|
Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 |
like. And besides, this would serve very little purpose in helping us to understand something about the appearance of the tesseract itself. Instead, let's try something else with the shadow. A shadow is made by using some sort of light source against the object. And naturally in the fourth dimension they will have a fourth dimensional light source, which would in some fashion be different from our three dimensional light sources. Picture this if you will: a transparent three dimensional cube being shined upon by a light source containing a lens as big as the cube itself. (That could be a small sized flashlight shining upon a transparent die.) The resulting shadow on the wall behind, assuming we have outlined all the edges of our transparent die in dark color, would be a box in a box connected at the corners. The front of the die would be the big box and the back of the die would be the little box, and the little box inside the big box would be connected to each other at the four corners. Now let's try something similar with the tesseract and a fourth dimensional light source. Let's use a transparent tesseract which has all its edges in dark color, and we could count 32 edges. (A square has four edges, a cube has 12 edges, and a tesseract has 32 edges) In order to understand the mathematical progression of edges we must remember the process we used to go from an object of any dimension to the object of the next dimension. We used the object of that dimension as a base, unfolded the sides 90°, and added a mirror image of the base at a top. (A square as the base, unfolded the four sides upward 90°, and added a mirror image of the square at the base as the top.) We always use the figure of the present dimension, extend the sides from all the corners in the new direction of the next dimension, and top it off with a mirror image of the figure used in the present dimension. And that gives us the figure of the next dimension. Which means that in order to get the number of edges of the figure in the next dimension, we will have the number of edges of the figure in this dimension times two, (we used the figure twice, once at the base and once at the top) plus the number of lines drawn from corner to corner from the figure used as the base to the mirror image used as the top. In the case of the cube, we had one square at the bottom, one square at the top and four lines drawn corner to corner between the two. Thus we had four edges on the first square, four edges on the second square, and four edges on the four lines drawn between the two connecting up the corners. From a cube to a tesseract, we have 12 edges in the bottom cube, 12 edges in the top cube and eight lines drawn corner to corner connecting up the two cubes, giving us a total of 32 edges. The mathematical progression mainly lies in the number of lines drawn between the two figures. From a finite line to a square, there are two lines drawn; from a square to a cube there are four lines drawn; and from a cube to a tesseract there eight lines drawn. That means that from a tesseract to a hypertesseract (fifth dimension) there would be 16 lines drawn and since a tesseract has ...... continued on next page Previous Page <-> Next Page |
|||||||||
|
||||||||||