Does the Middle Layer Graph have a nice embedding? Let the $70$ length $3$ and $4$ subsets of $\{1,2,3,4,5,6,7\}$ be vertices of a graph.  Two vertices are connected if the smaller set is a subset of the larger.  $\{1,2,3,4\}$ and $\{1,2,3\}$ are connected.  This is the middle layer graph.  Below is the best embedding I've found for it.

This graph can be made with the cycle $1$-$70$ around the edge and the following three cycles:
$\{\{5,24,43,28,47,66\}, \{3,56,9,62,15,68\}, \{1,42,51,12,25,34,7, 64,37,46,59,20,29,70, 17,50,27,36,19,32,63, 58,67,30,23,2,11,16, 31,26,65,14,49,38,61, 18,53,10,33,22,57,6, 45,40,55,60,69,48,41, 4,13,8,39,52,35,44, 21,54\}\}$
Is there anything better than mirror-symmetric embedding?
 A: Let $P_1, P_2, \dots, P_7$ be the vertices of a regular heptagon centered at the origin, and put vertex $S$ at the vector sum of $\{P_i : i\in S\}$.
Here is a diagram of the result; vertices with $3$ elements are red and vertices with $4$ elements are black, just for fun.

As a result of the construction, the central symmetry through the origin takes a vertex to its complement, and rotation by $\frac{2\pi}{7}$ is a cyclic shift of the elements of the vertex. Vertices whose elements are clustered together, like $\{1,2,3\}$, are placed further from the center; vertices whose elements are spread out, like $\{1,3,5\}$, are placed closer to the center.
Here is the Mathematica code I used to generate the diagram:
red = Subsets[Range[7], {3}];
black = Subsets[Range[7], {4}];
Graph[Flatten@Table[r <-> Union[r, {i}], {r, red}, {i, Complement[Range[7], r]}],
 VertexCoordinates -> Table[v -> Total[CirclePoints[7][[v]]], {v, Join[red, black]}],
 VertexStyle -> Join[Thread[red -> Red], Thread[black -> Black]], 
 EdgeStyle -> Black, ImageSize -> Large]

It looks like this embedding already appears in Wikipedia's article on Danzer's configuration, a point-line configuration whose Levi graph (or incidence graph) is precisely the middle layer graph of $Q_7$. Wikipedia points out that this is a unit distance embedding; in fact, each edge is a unit vector parallel to one of the seven vectors $\overrightarrow{OP_1}, \dots, \overrightarrow{OP_7}$.
