While I was away you posted a response to the triangle thing.
Triangle: "A three-sided polygon."
Polygon: "A closed plane figure bounded by three or more line segments."
H:”I have only formed triangles with the use of the ground.”
L: No, you have imagined that a part of the ground forms 1/3 of the total sides of your six triangles. The "use" of the ground requires an imaginary projection onto "reality."***
I take exception to the term 'imaginary'. The ground is just as real as the toothpicks. If the ground is imaginary, then so are the toothpicks.
***H: ”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.” L:
This contradicts itself! Either your “created” lines are imaginary, or they are not. If they are imaginary, then you have created eight triangles (and two polyhedrons); this is fine, as magic and illusion require imagination. If your lines are not imaginary, and you have therefore only invoked that part of the ground which exists as a "line," then your solution can’t be 3 dimensional, and you can not have created anything more than two triangles.***
No. I don't consider the ground to be 'imaginary'. That is, I can see the ground, I can feel it, even taste it if I choose, and even smell it if my sense of smell were that good. On the other hand, to form a non-imagined polyhedron with my toothpicks, I would need to use non-imagined lines. This would require that I have something 'real' (i.e., see, feel, taste, smell). As for me constructing real polyhedrans, I cannot do that. My 'lines' in this 3D case would not satisfy 3D requirements. A real 3D object should (theoretically) be able to be seen, touched, tasted, and smelled from any direction in space. If I turn my 'polyhedran' upside down, I cannot see any real triangle at the former 'bottom' (which is now 'top' since I turned it upside down). All I would see is a flat surface having no triangle shape. I do not have a real polyhedran that I can lay my eyes or hands on.
***You must recognize that the ground/table-top etc. is not just a polygon or a line, but it is actually a 3- (or more) dimensional object.***
Yes, the ground is 3-dimensional (not 2-D since there is elevation differences for a plane), but when asked to see a triangle, we are not looking at 3-dimensions, but only 2-dimensions. However, with respect to this problem, one leg of the triangle is a 1-dimensional object. We intentionally ignore other dimensions as part of the problem (remember, the problem is to find 2-D objects, not 3-D objects).
***So far, so good, but this cannot work; we must introduce a non-static third dimension (i.e., not just a point from which we can observe 2D planes). Otherwise some of the triangles disappear (the maximum number of triangles in your "solution," that can exist in a solitary 2d plane is two), and the triangles we see (from the above-mentioned vantage point) are not equilateral, unless viewed from an angle “above” the ground/tabletop. Clearly, no consideration of all six triangles can exist unless we can move about in three+ dimensions.***
I agree, you have to move around the structures to 'see' the triangles.
***Now, since each of your three-triangle pairs consists of only equilateral “triangles,” then the imagined (“created”) lines that form their collective bases establish two more equilateral triangles; but now we have something strange, indeed! You have created eight triangles. Somehow, you have created more triangles than axes with which you began the exercise! You have somehow created eight 2D planes from six 1D axes!***
I think you are concluding far too much. You have two pyramid-appearing structures. If you view each 'side' from eye level and straight on to each 'side' of the pyramid you see 6 triangles (i.e., the ground does complete one leg of each triangle that you see head on), and not 8 triangles. It is not required in the problem, that the triangles be made to appear from all directions. If that were true, then I could find a viewing obtuse direction which the object is not a triangle at all.
***You're not talking about logic here; you're talking about illusion! You're a magician!***
I'm talking about completing the task of the problem. If the solution turns out to be an illusion, then you have completed the problem unless the problem has restrictions that somehow eliminate illusions (e.g., the use of the ground, the use of 3D, etc).
***Either this, or you must prove that a partially imaginary object is just as valid as an entirely non-imaginary object, but an entirely imaginary object is completely invalid. Where is it?***
There's nothing imaginary about the triangles I constructed well within the boundaries of this problem. The solution simply has people hitting their heads because they didn't read the problem and realized the breadth of solutions available based on the lack of rules mentioned.
Can we move on from this toothpick thing?
Warm regards, Harv