \
6.416942163159983`*^28, 7.12993573616281`*^27, 7.922150817706228`*^26, \
8.802389797357865`*^25, 9.780433108140776`*^24, 1.0867147897921367`*^24, \
1.2074608775463436`*^23, 1.3416231972735392`*^22, 1.4906924414149782`*^21, \
1.6563249349055077`*^20, 1.8403610387838884`*^19, 2.0448455986487613`*^18, \
2.272050665165289`*^17, 2.5245007390725432`*^16, 2.8050008211917145`*^15, \
3.1166675791019056`*^14, 3.4629639767798938`*^13, 3.84773775197766`*^12, \
4.2752641688640674`*^11, 4.750293520960074`*^10, 5.278103912177859`*^9, \
5.864559902419845`*^8, 6.516177669355383`*^7, 7.240197410394871`*^6, \
804466.3789327634`, 89385.15321475148`, 9931.683690527943`, \
1103.5204100586604`, 122.6133788954067`, 13.6237087661563`, \
1.5137454184618115`, 0.16819393538464567`, 0.018688215042738408`, \
0.002076468338082045`, 0.00023071870423133835`, 0.000025635411581259828`}\)
In[47]:=
p=m[50]*r[50]/((3/2)*2*Pi*hbar/c)
Out[47]=
1.00916
(p-1)*137.03608
Out[48]=
1.25575
In[62]:=
1.083432147883765*10^12/1435795975383705177540497
Out[62]=
\!\(7.545864220675408`*^-13\)
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net
