I have not imagined a polyhedron. I have only formed triangles with the use of the ground. To form a polyhedron (a 3D shape), I need to create the human perception that I have done so (remember, we interpret real shapes as mathematical shapes - nothing in reality is an exact mathematical shape). The question is why I can say that I have created individual equilateral triangles and not formed a polyhedron. The answer is that in order to create the human perception of a polyhedron (a 3D object), I must create 'lines' visible from every direction that a 3D object can be perceptual viewed. In the case of what I have done, I cannot say that this is accomplished.
I think where you are getting confused is that you think that mathematical objects as they are seen in the world, are actually existing as mathematical objects. This is a mistaken notion. Humans interpret mathematical objects by what they see in their environment. A mathematical object no more exists in nature than 2 toothpicks are used in combination with the ground to form a mathematical triangle. All we can say is that we have formed a humanly acceptable equilateral triangle, or a humanly acceptable polyhedron, etc.
The weakness in that problem is what I exploited (knowing perfectly well what I was doing). I exploited the fact that mathematical forms as a pure object do not actually exist 'out here', so all I am required to do is satisfy human interpretation (a far less stringent requirement).
I understand why you are frustrated with my solution, and I cannot blame you for being frustrated. What I wish I could communicate though, is that when human interpretation is involved in a problem solving issue, then it is also your ally if you want to take advantage of it. I'm sorry if that makes me look weak or providing lame answers to presumably straight forward questions, but I'm eager to play with people's natural intuitions in this way. Afterall, the puzzle plays with those intuitions since most people think of that problem in a 2D fashion. I just upped the ante - to philosophical consequences. An alternative that I'm more than happy to do.
Warm regards, Harv