calamarain
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Not all forum games need to be long. This one's a quick one, it can be solved by googling of course, but try it in your head first. The answer surprises people, even those who're good at 3D visualising.

Take a cube. Balance it on a corner exactly (with the opposite corner pointing straight up)

Then slice the cube in half horizontally.

What shape do you get on the cross-section? Post your guesses/thoughts before you look up the answer. This one quite surprised me.

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CuriousShyRabbit
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My quick little sketch gives me

calamarain
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CuriousShyRabbit wrote:
My quick little sketch gives me

Click here to view the secret text

×a hexagon.

Quite correct. But even better...

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CuriousShyRabbit
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I stand nit-picked

You mentioned in another thread that you were studying chemical metallurgy. You didn't, by any chance, happen to be staring at crystal lattice structures when you thought of this puzzle? There are so many different ways to look at them...

calamarain
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CuriousShyRabbit wrote:
I stand nit-picked

You mentioned in another thread that you were studying chemical metallurgy. You didn't, by any chance, happen to be staring at crystal lattice structures when you thought of this puzzle? There are so many different ways to look at them...

Actually, no lattice structures have very little to do with my PhD, last time I looked at them was as an undergraduate. It's actually an old puzzle from a puzzlebook I read years ago. I just remember being impressed by how one shape was contained within another.

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Maurog
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Then you may be even more impressed if I tell you that if you take a cube, then calculate the middle of each side, and connect these points, you get a regular octahedron. What's more, if you take a regular octahedron, calculate the middle of each side and connect the points, you get a regular hexahedron, i.e. a cube.

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calamarain
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Maurog wrote:
Then you may be even more impressed if I tell you that if you take a cube, then calculate the middle of each side, and connect these points, you get a regular octahedron. What's more, if you take a regular octahedron, calculate the middle of each side and connect the points, you get a regular hexahedron, i.e. a cube.

Mmmm. It's quite an interesting property. Exactly the same applies to the dodecahedron and the icosahedron - the dodecahedron is the dual polyhedron of the icosahedron, as the cube is the dual polyhedron of the octahedron.

But a tetrahedron is its own dual polyhedron, connect up the middle of each side of a tetrahedron and you get another tetrahedron - known as being self-dual. It's the only regular/uniform polyhedron that's like that. There's infinitely many non-uniform ones of course, take a pyramid for example.

Oh, and I love your signature Visionaries, isn't it?

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[Last edited by calamarain at 09-19-2007 12:47 PM]

MartianInvader
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Well, I can't resist posting the follow-up question:

Take a four dimensional hypercube. Pass it through a 3-dimensional space starting with 1 corner (like you would move the plane through the cube standing on a corner), and stop halfway. What 3-dimensional shape intersects the space?



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Dex Stewart
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MartianInvader wrote:
Well, I can't resist posting the follow-up question:

Take a four dimensional hypercube. Pass it through a 3-dimensional space starting with 1 corner (like you would move the plane through the cube standing on a corner), and stop halfway. What 3-dimensional shape intersects the space?

Would I know you in real life, I would simply reply:

&#%$ you.

zonhin
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MartianInvader wrote:
Well, I can't resist posting the follow-up question:

Take a four dimensional hypercube. Pass it through a 3-dimensional space starting with 1 corner (like you would move the plane through the cube standing on a corner), and stop halfway. What 3-dimensional shape intersects the space?

A tetrahedron?

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[Last edited by zonhin at 09-26-2007 11:12 PM]

aztcg7
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Er...It's a cube, isn't it?

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zonhin
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aztcg7 wrote:
Er...It's a cube, isn't it?

Nope. It would form a tetrahedron that slowly grows then shrinks.

Edit: It's not really growing or shrinking. It would seem that way though/

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[Last edited by zonhin at 09-27-2007 01:42 AM]

aztcg7
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Would it be too much trouble to explain your reasoning? From what I understand, a 3-dimensional cross-section of a 4-dimensional object is similar to a 2-dimensional cross-section of a 3-dimensional object...

Oh, I now see my problem. I was thinking of when you do the same thing to a cube, you get a square, except when you do the corner, when you get the hexagon. So I was thinking if you did a 3-dimensional cross-section of a face-on hypercube. But that still doesn't adequately explain why it's a tetrahedron.

I guess I'm just not great at thinking in the 4th dimension of space.

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coppro
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I'm not that good with 4 dimensions, either.

I wish I were, because then I'd stop having to imagine complex function graphs while limiting myself to a single input axis.

zonhin
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aztcg7 wrote:
Oh, I now see my problem. I was thinking of when you do the same thing to a cube, you get a square, except when you do the corner, when you get the hexagon. So I was thinking if you did a 3-dimensional cross-section of a face-on hypercube. But that still doesn't adequately explain why it's a tetrahedron.

Okay, maybe a truncated tetrahedron. I dunno.

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I danced with the peanut butterflies
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Mikko
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This one is a lot easier to work out mathematically than by visualising. Let's say the hypercube has its corners at 0 or 1 on each axis, and we shove it through the 3D space starting with the (0,0,0,0) corner. Halfway through, this intersects six of the corners ((1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0) , (0,1,0,1) and (0,0,1,1)) and everything between them. It's easy to guess the answer from this, but it isn't too difficult to exactly work it out either.

By changing the coordinates to
x'=x-y-z+w
y'=-x+y-z+w
z'=-x-y+z+w
w'=x+y+z+w
it all becomes clear (technically, the results should then be scaled down, but that doesn't change the shape). The six corners are now located at (0,0,-2,2), (0,-2,0,2), (2,0,0,2), (-2,0,0,2), (0,2,0,2) and (0,0,2,2). These are clearly the corners of an octahedron.

calamarain
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Four dimensions make my brain hurt But it's a cool idea, and even harder to visualise than the 3D version.

I might be able to dig out an old game a genius friend of mine made. It was like Sokoban... but four-dimensional. It screwed with your mind.

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MartianInvader
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Rock on, Mikko. The answer is an octahedron, though a tetrahedron isn't completely off the mark. When you pass a plane through a cube, it starts as a growing triangle (3 faces meet at the corner), which then truncates itself bit by bit until it's a perfect hexagon (in the middle), and then 3 sides of the hexagon shrink away to make a triangle again (which then shrinks down to the point of the opposite corner).

For passing 3-space through a hypercube, you start with the point at the corner, then a tetrahedron grows (four hyperfaces touch each corner), then the corners of the tetrahedron start truncating themselves, which gives an octahedron, eventually giving a perfect octahedron, which then grows out 4 of its faces to points to remake a tetrahedron, which then shrinks down to a point.

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Chaco
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calamarain wrote:
It was like Sokoban... but four-dimensional. It screwed with your mind.

Jeebus, three dimensional Sokoban sounds hard enough, and I haven't even gotten close to solving some of the two-dimensional Sokoban puzzles

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Someone Else
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calamarain wrote:
It screwed with your mind.

It already has.