> For 4 dimensions and up, I can visualize [...]

Tell me more. I never quite managed to visualize 4D.

 > [...] that there could be more than one solution.

Already in 3D you can have exactly 2 solutions:
Circle A: center 0, normal y, radius 1
Circle B: center 0, normal z, radius 2


-- 
merge{#local i=-11;#while(i<11)#local
i=i+.1;sphere{<i*(i*i*(.05-i*i*(4e-7*i*i+3e-4))-3)10*sin(i)30>.5}#end
pigment{rgbt 1}interior{media{emission x}}hollow}//  Mark Weyer

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