The Petersen graph is a unit distance graph, and this embedding is shown below, where each edge of the graph is one unit in length.
Is there a ruler and compass construction for this embedding? If so, what is it?
It seems like the kind of thing that you can construct with a ruler and compass, but my brief attempt and google search failed.
I tried by constructing the outer pentagon, and then drawing unit radius arcs inside it, centered at each vertex. The inner star should have it's vertices on those arcs.
I did come fairly close by eyeballing the "radius" of the star and drawing the smaller circle, but I suspect this was cheating :p
Image credit: http://en.wikipedia.org/wiki/File:Petersen_graph,_unit_distance.svg
Best Answer
If the inner star's side are $1$ the radius of its circumflexing circle is $\frac1{2r}=\sin\frac25\pi=\sqrt{\frac{5+\sqrt5}8}$ which leads that $$r=\sqrt{\frac{5-\sqrt5}{10}}$$
So you can build $r$ with ruller and compass.
Update
Actually it might become simpler: First build your unit-sized pentagon $A_1B_1C_1D_1E_1$:
Then draw perpendiculars to each side at each vertex.
These perpendiculars will cut each-other at points $A_2,D_3,B_2,E_3,C_2,A_3,D_2,B_3,E_3,C_3$ (subindexes $3$ are not shown in the drawing). You dont need them to find them all, just one or two of them (v.g. $A_2$ and $C_2$).
The figure $A_2C_2E_2B_2D_2$ is a pentagram congruent to the one in the unit-distance Petersen graph, and they both share the same circumcircle (red in the figure)
So, take $A_1A_2$ and $C_1C_2$ they cut at the center $O$ of $A_1B_1C_1D_1E_1$. Draw the circle centered in $O$ with radius $OA_2$, and I supose you can now finish the problem.