GridGraph[{30,30}]
In[]:=
Out[]=
GridGraph[{30,30},VertexLabelsAutomatic]
In[]:=
Out[]=
HypergraphDimensionEstimateList[List@@@EdgeList[%]]
In[]:=
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
CenteredDimensionEstimateList
,GraphCenter

In[]:=
{2.32193,2.35658,2.27309,2.21694,2.17913,2.15229,2.13236,2.11699,2.1048,2.0949,2.08671,2.07981,2.07392,2.06884,1.99986,1.82378,1.62282,1.45375,1.30614,1.17319,1.05022,0.933832,0.821433,0.710955,0.600658,0.488999,0.374539,0.255868,0.131537,0.0339047}
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
Corner:
ListLinePlotCenteredDimensionEstimateList
,{900}
In[]:=
Out[]=
GraphNeighborhoodVolumes
,{900},Automatic
In[]:=
900{1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,276,300,325,351,378,406,435,465,494,522,549,575,600,624,647,669,690,710,729,747,764,780,795,809,822,834,845,855,864,872,879,885,890,894,897,899,900}
Out[]=
GraphNeighborhoodVolumes
,GraphCenter
,Automatic
In[]:=
Out[]=
MeshConnectivityGraph[DiscretizeGraphics[Sphere[]]]
In[]:=
Out[]=
HypergraphDimensionEstimateList[List@@@EdgeList[%]]
In[]:=
{2.8035±0.0011,2.4498±0.0032,2.289±0.006,2.193±0.009,2.120±0.011,2.056±0.013,1.994±0.013,1.930±0.012,1.861±0.012,1.789±0.013,1.709±0.015,1.624±0.018,1.531±0.021,1.427±0.022,1.311±0.022,1.182±0.020,1.031±0.018,0.861±0.017,0.657±0.017,0.406±0.016}
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
DiscretizeGraphics[Plot3D[Exp[-((x+1)^2+y^2)]+Exp[-((x-1)^2+y^2)],{x,-4,4},{y,-4,4},PlotRangeAll,BoxRatios{1,1,1/20}]]
In[]:=
Out[]=
Graphics3D[Point[RandomPoint[%132,1000]]]
In[]:=
Out[]=
ListSurfacePlot3D[RandomPoint[%132,10000],MeshAll]
In[]:=
Out[]=
Plot3D[Exp[-((x+1)^2+y^2)]+Exp[-((x-1)^2+y^2)],{x,-4,4},{y,-4,4},MeshAll,PlotRangeAll,BoxRatios{1,1,1/20}]
In[]:=
Out[]=

Gauss-Bonnet

Integrate Gaussian curvature over surface

Sphere

MeshConnectivityGraph[DiscretizeGraphics[Sphere[]]]
In[]:=
Out[]=
HypergraphDimensionEstimateList[List@@@EdgeList[%]]
In[]:=
{2.8035±0.0011,2.4498±0.0032,2.289±0.006,2.193±0.009,2.120±0.011,2.056±0.013,1.994±0.013,1.930±0.012,1.861±0.012,1.789±0.013,1.709±0.015,1.624±0.018,1.531±0.021,1.427±0.022,1.311±0.022,1.182±0.020,1.031±0.018,0.861±0.017,0.657±0.017,0.406±0.016}
Out[]=
BuckyballGraph[8]
In[]:=
Out[]=
GraphNeighborhoodVolumes[%152,{1},Automatic]
In[]:=
1{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666,716,768,820,874,928,980,1032,1082,1132,1180,1228,1274,1320,1363,1405,1443,1479,1511,1541,1567,1591,1611,1620}
Out[]=
First[Values[%]]
In[]:=
{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666,716,768,820,874,928,980,1032,1082,1132,1180,1228,1274,1320,1363,1405,1443,1479,1511,1541,1567,1591,1611,1620}
Out[]=
as=%154;
In[]:=
Differences[%]
In[]:=
{3,6,9,12,15,18,21,24,27,30,31,34,35,38,39,42,43,46,46,48,48,50,50,52,52,54,54,52,52,50,50,48,48,46,46,43,42,38,36,32,30,26,24,20,9}
Out[]=
Differences[%]
In[]:=
{3,3,3,3,3,3,3,3,3,1,3,1,3,1,3,1,3,0,2,0,2,0,2,0,2,0,-2,0,-2,0,-2,0,-2,0,-3,-1,-4,-2,-4,-2,-4,-2,-4,-11}
Out[]=
LogDifferences[%154]//N
In[]:=
{2.,2.25985,2.23112,2.19387,2.16461,2.14233,2.1251,2.11148,2.10047,2.09141,1.96774,1.98911,1.90369,1.93544,1.87024,1.90538,1.85239,1.8881,1.80572,1.80964,1.74388,1.75567,1.70092,1.71745,1.67041,1.68986,1.64848,1.55368,1.52504,1.4429,1.42289,1.34957,1.33563,1.26864,1.25911,1.16999,1.13803,1.02739,0.973303,0.866879,0.815845,0.711052,0.661162,0.555886,0.253472}
Out[]=
Fit[{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666,716,768,820,874,928},{1,r^2,r^4},r]
In[]:=
-3.48543+1.43388
2
r
-0.000316326
4
r
Out[]=
.0003/1.43
In[]:=
0.00020979
Out[]=
1/%
In[]:=
4766.67
Out[]=
as/Range[Length[as]]^2//N
In[]:=
{1.,1.,1.11111,1.1875,1.24,1.27778,1.30612,1.32813,1.34568,1.36,1.3719,1.36806,1.36686,1.35714,1.35111,1.33984,1.33218,1.32099,1.31302,1.3,1.28798,1.27273,1.25898,1.24306,1.2288,1.21302,1.1989,1.18367,1.16528,1.14667,1.12591,1.10547,1.08356,1.06228,1.04,1.01852,0.995617,0.972992,0.948718,0.924375,0.89887,0.873583,0.847485,0.821798,0.795556,0.765595}
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
as/(Range[Length[as]]+1)^2//N//ListLinePlot
In[]:=
Out[]=
GraphNeighborhoodVolumes[IndexGraph[TorusGraph[{20,20}]],{1},Automatic]
In[]:=
1{1,5,13,25,41,61,85,113,145,181,219,255,287,315,339,359,375,387,395,399,400}
Out[]=
First[Values[%]]
In[]:=
{1,5,13,25,41,61,85,113,145,181,219,255,287,315,339,359,375,387,395,399,400}
Out[]=
Differences[%]
In[]:=
{4,8,12,16,20,24,28,32,36,38,36,32,28,24,20,16,12,8,4,1}
Out[]=
Differences[%]
In[]:=
{4,4,4,4,4,4,4,4,2,-2,-4,-4,-4,-4,-4,-4,-4,-4,-3}
Out[]=
LogDifferences[%169]//N
In[]:=
{2.32193,2.35658,2.27309,2.21694,2.17913,2.15229,2.13236,2.11699,2.1048,1.99952,1.7491,1.47694,1.25615,1.06428,0.888187,0.71924,0.551077,0.378438,0.196432,0.051304}
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
FindFit[{1,5,13,25,41,61,85,113,145,181,219,255,287},ar^b(1-cr^2),{a,b,c},r]
In[]:=
FindFit
:Failed to converge to the requested accuracy or precision within 100 iterations.
{a0.19395,b-0.0166303,c-9.40837}
Out[]=
FindFit[{1,5,13,25,41,61,85,113,145,181,219,255,287},ar^b,{a,b},r]
In[]:=
{a1.84225,b1.97993}
Out[]=
as
In[]:=
{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666,716,768,820,874,928,980,1032,1082,1132,1180,1228,1274,1320,1363,1405,1443,1479,1511,1541,1567,1591,1611,1620}
Out[]=
FindFit[{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666},ar^b,{a,b},r]
In[]:=
{a1.76146,b1.897}
Out[]=
FindFit[{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666},ar^b(1-cr^2),{a,b,c},r]
In[]:=
FindFit
:Failed to converge to the requested accuracy or precision within 100 iterations.
{a0.377145,b-0.10059,c-4.63338}
Out[]=
FindFit[{1,4,10,19,31,46,64,85,109,136,166,197,231,266,304,343,385,428,474,520,568,616,666},ar^b(1-r^2/12),{a,b},r]
In[]:=
{a-29.2073,b-0.199945}
Out[]=
d/2
π
(d/2)!
d
r
1-
2
r
6(d+2)
R+O(
4
r
)
(1/2)!
In[]:=
π
2
Out[]=
Table[r^2(1-r^2/12),{r,20}]
In[]:=

11
12
,
8
3
,
9
4
,-
16
3
,-
325
12
,-72,-
1813
12
,-
832
3
,-
1863
4
,-
2200
3
,-
13189
12
,-1584,-
26533
12
,-
9016
3
,-
15975
4
,-
15616
3
,-
80053
12
,-8424,-
125989
12
,-
38800
3

Out[]=
Table[r^2(1-r^2/12),{r,0,1,.02}]
In[]:=
{0.,0.000399987,0.00159979,0.00359892,0.00639659,0.00999167,0.0143827,0.019568,0.0255454,0.0323125,0.0398667,0.0482048,0.0573235,0.0672192,0.0778878,0.089325,0.101526,0.114486,0.1282,0.142662,0.157867,0.173807,0.190477,0.207869,0.225976,0.244792,0.264307,0.284514,0.305405,0.32697,0.3492,0.372086,0.395619,0.419788,0.444582,0.469992,0.496005,0.522611,0.549798,0.577554,0.605867,0.634723,0.664111,0.694016,0.724425,0.755325,0.786701,0.818538,0.850821,0.883536,0.916667}
Out[]=
LogDifferences[%]//ListLinePlot
In[]:=
Out[]=

Hyperboloid

With[{a=1,b=1,c=1},ContourPlot3D[x^2/a^2+y^2/b^2-z^2/c^2-1,{x,-4,4},{y,-4,4},{z,-3,3}]]
In[]:=
Out[]=
https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Relation_to_the_hyperboloid_model
http://library.msri.org/books/Book31/files/cannon.pdf