4D visualization starting with 3d objects in 4D

Discussions about how to visualize 4D and higher, whether through crosseyedness, dreaming, or connecting one's nerves directly to a computer sci-fi style.

4D visualization starting with 3d objects in 4D

Postby 4Dspace » Tue Jul 24, 2012 3:05 am

I think that trying to visualize 4D starting with tesseract or 3-sphere is rather hard. To really appreciate 4D we have to start with our familiar 3d objects, like a cube, a sphere, a tetrahedron. The surprising thing one notices right away is that the 3d objects appear rather flat in 4D. But not completely flat, like a square is flat. They still maintain their 3d space.

This may take some time, but once visualizing of a cube is accomplished, one can see a cube made of, say 27 cubes (3 per edge), which I described on one of the current threads. Then, a cube made of 8 cubes (2 per edge) becomes interesting too, because a tesseract is made of 8 cubes. So, the task becomes to arrange the 8 cubes in such a way that they make up a tesseract. To this effect I bumped into a paper today describing unfolding of the tesseract: http://unfolding.apperceptual.com/

Image

The same goes for a tetrahedron. 5 tetrahedra make up a simplex.

I believe this is a superior method than using the analogies. This forum shows that analogies do not actually help people visualize 4D. Doing projections onto 3D can be misleading as well. This method works for me.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Tue Jul 24, 2012 4:11 am

Found a better site where tesseract is folded: http://www.herenow4u.net/index.php?id=cd6129

Image

When a tesseract is unfolded, 8 cubes are forming it.
The planes in the depiction, signified as a, b, c, d, e are laying on each other when folded.

Folding of a tesseract: One out of 8 cubes forming the tesseract stays now in 3D. The other 7 cubes of the tesseract are moved through 4D and folded

Another good one from wiki: http://www.search.com/reference/Tesseract

Image

The "vertex first" is sort of how I see it.


But this is the best, from wiki again. Here you can see how those cubes are folded. Projected onto 3D, some cubes sort of intersect.

Image
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Re: 4D visualization starting with 3d objects in 4D

Postby wendy » Tue Jul 24, 2012 7:32 am

In fact, all cubes intersect. It's a squashed polytope, after all. If the cubes don't intersect with each other, then they are projected onto planes, as in the first three projections (face first, margin first, and edge first). The hexagon at the top is a squashed cube.

The vertex-first view is a tetrahedral antitegum.

There are similar figures in four dimensions, such as \3\ = m3o3o3m, which consists of five forshortened tesseracts, arranged completely around a point, and so forth. The general class \2\, \3\, \4\, and so forth, will always tile in N space.

In 3d the rhombic dodecahedron fills both roles of 1\Q double-cube and \2\ dual of SPC. In four dimensions, these are different: 1\1Q is the double-tesseract or 24choron, while \3\ has 20 faces only. They present in the identical fashion as the tesseract vertex-first.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Tue Jul 24, 2012 10:17 am

Here is another good page with very good animations: http://mscoconut.blogspot.com/2009/10/tesseract.html Towards the bottom of the page, there is an animation that shows how the 8 cubes fold into tesseract.

The page has a good breakdown of the 8 cubes in color. Plus good animations that show each cube, like this one: http://www.flickr.com/photos/ai-momo/40 ... 6/sizes/o/

But this is the best by far! Exploring the 8 cubes of tesseract from inside out, a video from a gamer.
Could not embed here -? Here is the link: http://www.youtube.com/embed/lLg3oLbwRsQ
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Re: 4D visualization starting with 3d objects in 4D

Postby wendy » Wed Jul 25, 2012 10:03 am

The particular video is actually a topological map of the surface of a 4d surface. It is no more 4D than supposing that a cube, unfolded like a chess-board, is three dimensions. You can see, for example, that the same room neighbours two rooms which is unmayly in three dimensions. This is because there are only three cubes around a square, not four.

Still, it is hardly 'four dimensions'. You have in these no more left the surface of the tesseract.

I suppose you should say what side of hyperspace you are viewing these from?
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Thu Jul 26, 2012 12:14 am

So, this gamer video, http://www.youtube.com/embed/lLg3oLbwRsQ. You can explore the tesseract from inside. Note the small model with 8 colored cubes in the upper right of the screen. It shows you where you are. And so, from inside the 3-space, the cubes look like regular cubes. It is their relationship to each other, i.e. which ones are adjacent to each other is interesting. From which cube you can get into which and in what sequence.

Also I was looking at this stereo projection
Image
Note that there are 2 sets of 4 cubes each: one set is seen if you follow the blue central edges and the other set is seen if you follow red. And while on this projection the 2 sets intersect each other, the 4 cubes that make up each set do not intersect with each other in 3D. In fact, they are adjacent to each other. So, what we see are 2 hyperplanes, each consisting of 4 cubes. In reality they are parallel. Their intersection on the projection reflects the fact that one hyperplane is in front of another to our POV.

In any rate, the opposite pairs of cubes (right-most and left-most, or bottom and top) do not intersect each other and neither are adjacent to each other.

And so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. I.e. the cubes completely fill its 4d space. But they are 3d themselves. I find this fact fascinating.

Same goes for the tetrahedron. 5 tetrahedra, i.e. 5 3d things, make up one simplex, a 4d thing. How can it be? Only because they are arranged in 4D?

This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?

For example. If we cheat a bit and take 8 equal cubes with edge=8 units, then 8 to the 4th = 4096 and this is the 4-volume of a tesseract with edge = 8units. And 8 cubed = 512 == the 3-volume of one of the cubes. Since we have 8 of them, we multiply 512 by 8 and get 4096 as well. So, in this particular (and unique, lol) case where the edge = 8 units, the 8 3-volumes == 1 4-volume.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Thu Jul 26, 2012 12:39 am

PS
Of course, this is of the same category as 6 squares with edges=6 units will equal to 6 cubed.

Area and volume are such different things... they are not compatible. But still, where is that bulk, that 4-volume in tesseract? The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them. They are all adjacent to each other, cutting the 4D space into 8 3d subspaces. Which is different from faces of a cube in 3D. There is 3d space in between those squares.

Oh! is the bulk in between the 2 hyperplanes? In between the outer faces of the cubes? I can sort of see it in between the blue and red inner edges:
Image
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Thu Jul 26, 2012 1:14 am

PPS
Still, from what I read, it appears to be true in regard to simplex. It does consist of 5 tetrahedra. There is nothing in between the edges. All faces are triangles. There is nothing in between.

So, it may be the case with the tesseract as well. I am not sure yet. It appears that simple stacking of 3d things in 4D will make up a 4d object. And by stacking, I mean like tiles in 2d, like 8 cubes making a larger cube in 3D. The 3d subspaces of 4D somehow amount to 4D bulk.

That's right. You can break the 4D space into 4 3d subspaces, which is not analogous to planes in 3D.
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Re: 4D visualization starting with 3d objects in 4D

Postby Ovo » Thu Jul 26, 2012 8:09 pm

Hey it's me again. :)

I've just read this thread and, no offense, but the only thing I want to do is to put an image of a 4D facepalm.
You're missing a huge piece of the puzzle to understand 4D. And this piece is the 4th dimension itself.
I believe you have been partly misled by the commonly made mistake/missing step in the 3D->2D analogy that I have pointed out here.
Aside from human mistakes like that, the analogy is perfectly valid and you should stop being so wary of it. It's a great thing that we are able to use this downward analogy (if we were in a 2D or a 1D world, we couldn't since the analogy would involve 0D which is a special case) and it helps a lot in understanding 4D visualization.

So, this gamer video, http://www.youtube.com/embed/lLg3oLbwRsQ. You can explore the tesseract from inside.
Not at all, this is not a perspective or orthogonal projection of a tesseract. It's a self-wrapping 3D map of the surface of a tesseract. As the title of the video says too, you are exploring just the surface of the tesseract. The POV is in the surface. As if, by analogy with exploring the surface of a cube, the POV was in the faces, looking straight at the edges with a 1D display.

And so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. I.e. the cubes completely fill its 4d space. But they are 3d themselves. I find this fact fascinating.
It's so fascinating because you take a wrong assumption for reality. Though reality isn't much less fascinating. :) The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space. You can't picture it? Well this is 4D! We can't picture it because it's unknown to us. The same way a person who sees only in shades of red and blue can understand the notion of other colors but can't get a faithful mental image of a colorful scene.

This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?
A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.

For example. If we cheat a bit and take 8 equal cubes with edge=8 units, then 8 to the 4th = 4096 and this is the 4-volume of a tesseract with edge = 8units. And 8 cubed = 512 == the 3-volume of one of the cubes. Since we have 8 of them, we multiply 512 by 8 and get 4096 as well. So, in this particular (and unique, lol) case where the edge = 8 units, the 8 3-volumes == 1 4-volume.
x(3-volume unit) = y(4-volume unit) is a false equation. The unit for 3-volumes and 4-volumes is different.
The 4-volume of a tesseract with cube faces of volume 8 unit³ is... 8 units. Simple enough.

Area and volume are such different things... they are not compatible.
The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.

The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
Really, get this idea out of your head. Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.
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Re: 4D visualization starting with 3d objects in 4D

Postby quickfur » Thu Jul 26, 2012 8:57 pm

Ovo wrote:[...] I believe you have been partly misled by the commonly made mistake/missing step in the 3D->2D analogy that I have pointed out here.
[...] Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.

And the blame partly rests on me, for not making it clear that when we look at a (2D image of a) 3D projection of a 4D object, we aren't seeing the projection from a "native" 4D point of view at all, but from the side, as it were. A 2D image of a 3D projection of a 4D object is about as helpful in visualizing 4D as a 1D projection of a 2D projection of a 3D cube is in visualizing 3D.

When we draw a diagram of the cube, we may draw something like this (say on a piece of paper):

Image

But, seen from the side, it looks like this:

Image

This is about as helpful in understanding what a cube is, as a 2D image of a 3D projection of a 4D tesseract.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Sat Jul 28, 2012 12:55 am

Ovo wrote:Hey it's me again. :)
So, this gamer video, http://www.youtube.com/embed/lLg3oLbwRsQ. You can explore the tesseract from inside.
Not at all, this is not a perspective or orthogonal projection of a tesseract. It's a self-wrapping 3D map of the surface of a tesseract. As the title of the video says too, you are exploring just the surface of the tesseract. The POV is in the surface. As if, by analogy with exploring the surface of a cube, the POV was in the faces, looking straight at the edges with a 1D display.

The video underscores that point that 3d subspace of 4D feels the same as our homey 3D. When you get into it, it does not look flat at all.

Ovo wrote:The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space.

Yes, supposedly. While our router still worked, I spent whole night looking for cross sections of tesseract or simplex, or any 4d object. Hoping to see an arrow pointing at its middle with caption: here is where the 4d bulk lives. Everyone shows fancy projections and no one wants to do a boring cross-section through the middle of a tesseract to show its bulk.

If you see such a cross-section of tesseract or simplex, please let me know.

In 3D, the analogy is to cut a cube in the middle. If the cutting plane II to a pair of faces, we get a boring square. Just like a face of the same cube. Except that the edges of this square represent the cube's faces and its area is a section of 3d volume. That's where volume of a cube lives. Why no-one does the same for a 4-cube? Or a simplex? I want to see such a cross-section.

Ovo wrote:
This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?
A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.

See, here you're mistaken. That's where 4D is vastly different from 3D. Because you can break up a 4D space into 4 3d subspaces. So, take a tesseract and place its vertex at the origin of 4 axes. (It's like putting a cube's vertex at (0,0,0); thus its 3 adjacent faces coincide with XY, XZ and YZ planes.) With the tesseract the result is this: XYZ, XYW, XZW, YZW. Now, the difference with 3D is that you can put a cube into each of those 4 subspaces, and they will fill in the bulk. Because you will orient the additional cubes in the 4th direction.

It's like the 8 cubes of tesseract. The 8 cubes can be stacked in 3D or the same cubes can be stacked in 4D.

Draw a 4D graph. And break it into 4 3d subspaces for your convenience. (instead of drawing I recommend 4 corners of real boxes representing 3 planes; label them). Play with these real 3d subspaces and try to imagine all 4 of them fitting perfectly. Alas, not possible in 3D, but you can examine each pair of 3d subspaces separately, where their planes coinside, say XYZ and XYW are adjacent at XY plane. You realize that there is a cube sitting in each of subspaces: XYZ, XYW, XZW, YZW, with its vertex pointing at the origin of the XYZW axes. The faces of all those 4 cubes are adjacent in 4D and they are stacked continuously.

That's because volume IS a volume. But area is not volume. And even though, granted, 3-volume is not the same as 4-volume, and mathematically in both cases, going 2D->3D or 3D->4D it's just another power, but the result is not the same. You can stack n-volumes and they will make up (n+1) volumes for n>2. You simply stack them into the n+1 direction.

Ovo wrote:
Area and volume are such different things... they are not compatible.
The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.

Here you are too quick to draw wrong conclusions. I am off in search of a theorem that n-volumes can add to (n+1) volumes for n > 2.

Ovo wrote:
The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
Really, get this idea out of your head.

Show me where the bulk lives. If it is there, you can do a cross-section of a tesseract to demonstrate it.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Sat Jul 28, 2012 1:58 am

PS
see, it turns out that 3D ≈ 0.25 of 4D for the same length l. But with 2D and 3D, 2D = 0 of 3D.

With 5D, 4D ≈ 0.2 of 5D
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Sat Jul 28, 2012 2:46 am

Here is the stereo 3D projection of 4 cubes stacked in 4D. I simply removed 4 edges from the tesseract projection above. What is left is 4 adjacent cubes that sit in the 4 3d subspaces of 4D, as discussed above:

Image

See how all 4 cubes are stacked continuously? Two rows of this make up a tesseract. Unless there is 4-volume in between those 2 rows, which I don't think so. Because tesseract has 8 sides. Which means that I can take any set of continuous 4 cubes and call it a row. Just as there is no space in between the 4 cubes above, the same will apply for any other combination. Thus the is no 4d bulk sitting somewhere in between there. 4 3-volumes add up to a 4-volume. You just have to stack them right. That's because 3-volume is never 0. An area has 0 volume though. Thus there is a huge difference between 3D and 4D, and most analogies are moot.
Last edited by 4Dspace on Sat Jul 28, 2012 5:33 am, edited 3 times in total.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Sat Jul 28, 2012 4:42 am

Here is another "row":
Image


So, we stack 2 cubes one on top of another, just like in 3D (that's the front cubes on the projection, a smaller one on the bottom and a larger one on top). The 3rd cube is tricky: its 2 adjacent faces, which are perpendicular to each other, go along the straight wall made up of 2 continuous faces of the 2 stacked cubes (that's the largest cube behind the front 2; you can see one of its faces on the right). The 4th cube simply fits on the left into the concave niche that results after you put the first 3 cubes right (that's the squashed cube on the left, and all of its faces are obscured by the 2 front cubes. You can see its 3 edges though: it shares the bottom visible edge with the bottom front cube and its other 2 visible edges it shares with the top front cube).

Nice? :D

But the first, "red" row is easier. It's projection is less distorted than this one.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Thu Aug 02, 2012 5:43 am

And here is some quotes from http://books.google.com/books?id=hZULAAAAYAAJ Geometry of Four Dimensions, in regard to a simplex (called pentahedroid there, which I changed) and its 5 tetrahedra, which is essentially the same as tesseract and its 8 cubes above, which confirms what I was saying above:

p 59

In a simplex each tetrahedron is adjacent to each of the other four.

It may not be very difficult to think of two adjacent tetrahedrons, even though they lie in different hyperplanes...

p 66

In a simplex, for example, there are 5 tetrahedra whose 20 faces fit together in pairs, each tetrahedron having a face in common with each of the other 4. We can take any one of these tetrahedra and place the other 4 upon it, all in one hyperplane, and then we can turn the 4 outside tertrahedra away from this hyperplane without separating them from the 5th or distorting them in any way, until we have brought together every pair of corresponding faces. The 5 tetrahedra together with their interiors now enclose a portion of hyperspace.


Another quote from the same book, underscoring the differences between spaces with even or odd number of dimensions, i.e. 3D and 4D:

p 14

There are also many properties in which space of an even number of dimensions differ from spaces of an odd number of dimensions, and these differences would hardly be recognized if we had only the ordinary geometries. Thus in space of an even number of dimensions roatation takes place around a point, a plane, or some other axis-space of an even number of dimensions, while in space of an odd number of dimensions the axis of a rotation is always of an odd number of dimensions (chap.IV)

So, there are more similarities between 3D and 5D than between 3D and 4D. That's why I say that most of the analogies used are moot.
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Re: 4D visualization starting with 3d objects in 4D

Postby wendy » Thu Aug 02, 2012 7:24 am

I have this book, too.

The main similarites between odd dimensions vs even dimensions, do not affect most of the analogies.

What happens, for example, is that the 'hairy ball', must have a crown or vortex in an odd dimension, but not in an even dimension. Another difference is that solid space, for having an even dimension, is of the oppostit sign to its dual nulloid (of -1 D), so Euler's characteristic equation reduces to 0, rather than 2, eg

cube 8v - 12e + 6h = 2 vs tesseract 16v - 32e + 24h - 8c = 0.

One can not use euler's equation to solve the number of faces of a 4d polytope, since this equation for the tesseract is 2Fv - 4F e + 3F h - Fc = 0, for all values of F. This is similar to the polygon equation, where P v - P e = 0 for all P, and different to 3d, where one can derive v,e,h from the equity of 2.

On the other hand, the application of equalities to define planes (ie a N-1 space has one equal sign, and thus divides space, while an N-2 space has two equal signs, and doesn't divide allspace), applies throughout.
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Re: 4D visualization starting with 3d objects in 4D

Postby 4Dspace » Mon Aug 06, 2012 1:08 am

Ovo wrote:Hey it's me again. :)
And so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. I.e. the cubes completely fill its 4d space. But they are 3d themselves. I find this fact fascinating.
It's so fascinating because you take a wrong assumption for reality. Though reality isn't much less fascinating. :) The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space. You can't picture it? Well this is 4D! We can't picture it because it's unknown to us. The same way a person who sees only in shades of red and blue can understand the notion of other colors but can't get a faithful mental image of a colorful scene.

This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?
A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.

Area and volume are such different things... they are not compatible.
The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.


The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
Really, get this idea out of your head. Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.


Here, Ovo, I found the reference I was looking for in the same great book, that's Geometry of 4D, 1914, available from google books. I have not found the theorem itself yet. But the quote is very interesting and totally in line with what I was saying. The quote about 8 3d volumes making up a hypervolume, just as I saw.
http://books.google.com/books?id=hZULAA ... &q&f=false

p 210 - 211

A spherical tetrahedron has 6 edges, each lying in the edge of a spherical dihedral angle whose interior contains the interior of the tetrahedton. The interior of one of these spherical dihedral angles contains also the interiors of 3 of the 15 tetrahedrons associated with the given tetrahedron as explained above, and its volume is = to the sum of the volumes of the 4 tetrahedrons whose interors are within it.

Writing A' for the opposite point to A, the other extremity of the diameter to A, and so for other points, we let T denote the volume of the tetrahedron ABCD, T1 the volume of A'BC,T12 the volume of A'B'CD, and so on. ABC'D' is congruent to A'B'CD, and we have T34 = T12 etc.

The interior of the dihedral angle C-AB-D contains the interiours of the 4 tetrahedrons whose volumes are T, T1, T2 and T12. If Ф12 is the measure of the dihedral angle AB in terms of a right dihedral angle, and if we take fo unit of volume 1/16 of the volume of the hypersphere, we shall have the relation

T + T1 + T2 + T12 = 4 Ф12

There are 6 if these equations, and in addition one equation expressing the fact that the sum of the 8 different volumes is equal to the volume of a half-hypersphere, namely,

T + Sum(T1) + Sum(T12) = 8

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Re: 4D visualization starting with 3d objects in 4D

Postby wendy » Mon Aug 06, 2012 7:25 am

The quote on 211 to 212 is actually a reference to solid angle, which is measured as surface in all dimensions. In 4D, the surface is a 3-space.

It's like saying 3d things are 2D, because the sphere and photos are 2d surfaces. It entirely misses the point. I seriously think that you're trolling the list. In any case, what you're presenting as 4D is not what is commonly recieved as 4D.
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