attempted drawing a tetracube, opinions?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

attempted drawing a tetracube, opinions?

Postby Atropos » Tue Dec 21, 2004 7:55 am

http://tinypic.com/ycdhv
my interpitation anyway...would this be correct?
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Postby jinydu » Tue Dec 21, 2004 10:47 am

You should probably also say that all line segments are of equal length and that they intersect only at right angles. But the drawing seems to be technically correct.

Unfortunately, human brains are generally trained to see 3D objects. Thus, while I know that the picture is supposed to represent a tetracube, my brain sees two connected 3D rectangular prisms.
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Postby Keiji » Tue Dec 21, 2004 12:16 pm

I only see connected prisms because the connections are green, a relatively light color. If the green lines were made into dark red or something, it'd be easier to visualise. ;)
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Re: attempted drawing a tetracube, opinions?

Postby quickfur » Tue Jan 04, 2005 8:16 pm

Atropos wrote:http://tinypic.com/ycdhv
my interpitation anyway...would this be correct?


It's correct as far as the connectedness of the lines are concerned. It would correspond to a parallel projection of the tetracube into 2-space.

For me, though, I still prefer to visualize the tetracube using its perspective projection into 3-space, i.e., a "cube within a cube". That would be the cell-first projection of the tetracube. Another favorite projection of the tetracube of mine is the vertex-first projection, which is a rhombic dodecahedron with the closest tetracube vertex at its origin.
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Postby 3l3ctr0 » Wed Feb 16, 2005 10:35 pm

the drawing of ur 4d tetracube was cool but what i understood of the 4D cube that all the lines had to be parelell. am i wrong or...
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Postby wendy » Wed Feb 16, 2005 11:01 pm

Although the tesseract does have its 32 edges parallel in four groups of 8, one also notes that a picture of a four-dimensionl thing would be three-dimensional with a good deal of prospective built in.

The prospective of these lines cross at some real point, being the "line at infinity".

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Postby jinydu » Thu Feb 17, 2005 3:43 am

3l3ctr0 wrote:the drawing of ur 4d tetracube was cool but what i understood of the 4D cube that all the lines had to be parelell. am i wrong or...


If all the lines were parallel, they would never intersect each other, and there would be no vertices! No, lines are parallel or perpendicular (or skew).
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Tetra-rectangle

Postby Gilles » Fri Feb 18, 2005 9:39 pm

The problem of this tetracube is (besides the fact that it's a tetrarectangle) that the movement you'd see, if you take the ABCDLIJK rectangle, and displace it until you get the EFGHPNMO rectangle, is a 3-d movement.
As 4-d is refered to as being spacetime, with time being defined by the second law of thermodynamics, you'd expect the movement of the rectangle (or cube) to be towards the in- or outside of the rectangle.

Unfortunately, human brains are generally trained to see 3D objects. Thus, while I know that the picture is supposed to represent a tetracube, my brain sees two connected 3D rectangular prisms.


Yes, jinydu, it's supposed, but it doesn't, that's why you can't see it

To draw a tetracube, i'd draw a cube inside another cube. Then, connect the inside corners with the outside corners.

If you realy want to give yourself an impression of the 4th, imagine the inside cube is the outside cube, and that everything in between, represented by the lines connecting the cubes, is an infinite number of new cubes, wich actualy are the same cube.

It's hard to draw on a pc, but it's doable on a piece of paper
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Line of infinity

Postby Gilles » Fri Feb 18, 2005 9:43 pm

My cube solves the problem of the point where the prospective of the interconnecting lines meet. The point is the center of both cubes.
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Re: Tetra-rectangle

Postby houserichichi » Fri Feb 18, 2005 11:10 pm

Gilles wrote:The problem of this tetracube is (besides the fact that it's a tetrarectangle) that the movement you'd see, if you take the ABCDLIJK rectangle, and displace it until you get the EFGHPNMO rectangle, is a 3-d movement.


You're thinking too close-minded. If you slide that first rectangle into the second you necessarily have a self intersection. Think of that picture as being a four dimensional box...you can't slide one wall through another in reality, so a tetracube would require 4 spatial dimensions to work. That's called a representation, it's not how the things actually look.

Look up Klein Bottle on the internet. You'll see a representation of one where it's a big bulb on the bottom and a hooked top that goes back inside itself - a real Klein bottle is four dimensional and DOES NOT INTERSECT ITSELF. Same goes for the hypercube.

Gilles wrote:As 4-d is refered to as being spacetime, with time being defined by the second law of thermodynamics, you'd expect the movement of the rectangle (or cube) to be towards the in- or outside of the rectangle.


No again. Spacetime is something to do with physics. This is a geometry forum - and while spacetime and geometry are closely related, the tetracube in question is a mathematical one. It requires four SPATIAL dimensions. If we were able to see one in "real life" then spacetime would necessarily be AT LEAST five dimensional.

Gilles wrote:To draw a tetracube, i'd draw a cube inside another cube. Then, connect the inside corners with the outside corners.


Again, this is a representation. If you could actually see a four dimensional cube it would not look like this - in four dimensions a tetracube does not look like a 3-cube inside another. In fact you wouldn't even be able to see through the outter cube as it would have a 2-dimensional boundary blocking the way.

Gilles wrote:It's hard to draw on a pc, but it's doable on a piece of paper


Drawing it is not seeing it. You are representing what the hypercube would look like if it was projected into two dimensions.

When you draw a cube on a sheet of paper it is not three dimensional. It may look it, but it is not. You can't turn it around, you can't touch a particular face of it. Same goes for your hypercube ideas - they are representations.

You do not, have not, and will not see a hypercube as much as popular (and unpopular) literature, websites, and people will claim.
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Postby Gilles » Sat Feb 19, 2005 4:04 pm

Too close-minded, :D

I know it's just a 2-d representation, and that we're talking geometry here, but I only try to show you an impression.

When you draw a cube on a sheet of paper it is not three dimensional. It may look it, but it is not. You can't turn it around, you can't touch a particular face of it. Same goes for your hypercube ideas - they are representations.


That's why i also explain how to visualise the "real" thing :

If you realy want to give yourself an impression of the 4th, imagine the inside cube is the outside cube, and that everything in between, represented by the lines connecting the cubes, is an infinite number of new cubes, wich actualy are the same cube.
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Postby houserichichi » Sat Feb 19, 2005 4:28 pm

If you realy want to give yourself an impression of the 4th, imagine the inside cube is the outside cube, and that everything in between, represented by the lines connecting the cubes, is an infinite number of new cubes, wich actualy are the same cube.


Again, this is a representation. It's not "the real thing." It's like a being in a 2D universe saying that a 3D cone looks like a line that grows from big to small or small to big depending on what point passes through his plane of vision. The 2D person can only visualize what the cone looks like, he can't see it - and the same logic extends to us. We don't live in 4 spatial dimensions, we can't see in 4 spatial dimensions, and thus we can never see a 4 dimensional shape.

As per the close-minded comment it was in relation to "seeing what you want to see". Maybe I should have said "too open minded" :wink: [/code]
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also

Postby zoralink » Tue Mar 08, 2005 4:45 pm

one thing to think about in an attempt to draw a teracube is something that has bothered me, but perhaps it has been explained and I haven't come across it yet;

if you imagine a 2d world, there are two ways to connect two points, straight through the 2d world, which is the shortest way, I believe, and also through the third dimension, where two points can be connected without the knowing of the 2d person, they are linked through the third dimension, right?

but doesn't that mean that the link through the third dimension is longer than the regular one? Why am I given the understanding then, that two 3d objects linked through the fourth dimension is shorter than linkage through 3d space? Do I understand something wrong?

I have even heard references to such a link being an ultimate shortcut, but....
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Postby houserichichi » Tue Mar 08, 2005 5:10 pm

Do you mean if you have, say, a sheet of paper and on it two points the two ways to connect the points would be

1. Stay on the paper and connect them via a straight line
2. From the first point start drawing out into space then eventually end up again at the second point

Is that what you meant? If so then obviously the 1st method would produce the shorter distance (basic geometry), but I feel as though I missed a point in your post.

As far as distances being shorter through the fourth dimension it depends on what "shape" the four dimensions are. Think back to our paper exame again. Roll that piece of paper into a tube and on the top mark a dot and on the bottom mark a dot. It would actually be shorter for one to connect the two points via a line OFF of the paper (thus in the third dimension) rather than traverse the entire line ALONG the paper. Sometimes it makes more sense to travel into a "hyperspace" to minimize distances.

Did that help?
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Postby zoralink » Tue Mar 08, 2005 5:54 pm

that helps, thanks

hey, does that mean we might live in a 3d world folded like that paper tube?

but in a different, more complicated foldy 3d way?
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Postby jinydu » Tue Mar 08, 2005 6:31 pm

zoralink wrote:that helps, thanks

hey, does that mean we might live in a 3d world folded like that paper tube?

but in a different, more complicated foldy 3d way?


Maybe. But so far, experimental observations have not uncovered any evidence that the universe is curved on the largest scales.
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oh, I see

Postby zoralink » Wed Mar 09, 2005 4:18 am

I heard references to space time being discovered as curved, the only thing that confused me is how a three dimensional world can curve through the fourth without moving, but that's just something we can't inheritly conceive.

thanks for the help; I am enlightened. What about this thought?

the number of corners of a square is 4, I don't know if a "two cornered" line would be applicable, but it might, but in 3d, a cube has 8 of such corners, might the increase in corners have a mathematical pattern as you go up dimensions, but going from 4 to 8 might not help, you can either make it have 16 corners, or the increase might be exponential.
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Postby jinydu » Wed Mar 09, 2005 4:29 am

Yes, a 4D hypercube does have 16 corners.

This is not just speculation. There's a very detailed mathematical theory on higher-dimensional geometry. The proofs are as rigorous as those in 2D and 3D geometry.

http://mathworld.wolfram.com/Hypercube.html
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Postby houserichichi » Wed Mar 09, 2005 12:43 pm

how a three dimensional world can curve through the fourth without moving


It's a strange concept at first and requires some pretty heavy math to describe properly, but an n-dimensional shape doesn't necessarily require an (n+1)-dimensional embedding space to curve in, in which case we say it has intrinsic curvature. Our Earth is three dimensional with its two dimensional surface curved into the third dimension...but for all intents and purposes if there was no "space" then there'd be nothing wrong with a curved Earth still, it just means we could never leave it!
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Postby Argannon » Wed May 25, 2005 1:58 am

That would be really hard to render.
A forth dimensional object rendered for a three dimensional mind that's using a two dimensional display for viewing a two dimensional rendering of forth dimensional object.

I think that may get it. It's a bit redundent, though.
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