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Stella image in New York Review of Books

 
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robertw
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PostPosted: Mon Apr 12, 2010 2:41 pm    Post subject: Stella image in New York Review of Books Reply with quote

Hey guys,

An image of the tesseract (4D cube) made using Stella4D has just appeared in an article in the New York Review of Books. You can see it here: http://www.nybooks.com/articles/archives/2010/apr/08/he-conquered-the-conjecture/?pagination=false

It's the last image in the article.

[Edit: let me know if that link doesn't work. They've changed it three times now!]

Rob.
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oxenholme



Joined: 16 Jan 2008
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PostPosted: Sat May 15, 2010 5:22 pm    Post subject: Reply with quote

You can view a polygon (two dimensions) as a line (one dimension) by looking at it edge on.

But you cannot view it as a point (zero dimensions).

You can view or draw a two dimensional image of a polyhedron (three dimensions)

But you cannot view or draw it as a line (one dimension).

I wonder, therefore, whether you can draw or represent a four dimensional polytope in two dimensions.

You might be able to in three, but in two all you can show would be a two dimensional projection of a three dimensional representation of the four dimensional polytope. But, in so doing you will lose all concept of its hypershape.

A flat two dimensional net for a three dimensional polyhedron gives little idea of the eventual shape of said polyhedron.

I wonder how much idea the three dimensional net for a four dimensional polytope gives as to its eventual nature when assembled?
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robertw
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PostPosted: Mon May 17, 2010 12:54 pm    Post subject: Reply with quote

oxenholme wrote:
You can view a polygon (two dimensions) as a line (one dimension) by looking at it edge on.

But you cannot view it as a point (zero dimensions).

You can view or draw a two dimensional image of a polyhedron (three dimensions)

But you cannot view or draw it as a line (one dimension).

Actually I'm not sure I agree with this. I think rather it is just something our human instincts (always unreliable Smile) tell us. For example, you can view a polygon in two dimensions, right? But can you view a polyhedron in three? It's three dimensional, but we always view everything in 2D really. You would need to view it from a 4th dimension to be able to view it's full 3D structure all at once.

You can project something from any number of dimensions onto any other number of dimensions. But if going to a lower number, then you will always lose information. We regularly project 3D stuff onto our 2D screens, but our brains have evolved great software for reconstructing a mental model of 3D items from 2D images (the source for many optical illusions too). Our brains have not evolved any intuition for dealing with 4D though, so it's always more of a struggle.

Quote:
You might be able to in three, but in two all you can show would be a two dimensional projection of a three dimensional representation of the four dimensional polytope. But, in so doing you will lose all concept of its hypershape.

Don't forget that you could make a physical model of a 3D projection of a 4D polytope, thus only losing one dimension as in your other examples (I've made a few, eg http://www.software3d.com/Grit.php), but looking at these does not suddenly make 4D more intuitive. It is our brains that have trouble with 4D.

Quote:
A flat two dimensional net for a three dimensional polyhedron gives little idea of the eventual shape of said polyhedron.

I wonder how much idea the three dimensional net for a four dimensional polytope gives as to its eventual nature when assembled?

I've made some paper models of these too. Eg see http://www.software3d.com/TatNet.php. They're interesting brain twisters to think about.

The other way to view 4D shapes is to view their 3D cross-sections, and most interestingly, watch a 3D animation as the sectioning hyperplane moves through the 4D model. Can't make a paper model of it changing, but you can watch that in Stella4D Very Happy

Rob.
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guy



Joined: 11 Feb 2008
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PostPosted: Tue May 18, 2010 8:53 pm    Post subject: Reply with quote

Around a century ago, when extra dimensions first became popularised, people investigated these issues.

There is some evidence that Picasso used time and cubism as analogues of the fourth dimension - he was certainly fascinated by it. One painting has a succession of figures as if caught in multiple exposures while moving, and as each figure becomes further away in time it becomes a little more abstract and cubist. So position and cubism between them add third and fourth dimensions to the two-dimensional painting.

Coxeter tells of his friend Flinders Petrie who, if given a problem with a four-dimensional polytope, would spend a while visualising the figure, and then come up with the correct solution.

I have slowly developed an ability to visualise some four-dimensional aspects of a hypercube or a pentachoron (4-simplex), but I am a long way short of Petrie. It's kind of like visualising depth in a 2-D image and then doing it again independently of but alongside the first time.

Meanwhile, the human brain is remarkably adaptable. An area can be retrained to work in strange ways, If an important area is damaged, then nearby areas will start to help out. More weirdly, if a person is blinded, then the vision centres of the brain can apparently begin to retrain themselves to process other spatially related sensory inputs, giving a form of blindsight. There is also evidence that people blind from birth use their vision centres in this way.

It might yet prove possible for someone to methodically re-train some part of their brain to develop an ability to visualise in four dimensions.
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oxenholme



Joined: 16 Jan 2008
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PostPosted: Sat May 22, 2010 6:08 am    Post subject: Reply with quote

My brain is obviously too ingrained with 3D!

I can understand (for example) all the stages of a draw-polytope from zero through to three dimensions, but I cannot visualise how to effect the stage beyond three.

You have your X, Y and Z axes. How do you go beyond these?

You can convert a tessellation into three dimensions by removing shapes and folding the edges. In so doing the remaining shapes or polygons are not distorted in any way. They are merely re-oriented.

Is the four dimensional equivalent of a polygon or shape a cell or polyhedron? What is the "fold" like that will take it into four dimensions? To go from two to three only two polygons are necessary. How many cells are necessary to go from three to four?

Or am I being completely thick?
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robertw
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PostPosted: Sat May 22, 2010 7:30 am    Post subject: Reply with quote

oxenholme wrote:
My brain is obviously too ingrained with 3D!

Actually having thought some more, I think you were more right than I first thought! Here's my current thinking...

Viewing an object in any number of dimensions presumes light (or similar) travelling from the object to the viewer, and light travels along 1-dimensional lines. We can't get a sense of depth along that line, so the viewer always loses one dimension. This is why 3D objects only appear to us as 2D projections.

When this N-dimensional viewer looks at a M-dimensional object, they still lose one dimension. If the viewing direction lies in the hyperplane of the MD object, then we lose the ability to perceive that dimension, and the object appears a (M-1)-dimensional. Eg, when we view a ploygon side on so it appears as a line, or when we view a line end on to appear as a point. And as you said, you can never view a polygon to appear as a point. You can only lose at most one dimension.

On the other hand, if the viewing direction is not aligned with any of the M-dimensions, or even better when it is orthogonal to all those dimensions, then we can see the full structure of the object. This however is not possible when M = N, since there are no spare dimensions for our viewing direction. So N dimensional objects always appear as (N-1)-dimensional.

You can of course project an N-D object onto any number of dimensions, but this is a bit different from how a viewer might be able to see it.

Quote:
You have your X, Y and Z axes. How do you go beyond these?

Well, in our universe, there are only 3 dimensions, so it is not possible to build anything with more dimensions. Mathematically there is nothing special about 3 though. No reason to stop at 3. It's just our physical world that disallows it. The fact that it seems so intuitive that there can't be more than 3 is a human issue, not a mathematical one Smile

Quote:
You can convert a tessellation into three dimensions by removing shapes and folding the edges. In so doing the remaining shapes or polygons are not distorted in any way. They are merely re-oriented.

Is the four dimensional equivalent of a polygon or shape a cell or polyhedron? What is the "fold" like that will take it into four dimensions? To go from two to three only two polygons are necessary. How many cells are necessary to go from three to four?

One of the most difficult things to visualise in 4D is rotation. Our initial intuition says that since orientation in 3D requires 3 values (eg heading, pitch and roll), that it should require 4 values in 4D, right? Actually no, it requires 6! Think of 2D. There you only need 1 value. These are triangular numbers: 1, 3, 6 (the difference between successive numbers increasing by one each time).

The reason is that we think of rotation as being "about a line", ie we concentrate on what is not changed by the rotation rather than what is. We imagine rotation is still about a line in 4D, but no. In 4D rotation is about a plane! Think of 2D again, rotation is not about a line there (unless you think of a line that extends out of the 2D plane, but then it's 3D). In 2D we rotate about a point. The key is that rotation always happens within a plane. Think of what is changing rather than what is not.

Now you're ready to think about 3D nets for 4D shapes. The nets are like 3D shapes all stuck together face-to-face. It's hard for us to imagine, but the folding happens about the planes of those shared faces, by rotating in the plane of the two remaining dimensions. Just like 2D faces in 3D, each 3D cell rotates into 4D without distorting, but still maintaining full contact at their shared face. In 3D it would be locked solid, but the extra dimension allows it to rotate. Hard to imagine!

Rob.
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guy



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PostPosted: Wed Jun 09, 2010 3:42 pm    Post subject: Reply with quote

oxenholme wrote:
You have your X, Y and Z axes. How do you go beyond these?

Mathematically it is really easy, you just add a W axis at right angles to all the others. When you project a 3-D object onto the X-Y plane (say a computer screen), you just forget the Z dimension. Similarly, to project a 4-D object onto the plane, you just forget the W dimension as well. Harder to imagine is projecting the 4-D object into 3D by forgetting only the W dimension.

You can convert a tessellation into three dimensions by removing shapes and folding the edges. In so doing the remaining shapes or polygons are not distorted in any way. They are merely re-oriented.

Quote:
Is the four dimensional equivalent of a polygon or shape a cell or polyhedron? What is the "fold" like that will take it into four dimensions?

Yes, a polychoron is made up of polyhedra in just the same way a polyhedron is made up of polygons. You can even make an unfolded net of a polychoron, because the net is only 3-D. For example you can make the net of a cube by starting with a square, sticking four more on all round to form a cross and then adding one more to make a "lid". In just the same way you can make the net of a hypercube by starting with a cube, sticking six more on all round and then adding one more to make a "hyper-lid". The net of the cube folds along the joined edges to form the cube. The net of the hypercube folds along the joined faces - as Rob points out, rotation in 4-D is about a plane, not a line.

Quote:
To go from two to three only two polygons are necessary. How many cells are necessary to go from three to four?

Just two again - join two cubes and "hyper-fold" along the joined face to rotate one cube into the fourth dimension.
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oxenholme



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PostPosted: Sat Jun 12, 2010 6:10 pm    Post subject: Reply with quote

guy wrote:
...join two cubes and "hyper-fold" along the joined face to rotate one cube into the fourth dimension.


Back in the mid 60s I, as a third former, walked proudly into the sixth form mathematics room with the hypercube that I'd made from acetate sheet only to have it dismissed by a sixth former who told me it had three dimensions and that they could produce an equation for it.

They were correct.

The net for a cube is rigid in a two dimensional world.

The net for a hypercube is rigid in a three dimensional world.

Maybe one day I will be able to visualise hyper-folding.

Meanwhile, this arrived through the post this morning. I thought about this thread as I looked at the cover!

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guy



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PostPosted: Sat Jun 12, 2010 8:00 pm    Post subject: Reply with quote

You must be feeling eight miles high (though I prefer the 20 minute live version on the untitled album).
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