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garrick92
47389.  Tue Jan 24, 2006 7:11 am Reply with quote

Q: How can you make your left hand fit your right glove?

A: Rotate it in the fourth spatial dimension

Here's a 4D cube (AKA hypercube or tesseract).



The 'inner' cube that you can see is its back.

As you can see, rotating it makes the 'outer' and 'inner' cubes swap places.

Each of the six cavities between the 'inner' and 'outer' cubes is also a cube, what you're looking at is a 2D rendering of how it fits together, so none of them look like cubes.

In 4D space, you can fit four right-angles into a corner.

Here's how it looks when you 'unfold' it:



Sounds baffling? Watch it folding as an animation here

Anyway, whether you understand it or not, you don't get any points for answering "Turn the glove inside out."

 
Gray
47399.  Tue Jan 24, 2006 8:04 am Reply with quote

I like it when you slice the end off a 4-dimensional salami with a spherical blade: You get a nice sphere of salami to put in your hyperbap.

 
Jenny
47421.  Tue Jan 24, 2006 9:54 am Reply with quote

That is totally weird to watch.

 
Celebaelin
47436.  Tue Jan 24, 2006 11:08 am Reply with quote

It's a head-fryer alright. I think the LHS animation of the shadow of the cube folding helped my understanding of the process better than the RHS animation. I can now accept it without of course being able to claim that that I have the vaugest idea of what is being depicted.

I have this strange notion at the back of my mind that the point source is inappropriate and the shadows should be more complex (yes, even Hank).

 
mckeonj
47455.  Tue Jan 24, 2006 12:08 pm Reply with quote

I have an ornamental basket made of bent wires joined by beads that does something like that, but you have to push it around. I'll set about photographing it.

 
Ponderer
47981.  Fri Jan 27, 2006 9:28 am Reply with quote

Surely there's an easier way to get your left hand to fit your right glove-go through a mirror for instance.

(Ok it's not practical, but it is easier)

Or is going through a mirror rotating yourself through the fourth spacial dimension? I don't think so, as they're barely three dimensional, but still...

 
samivel
47988.  Fri Jan 27, 2006 10:21 am Reply with quote

That image has a powerful hypnotic effect

 
garrick92
48010.  Fri Jan 27, 2006 12:34 pm Reply with quote

Ponderer wrote:
Surely there's an easier way to get your left hand to fit your right glove-go through a mirror for instance.

(Ok it's not practical, but it is easier)

Or is going through a mirror rotating yourself through the fourth spacial dimension? I don't think so, as they're barely three dimensional, but still...


But if you reorientate yourself from the mirrror's point of view, that's not your left hand any more, and nor is it your right glove. It would just be the, er, reverse of the original dilemma for your reflection (imagining (s)he's a conscious 3D entity, which (s)he isn't ... we hope ... <GRAMS: Creepy musicbox>)

Unless you could somehow 'reflect' the glove (or the hand) and still keep it on this side of the mirror, which would amount to a 4D rotation.

But the similarity between your reflection and you is that even though you're facing each other and his left hand is your right hand (and vice versa, if you're still following this train of thought, and I'm not sure I am), if your right hand points (say) west, your reflection's hand also points west. Even if 'west' is judged by a reflected compass. This seems paradoxical at first glance, but I don't suppose it is.

Which brings me to a totally unrelated question, to which I do know the answer, but still cannot bring myself to fully accept: Why is it that mirrors reflect left-to-right but not top-to-bottom?

<brain implodes>

 
garrick92
48012.  Fri Jan 27, 2006 12:57 pm Reply with quote

Incidentally (yes, folks - more trippy graphics!), readers of Edwin Abbott's Flatland (one of my all-time favourites) will be familiar with what a sphere looks like to the inhibitants of a two-dimensional world as it passes through their plane of existence.

The sphere starts as a dot, grows into a rapidly-expanding circle, which slows as it approaches its maximum diameter - after which it repeats the process in reverse, dwindling away to a dot and finally vanishing. (You can imagine this by mentally slicing a sphere into very thin layers and then considering them one after the other).

In the same way, a cube passing face-first through flatland at a steady speed would appear instantaneously as a square, the flatland appearance of which would last a period determined by the speed at which it passed through. (Of course, a square prism passing through (square end first) would be indistinguishable from a cube: but passing through at the same rate, the prism would appear as a longer-lasting square).

Of course, if the cube hit Flatland perfectly corner-first (with a diagonal length as a vertical) it would appear as a swelling triangle at first, then a hexagon (imperfect, briefly perfect, then imperfect again), then a triangle before shrinking and vanishing. (You could give yourself a migraine by thinking about this too much, couldn't you?)

Anyway. By analogy with the experience of the mystified Flatlanders (and this is probably the only way it's comprehensible) here's what a hypercube looks like if it falls corner-first through our three-dimensional world.



If you concentrate on one face of the initial tetrahedron that appears, you'll see that each triangular face becomes an imperfect hexagon, then (briefly) a perfect one, then an imperfect one, before becoming a triangle again and shrinking to nothing.

So you're seeing the same thing as a flatlander would from a corner-first cube ... but happening in three dimensions.


Last edited by garrick92 on Fri Jan 27, 2006 1:21 pm; edited 2 times in total

 
garrick92
48013.  Fri Jan 27, 2006 1:09 pm Reply with quote

Heh, heh, heh ... a few more of these weird animations and my army of hypnotised zombies will be under my complete control and be ready to wipe out the scourge of woolly thinking FOREVER ... <rubs hands together>

 
Colonel Krummhorn
48023.  Fri Jan 27, 2006 2:02 pm Reply with quote

Quote:
Anyway, whether you understand it or not, you don't get any points for answering "Turn the glove inside out."


Does that work?!?! WOW! I must find a glove.

 
Gray
48027.  Fri Jan 27, 2006 2:12 pm Reply with quote

Here's our previous debate on why mirrors do what they do.

 
Zaphod Beeblebrox
48034.  Fri Jan 27, 2006 3:17 pm Reply with quote

Quote:
Heh, heh, heh ... a few more of these weird animations and my army of hypnotised zombies will be under my complete control and be ready to wipe out the scourge of woolly thinking FOREVER ... <rubs hands together>


No, I'm sorry but I just can't let you do it. <folds arms>

 
mckeonj
48035.  Fri Jan 27, 2006 3:53 pm Reply with quote

Regarding the perception of the mirror image, it is possible to make a simple device known as the 'magic mirror'. When you looks into this, the mirror image is reversed laterally, so that if you wink the left eye, the image winks its left eye, opposite your right eye. It is the same effect as produced by a video camera set alongside the view screen. The effect of this is interesting; you see yourself as others see you. The full version of this device is a cube corner made from three mirrors, as illustrated here:

but a satisfactory one, which only reverses laterally, can be made with two unframed mirrors set into the corner of a room, or the reveal of a window, so that the mirrors are at right angles. The sliding mirror doors from a bathroom cabinet are about right for size. Look into the mirror so that the dividing line bisects your image, and enjoy. The half of the human race that spends a lot of time looking at themselves in mirrors finds it either fascinating or horrifying.

 
Gray
48065.  Fri Jan 27, 2006 10:27 pm Reply with quote

You get this effect in mirrored lifts (elevators) as well, although because you're so close to the mirror junction, your mirror image appears to be looking at someone much more interesting over your shoulder...

 

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