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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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A metric space having this property is ultrametric, hence it is totally disconnected, hence reduced to a point. –  t.b. Sep 8 '11 at 12:33
    
How would non-smoothness make this possible? Do you know of some non-smooth surface on which this happens? And as for definitions of the length of a curve, what's wrong with the usual one (the smallest upper bound of the lengths of polygonal paths whose vertices are on the curve). –  Michael Hardy Sep 8 '11 at 12:35
    
@Michael: that only works for rectifiable curves. If the surfaces is sufficiently crinkly, no rectifiable curves exist. (Of course that is even worse for constructing an example, since it doesn't even make sense to talk about isosceles triangles in that case.) –  Grumpy Parsnip Sep 8 '11 at 12:45
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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. –  Henning Makholm Sep 8 '11 at 12:49

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.

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