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The Euclid's definition of similar solid figures is

Similar solid figures are those contained by similar planes equal in multitude.

And the Euclid's definition of equal solid figures is

Equal and similar solid figures are those contained by similar planes equal in multitude and magnitude.

This definition was a lot criticized (see the commentary of Heath of the XI book of the Elements for a summary of the critics), because actually there exists solid figures contained by similar planes equal in multitude that are not similar: a typical example is that of a pyramid such that on its base, on opposite sides of it, are erected two equal pyramids smaller than the first. The addition and subtraction of these pyramids respectively from the first give two solid figures which satisfy the definition but are clearly not similar (the smaller having a re-entrant angle).

It was proposed to change the definition in this way: Similar solid figures are such as have all their solid angels equal, each to each, and which are contained by the same number of similar planes.

What if Euclid had actually in mind only the CONVEX solid figures?

Well, there is a counter example for two convex solid figures with similar faces equal in moltitude: Let's take a frustum with squared as bases, not too much flat in order to mantain the convexity of the final figure (wait a moment to have this more clear). Now let's consider the two solids made by erecting a pyramid with equilateral faces respectively on the two bases of the frustum. We have two convex solids made of 4 equilateral triangles, a square and 4 similar isosceles trapezoids.

Now my question is... And if we want only EQUAL faces (not just similar) equal in moltitude?

I can't figure out any counterexample. It seem to me that the definition of similar solid figures is wrong, but that of equal solid figures may be correct (even thou is not proved that two convex solid figures contained by similar planes equal in multitude and magnitude, are equal).

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I've found a counterexample

Here it is a counterexample! It was a little hard to imagine but I made it :-)

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