Given the immense popularity of my last blog, I thought it only prudent to post another. I actually have another planned for tomorrow, but this seemed too good to pass up now. Listen up kids, this may actually raise your IQ.
Specifically, I want to introduce you all to a new friend of mine. It is part of the hypercube family, but not the two-dimensional square, nor the three-dimensional cube we are all aware of. This one is also projected into a fourth
spacial dimension (none of that "time is the fourth dimension" stuff today).
Its name is the Tesseract. I would explain it in detail myself, but I cannot do so more effectively than the master already has. Please give a warm welcome to the stage, Dr Carl Sagan!
More- much more- on him later.
No doubt Carl's exquisite way with words has been universally understood by my paltry few readers.
Now, one thing that Carl's description and model cannot demonstrate is the true nature of the tesseract's three-dimensional shadow.
We all understand the two-dimensional shadow of the cube, even the video image of Carl's actual cube follows the same principle- but we understand. Even though the 2D representation must have sides that are not at right angles to the others, nor the same length, we understand it perfectly well.
It is not quite the case with the tesseract. We are no used to seeing three-dimensional shadows, we are not accustomed to diagrams that reduce the number of dimensions yet are still 3D. The limit of Carl's tesseract model was that it was static, a 3D snapshot of the shadow. This is the shadow when the object casting it is rotating:
Absolutely bizarre, is it not? Keep watching until you see the pattern. Try watching a single four-sided face as it passes through space, looping and twisting in the fourth dimension. Keep track of the middle cube, or another six-sided shape: where does it go? What does it become before it reaches the same point again?
A simpler rotation:
Okay, that's weird, right? Let's see if I can explain. Go back to the shadow of Carl's 3D cube. If he were to rotate it- indeed, if you think of the image of the actual cube as a shadow in the same way, he does- you would see the same thing. The edges and the faces warp and twist on the flat surface as the pass through the dimension above- the third dimension.
It is perhaps difficult to see the comparison, because our brains can automatically reverse the loss of the third dimension and construct a mental 3D model of the object the 2D image represents. This is not the case with the tesseract, unfortunately.
But see that in the same way the square sides of the cube are warped into rhombi (diamonds) in 2D, the cube faces of the tesseract are warped into (among other things) quadrilateral frustums, or truncated square pyramids; which continue to change shape and intersect one another. Such is the penalty of losing a whole dimension for an otherwise solid, sensible, regular polychoron.
