• "{A,B,C,D}"
  where A,B,C and D are subsets of {1,2,3,4,5,6,7}. The matroid is the graphical matroid given by the multi-graph of a four cycle with edge repeated and named according to the four sets. Notice that the ordering of the sets is unimportant.
  Example:
  Matroid for "{1234,5,6,7}"
  

  	5 6 7 4 6 7 4 5 7 4 5 6 3 6 7 3 5 7 3 5 6 2 6 7 2 5 7 2 5 6 1 6 7 1 5 7 1 5 6
  Graph	Bases

  We remind the reader that we go from a graph to a matroid by letting every spanning tree define a basis.
• "[A,B;C,D](E)"
  where A, B, C, D and E are subsets of {1,2,3,4,5,6,7}. The matroid is the graphical matroid given by a four cycle with a diagonal edge. Edges are repeated and named acoording to A, B, C, D and E. Here E corresponds to the diagonal edge while (A,B) and (C,D) correspond to the two paths connecting the two valency three vertices.
  Example
  Matroid for "[1,5;23,67](4)"
  

  4 5 7 4 5 6 3 5 7 3 5 6 3 4 5 2 5 7 2 5 6 2 4 5 1 5 7 1 5 6 1 4 7 1 4 6 1 3 7 1 3 6 1 3 5 1 3 4 1 2 7 1 2 6 1 2 5 1 2 4

• "<A;a;(b,c,d,e)>"
  where A is a subset and a,b,c,d and e are elements of {1,2,3,4,5,6,7}. The matroid is the graphical matroid given by the complete graph on four vertices with one edge repeated and named according to A and its opposite edge named a. The remaining four edges form a cycle with names b, c, d and e. Our convention for writing down this cycle is that edges a,b and c all share a vertex.
  Example:
  Matroid for "<15;4;(2,3,7,6)>"
  

  	4 6 7 4 5 7 4 5 6 3 6 7 3 5 7 3 5 6 3 4 6 3 4 5 2 6 7 2 5 7 2 5 6 2 4 7 2 4 5 2 3 7 2 3 6 2 3 4 1 4 7 1 4 6 1 3 7 1 3 6 1 3 4 1 2 7 1 2 6 1 2 4
  Graph	Bases

• "Non-Fano Matroid"
  This text string represents a non-Fano matroid. It only appears once in our drawings and is the matroid of the affine point configuration given below.

  	4 6 7 4 5 7 4 5 6 3 6 7 3 5 7 3 5 6 3 4 6 3 4 5 2 6 7 2 5 7 2 5 6 2 4 7 2 4 5 2 3 7 2 3 6 2 3 4 1 6 7 1 5 7 1 5 6 1 4 7 1 4 6 1 3 7 1 3 5 1 3 4 1 2 7 1 2 6 1 2 5 1 2 4 1 2 3
  Affine point configuration	Bases

• A caterpillar tree is represented by a string of the form C(ab,...,yz) where the leaves ab and the leaves yz form cherries. The dots describes the leaves in between in order from ab to yz. Here the letters are all names of vertices of the tree, that is, numbers between 1 and 7.
• A snowflake tree is the unique tree with all interior vertices tri-valent, with 6 leaves and with 3 cherries. It is represented by a string of the form C(ab,cd,ef) where the leaves ab, the leaves cd and the leaves ef form cherries. Here the letters are all names of vertices of the tree, that is, numbers between 1 and 7.

Less generic planes in TP6

Dimension 12

Dimension 11