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capri0mni(An edited repost of a thread originally posted to rec.arts.drwho, May 1st 2001 -- which I am putting up here to explain why this icon is a Doctor Who homage)
Well, okay... what you'll get is not *really* dimensionally transcendental, since your finished project will not be seamless. But it will be a passable model for the idea of D-T-ness -- and it is fun (IMO. But then again, I'm easily amused.) and easy to make. And it's legal.
Step one: cut out a rectangle of paper, length and width of your choice, but to make things relatively easy, I'd say make it at least 4 times long as it is wide.
Step two: Twist one end a half turn, so that the "underside" of your rectangleis facing up. Don't worry if you crease it, but it looks better if you don't.
Step 3: Keeping the twist in the paper, tape or glue the ends of your rectangle together, making a ring.
That's it. You've done it. This is a Möbius strip, and, as far as I can figure it, it's dimensionally transcendental. It's clearly three dimensional, but because the underside of the strip is connected end-to-end with the overside (Is that a word? :::Shrugs::: It is now.), it has only one side, twice as long as before, meaning that it is also two dimensional (or at least has properties of a two dimensional object). In other words, its properties "transcend" the limits of either dimension: "Dimensionally trancendental".
If you don't believe that it really has only one side, try this: trace a line down the middle of the outside of the ring. And then, if you want to see something really freaky, try cutting the ring along that line.
. . . . . . . . .
A Möbius strip is a three dimensional object, which, having only one side, is an object which a two dimensional being, in theory, could interact with, thanks to that simple half twist.
Now, imagine the TARDIS as a four dimensional object, with an analogous "half twist" through four dimensional space, which would make it something three dimensional beings (such as ourselves) could interact with; the point where the twist occurs is the threshold through which you pass from the "inside" to the "outside" (In quotes because that is not exactly right, but I have no other words for these ideas in my head right now).
Now, because of that simple twist in the original Möbius strip, the useable two dimensional area has been doubled. With a 4d mobius strip (as this theory imagines the TARDIS to be [atm]), the usable three dimensional space
could be tripled or cubed... I can't decide which.
Now, if the TARDIS were actually a 5d ship (taking the clue from Susan's comment in science class), than it could be considered a sort of double Möbius shape -- first twisted through 5d space, to make a 4d interface, than twisted through 4d space to make a 3d interface, each twist multiplying the usable space "inside".
Now, AIUI, when you cut an ordinary mobius strip, there is only one side for your scissor blades to sever, so rather than cutting, you are actually "unfolding" the strip (much like one of those folding rulers), so you get a single loop of double length... Theoretically, this can be continued infinite times... If the living space of the TARDIS is not the matter itself, but the space defined by the matter, than that could explain the infinite space in the TARDIS -- maybe...
Also, the non-orientable characteristic of the Möbius strip and related klein bottle (there is a rumor that you can make a klein bottle out of two Möbius strips) may explain why companions are always getting lost in the TARDIS corridors... ;-)
Well, okay... what you'll get is not *really* dimensionally transcendental, since your finished project will not be seamless. But it will be a passable model for the idea of D-T-ness -- and it is fun (IMO. But then again, I'm easily amused.) and easy to make. And it's legal.
Step one: cut out a rectangle of paper, length and width of your choice, but to make things relatively easy, I'd say make it at least 4 times long as it is wide.
Step two: Twist one end a half turn, so that the "underside" of your rectangleis facing up. Don't worry if you crease it, but it looks better if you don't.
Step 3: Keeping the twist in the paper, tape or glue the ends of your rectangle together, making a ring.
That's it. You've done it. This is a Möbius strip, and, as far as I can figure it, it's dimensionally transcendental. It's clearly three dimensional, but because the underside of the strip is connected end-to-end with the overside (Is that a word? :::Shrugs::: It is now.), it has only one side, twice as long as before, meaning that it is also two dimensional (or at least has properties of a two dimensional object). In other words, its properties "transcend" the limits of either dimension: "Dimensionally trancendental".
If you don't believe that it really has only one side, try this: trace a line down the middle of the outside of the ring. And then, if you want to see something really freaky, try cutting the ring along that line.
A Möbius strip is a three dimensional object, which, having only one side, is an object which a two dimensional being, in theory, could interact with, thanks to that simple half twist.
Now, imagine the TARDIS as a four dimensional object, with an analogous "half twist" through four dimensional space, which would make it something three dimensional beings (such as ourselves) could interact with; the point where the twist occurs is the threshold through which you pass from the "inside" to the "outside" (In quotes because that is not exactly right, but I have no other words for these ideas in my head right now).
Now, because of that simple twist in the original Möbius strip, the useable two dimensional area has been doubled. With a 4d mobius strip (as this theory imagines the TARDIS to be [atm]), the usable three dimensional space
could be tripled or cubed... I can't decide which.
Now, if the TARDIS were actually a 5d ship (taking the clue from Susan's comment in science class), than it could be considered a sort of double Möbius shape -- first twisted through 5d space, to make a 4d interface, than twisted through 4d space to make a 3d interface, each twist multiplying the usable space "inside".
Now, AIUI, when you cut an ordinary mobius strip, there is only one side for your scissor blades to sever, so rather than cutting, you are actually "unfolding" the strip (much like one of those folding rulers), so you get a single loop of double length... Theoretically, this can be continued infinite times... If the living space of the TARDIS is not the matter itself, but the space defined by the matter, than that could explain the infinite space in the TARDIS -- maybe...
Also, the non-orientable characteristic of the Möbius strip and related klein bottle (there is a rumor that you can make a klein bottle out of two Möbius strips) may explain why companions are always getting lost in the TARDIS corridors... ;-)
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no subject
Date: 2005-01-09 04:42 am (UTC)Heh...do that enough times to Source (which has no beginning and no end), maybe that's what makes them 'strings' they're always going on about...
...or maybe not. 'Tis a thought.
no subject
Date: 2005-01-09 05:38 pm (UTC)no subject
Date: 2008-03-16 03:55 am (UTC)For more fun still, think about this: the paper loop isn't really two-dimensional. It has a thickness, just a very short one compared to the width. Imagine if the thickness were the same as the length. So you'd have a kind of twisted Moebius... noodle. A square noodle-ring of Play-Doh with a half-twist in it. It'd still function more or less the same as the paper strip, with that half-twist in it, yes? Still unfold into one long full-twisted noodle-ring when sliced lengthwise?
But what if you only gave it a quarter-twist? Which is possible, now that it's as thick as it is wide. Now when you cut through it lengthwise, your virtual scissors come back at a 90 degree angle to the cut they made before they went, erm, round the twist. My brain is failing to visualize the kind of solid shape that would emerge from the unfolding of this sliced strip.
But I find myself wondering about other solid tube shapes. Say f'rinstance an extruded hexagonal noodle made of Play-Doh, given a 60-degree twist and then formed into a loop and sliced...
Ow! My head!
I fail at visualizing.