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I was thinking today, what the best definition of a cube is. Google defines it as such:

a symmetrical three-dimensional shape, either solid or hollow, contained by six equal squares.

And I was wondering if the "squares" part of the definition is necessary?

a symmetrical three-dimensional shape, either solid or hollow, contained by six equal sides

Would this definition be equivalent?

Essentially the question is, is a 3 dimensional shape consisting of 6 equal sides, possible to be a non-cube? Is this provable?

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    My first thought is that "bulging" cube would be a contradiction. OTOH: A cube has a group of symmetries, $G$. I think the heart of your idea is that "if a shape has group of symmetries $G$, then it has to be a cube." If curvy surfaces are allowed, my first comment says "no" to your question. If not, then I think that $G$ may well define the shape. – 2017-01-02
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    See also [these dice](https://www.youtube.com/watch?v=uAnCL3vhVIs) – 2017-01-02
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    Two regular tetrahedra attached along a face give a solid with six regular triangular faces. (The vertices are not all identical; three triangles meet at two vertices, and four triangles meet at the other three vertices. This asymmetry may disqualify the solid, depending what you mean by "symmetrical".) – 2017-01-02
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    It depends upon which concepts you wish to consider more basic. For example: A cube is a right cylindrical solid with square base and height equal to the side of the square. – 2017-01-02
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    It is well known that only $5$ regular solids exist , and one of them is the cube. So, you could define a cube via regular solids – 2017-01-03
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    "Side" isn't really appropriate. You should say "face". – 2017-01-03

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No, it's not equivalent at all. You can for example fuse two triangular pyramids together to form a triangular bipyramid. This polyhedron has six equal faces which happens to be triangles instead (they could be equilateral). Since triangles are congruent if their side are the same and we can assign the edge lengths quite freely on such figure we don't need the triangles to be equilateral or even isosceles.

Another counter example is a rhombohedron. Here all the edges are the same, but the faces are rhobi. Given the side length of a rhomb it's determined up to congruence by the angle at one corner. In the same way the rhombohedron is determined by its edge length and the angles at one corner and the angles at the oposite corner will match due to parallellity making all faces congruent.

A more elaborate counter example could be to start with a rhombohedron and then pick two opposite corners and then for each of the other we move them perpendicularily to or from the plane halfway the first corners (perpendicularily to the line connecting them) by equal amount.

Apart from this there are vaugenesses in your proposed definition. The concepts of "equal", "side" is not as precise as one might think. Normally one talks about congruence in geometry (or "similar"). In addition what a "side" is is vaugue: do you mean surface or edge? do you require it to be flat or may it be curved? in that case if it may be a curved surface wheres the border of the surface (you could have a sphere and state that it's made up of six congruent surfaces)?

Having that sorted out one could actually prove that there are basically only these two alternatives (the second is a special case of the third, and the cube is a special case of the second) and counter examples. Such a polyhedron would either have to be assembled the same way as a cube consisting of kite (quadrilateral where each edge has an adjacent edge of the same length) surfaces or assembled like a triangular bipyramid.

The proof is by considering a corner and then rule out the possibility that other than three faces meet there. Then one rules out that they be anything else than triangles or quadrilateral. Last one rules out that the quadrilateral can be anything else than kites (by ruling out anything else than pairs of edges with the same length and that they can't be parallellograms unless they are rhomboids and thereby kites).

After realizing the possibilities one could complete your proposal to apply only to cubes. Going along on the same theme it would be to require twelve equal edges and that all angles are the same.

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If you talk about a “shape”, then I'm not sure the term “face” is well defined in this concept. In particular, there is no guarantee that it will be flat. So I'd say you need to define the shape as a polytope.

The “symmetrical” in the definition you quote is pretty vague, too. You could easily come up with an object which has a mirror reflection but is pretty unsymmetric otherwise.

If you require your polytope to be vertex-transitive and face-transitive, in addition to specifying the number of faces, then I guess you should be fine. One does usually require edge transitivity and convexity, too, to obtain the platonic solid, but in the case of the cube I guess those properties might be derivable from the others. Don't have a proof at hand, though.

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Your argument is good but the problem is that the term $Side$ is defined for left or right position of a body (Even for mathematics), as google dictionary says:

A side is an upright or sloping surface of a structure or object that is not the top or bottom and generally not the front or back.

OR

A side is a position to the left or right of an object, place, or central point.

Another advantage of using the square is that $Square$ is a well defined mathematical shape, but the term $Side$ can have different meanings under different circumstances.