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Are there any solids in $R^{3}$ for which, for any 3 points chosen on the surface, at least two of the lengths of the shortest curves which can be drawn on the surface to connect pairs of them are equal?

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There can be no smooth surface with this property, because a smooth surface looks locally like a plane, and the plane allows non-isosceles triangles.

As for non-smooth surfaces embedded in $\mathbb R^3$ -- which would need to be everywhere non-smooth for this purpose -- it is not clear to me that there is even a good general definition of curve length that would allow us to speak of "shortest curves".

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    @Michael, I _don't know_ whether non-smoothness would make this possible or not. Only that my particular argument against it assumes smoothness.2011-09-08
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If one has a plane 3-connected graph in the plane all of whose faces are triangles it is known that one can not always realize this graph by a convex 3-dimensional polyhedron all of whose faces are congruent strictly isosceles triangles. There are exactly 8 types of such graphs which can be realized with equilateral triangles - the so called convex deltahedra. It is however an open problem whether one can always realize such a graph so that all of the faces are isosceles but have faces with edges of different lengths. Your question goes beyond these considerations in its requirements.