Theoretical Predictions
Theoretical Predictions
Reproduce General Relativity
Reproduce General Relativity
Yes for the structure of the Einstein equations
Matter vs. background of space [ probe: scalar field ]
Not yet: trajectories of test particles [ topological soliton as a faux particle ]
New Phenomena
New Phenomena
Early Universe
Early Universe
Dimension change / inflation
Dimension change / inflation
In[]:=
ResourceFunction["WolframHausdorffDimension"][RandomGraph[{200,600}],1]
Out[]=
3.28979
In[]:=
Table[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],1],400];
In[]:=
Histogram[%]
Out[]=
In[]:=
ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],All,"AllDimensions","VertexMethod"Mean]
Out[]=
3.30127
In[]:=
Histogram[ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],All,"AllDimensions","VertexMethod"Mean],1000]]
Out[]=
In[]:=
Histogram[ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,800}]]],All,"AllDimensions","VertexMethod"Mean],1000]]
Out[]=
In[]:=
ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,800}]]],1],4000];
In[]:=
Histogram[%]
Out[]=
In[]:=
RandomGraph[{200,800}]
In[]:=
init=
;
In[]:=
ResourceFunction["WolframModel"][{{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}}},List@@@EdgeList[init],5,"StatesPlotsList"]
Out[]=
,
,
,
,
,
In[]:=
ResourceFunction["WolframModel"][{{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}}},List@@@EdgeList[init],10,"StatesPlotsList"]
Out[]=
,
,
,
,
,
,
,
,
,
,
Is the dimension “cool down” continuous or does it have phase changes?
Is the dimension “cool down” continuous or does it have phase changes?
Phase change ⇔ percolation ?
What is the effect on dimension of random reconnection?
What is the effect on dimension of random reconnection?
Not even local, etc. If it’s completely random, the dimension can increase.
Is dimension cool-down an “approach to equilibrium” that does not show universal behavior?
Is dimension cool-down an “approach to equilibrium” that does not show universal behavior?
Relationship to variable speed of light
Relationship to variable speed of light
r^d vs. c t
Projecting a causal cone to an instantaneous spatial structure
Events in the causal cone (c t)^d t
If c ~ t^α, then dimension is d + α
Cosmological constant : events associated with maintenance of space / events associated with matter
Cosmological constant : events associated with maintenance of space / events associated with matter
Grid-graph isotropy?
Grid-graph isotropy?
While geodesics are preserved by rotations; the whole space also has reflections...