LayeredGraphPlot[GeneralizedGridGraph[{10"Directed",10"Directed"}],AspectRatio1]

In[]:=

Out[]=

#### What is this invariant under?

What is this invariant under?

Coordinates of points are {t,x} or {t,x1,x2}

Isometry group:

What matrices preserve the arrows here?

#### Undirected graph case

Undirected graph case

Positions are {x,y}

What matrices leave infinite plane invariant?

{{0,1},{1,0}}

What is ISO(2Z) ?

Rotations + translations

Select[Partition[#,2]&/@Tuples[{-1,0,1},4],OrthogonalMatrixQ]

In[]:=

Out[]=

{{1,0},{0,-1}}.{x,y}

In[]:=

{x,-y}

Out[]=

Select[Partition[#,2]&/@Tuples[{-2,-1,0,1,2},4],OrthogonalMatrixQ]

In[]:=

Out[]=

### Claim: for the 3-direction case, you have to use complex numbers in the coordinates in order to have a matrix representation

Claim: for the 3-direction case, you have to use complex numbers in the coordinates in order to have a matrix representation

#### In the general case:

In the general case:

{t,b,x}