For polyhedra of side 1m, that "inradius" is

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`tetrahedron 0.204124m`

cube 0.500000m

octahedron 0.408248m

dodecahedron 1.113516m

icosahedron 0.755761m

However, if we want to compare equal sized coils, a coil of radius 1m fits inside a triangle of of side 3.464102m, a square of side 2m or a pentagon of side 1.453085m.

So for a coil of 1m radius the equivalent "inradius" is

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`tetrahedron 0.707106m`

cube 1.000000m

octahedron 1.414213m

dodecahedron 1.618033m

icosahedron 2.618033m

This makes a big difference to the strength of field applied at the centre of the polygon.

For a Helmholtz coil of radius R, made of n turns of wire carrying a current I, the axial field B at distance x is

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` mu_zero*n*I*R^2`

B = -----------------

2*(R^2+x^2)^(3/2)

If R = 1m as above, then the relative field strength b = B/(mu_zero*n*I) is

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`tetrahedron 0.272166`

cube 0.176777

octahedron 0.096225

dodecahedron 0.072654

icosahedron 0.022716

and the relative field applied by all the coils at the centre of the polygon is

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`tetrahedron 1.088663`

cube 1.060660

octahedron 0.769801

dodecahedron 0.871852

icosahedron 0.454314

From which it appears that a tetrahedral configuration is slightly preferable to a cube, and probably somewhat cheaper.