Harvey,
It’s funny we’re arguing about this, but I’m having a good time -- and as long as you are too then I guess our ability to laugh at ourselves makes it all worthwhile. I know I can be a real S.O.B., but I wouldn't come back if you weren't an excellent "opponent."
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Triangle: "A three-sided polygon."
Polygon: "A closed plane figure bounded by three or more line segments."
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>>>”I have only formed triangles with the use of the ground.”
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."
>>>”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.”
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.
(Remember, a triangle consists of three sides; as a polygon it is a two-dimensional object. A triangle is not "an intersection of two, perpendicular polygons.")
Let’s think about this:
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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. It follows, you might then say, that for the sake of your "solution" you are only considering the 1-dimensional plane constituting the 'horizon' of the ground/table-top itself. Indeed, this singular dimension is what you require.
But any consideration of the tabletop/ground that restricts itself to only the one dimension that constitutes its “horizontality” (sic) is logically bound to those considerations only from the perspective of its one-dimensionality. You might therefore conclude that your "ctreaed" line is a function of this "horizontality," and that this line is broken into segments, in order to complete the six triangles.
From this point the triangles can be seen if you include a second dimension – a dimension 90 degrees perpendicular to this horizon -- the first dimension. After all, you might remind me, a triangle is a two-dimensional object.
You now step up to two dimensions, and any subsequent consideration of the allowed, second dimension -– the one 90 degrees removed from the first -- is equally logically bound to considerations of the resultant 2D surface and subordinate polygons.
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.
”Well,” you might object, ”I know -- and JUST DISCUSSED my solution as a 3-dimensional object!”
But a 3d solution that contains only non-imaginary objects must include the entire ground itself. If you attempt to reduce the ground to less than its entire non-imaginary "real" presence, for the sake of removing the status of imaginary from your lines, then you cannot define anything more than two triangles!
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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!
You're not talking about logic here; you're talking about illusion! You're a magician!
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?
;)
LH |