We know the square cutting of the regular tetrahedron (and more generally of a tetrahedron with two opposite edges orthogonal and of same length): the plane goes through the midpoints of four edges and the side of the square is the half of the edge.
Curiously there is also a regular pentagonal cutting of the square pyramid with edges of same length e. The three lengths in green are equal with value e×(3√5)/2 = e×0.382...
Let's continue... Is there a regular hexagonal cutting for a regular pentagonal pyramid with edges of same length? No! It is even not possible to get an equilateral hexagon (the two red sides don't have same length as the blue ones).
The case of the square pyramid is thus quite "miraculous".



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