There exists a parallel projection of an $N$ dimensional cube from $N$ dimensional (Euclidean) Hilbert space to the $x-y$ plane whose projected perimeter approaches a circle as $N$ increases. The interior angles between the lines intersecting at one of the perimeter points in this projection are all equal to $180/N$.
Assuming the cube center remains at the origin of the Hilbert space, and that it was oriented parallel to the basis vectors of the Hilbert space, and that the parallel projection is merely accomplished by discarding the $N-2$ highest numbered coordinates, and that successive rotations are applied one degree-of-rotational-freedom at a time, what would be one such sequence of rotations to achieve the desired orientation?
Or, as an alternative answer, what would be the single rotation matrix that would accomplish this in one rotation?
So far I have tried to discover this by trying different sequences using my simulator. 
In case you are wondering how I was able to make these movies if I did not already know the answer, it is simple. I do not know the answer. The Eulerian circuit sequence is calculated by my program. I start with the $N$, two dimensional basis vectors $e$ to the power of $n$ times $pi$ divided by $N$, where $n$ goes from zero to $N-1$. Then I just draw them in two dimensions applying them in the Eulerian walk sequence.
That is why I only work with even numbered dimensions in this case. I suspect the answer will depend on if $N$ has an odd factor or not.
So you see, another requirement for the projection is that it render ALL edges equal length, but not necessarily the same length as the full object in N-space.
Just researched and found that these figures have an exact name: N-dimensional Hypercube Petrie polygon Orthographic projections. I have the projection; I need the complete orientation of the hypercube that yields this projection.
for 4 dimensions five sequential rotations by 45 degrees orient the 4-cube in the desired orientation: xy xz xy xy xw
This can be simplified to three rotations if we allow rotation in the yw plane. However, my original hardware only had three switches available so all allowed rotations involve x,
Still seeking the general solution for $N$ an integer larger than 4.
The numbers for the 4 cube rotating xy xz xy xy xw orient into the Petrie form desired as shown below. Note the basis vectors are nicely separated at equal angles, required for the near circle shape of the projection.
Since we can write down the $x$ and $y$ components for the rotated basis vectors by noticing the angle between them is $180/N$, and they are all equal length, it should be possible to compute/solve for the other components using the other constraints of the geometry. This can be done for, 5, 6, 7, and 8, after which a pattern may emerge that can be tested with induction.