notes-math-visualizing4d

" brossa

Charles Howard Hinton came up with a system of colored cubes in 1888 that were supposed to help you teach yourself to grasp the fourth dimension. The system is outlined in his book "A New Era of Thought". The 125 cubes have different colored faces, edges, corners, and interiors, and by manipulating the cubes you're supposed to get a sense of the 3D projection of a 4D cube. I don't know how many hours you'd have to spend fiddling with cubes and trying to keep ten shades of green separate in your mind, but I remember reading a contemporary reviewer claiming that the effect was disturbing. " -- http://boards.straightdope.com/sdmb/showthread.php?t=538948

" Washoe Washoe is online now Guest

Join Date: Aug 2004 I read somewhere once that Minkowski claimed that he could visualize four-dimensional space. I’ve also read that Einstein claimed to have tried many times and failed. Reply With Quote "

 Nobody Nobody is online nowGuest

Join Date: Aug 2000 I really got into reading up on the fourth spacial dimension a few years ago. I have 12 links, and I'll share a few. Hope they help: 4D notes Hypersphere Higher Dimensional Geometry Steroscopic Animated Hypercube Thomas Banchoff

http://dogfeathers.com/java/hyprcube.html

 chorpler chorpler is online nowCharter Member Charter Member

Join Date: Apr 2002 Location: Vegas, baby! Posts: 3,083 In the introduction to his book The Fourth Dimension, Rudy Rucker said something that totally blew my mind when I read it as a young teenager. He said:

"What is really needed here is the concept of a fourth space dimension. It is very hard to visualize such a dimension directly. Off and on for some fifteen years, I have tried to do so. In all this time I've enjoyed a grand total of perhaps fifteen minutes' worth of direct vision into four-dimensional space. Nevertheless, I feel that I understand the fourth dimension very well."

(The Fourth Dimension, pp. 7-8)

I was greatly thrilled by the prospect that it might actually possible to get a vision of four spatial dimensions. And now it seems like Chronos can do it as well. And a few others here.

I suspect this is the book Fenris was talking about. There's also The Planiverse by A.K. Dewdney (who has unfortunately gone off into 9/11 conspiracy theories these days) which posits a much more "realistic" 2D universe than Flatland. It goes into detail about how the 2D creatures eat, excrete, live, travel, communicate, build technology, etc. Quite interesting. Last edited by chorpler; 11-07-2009 at 02:40 PM.

Reply With Quote #49 Old 11-07-2009, 06:07 PM Ponderoid Ponderoid is offline Guest

Join Date: Jan 2008 Quote: Originally Posted by Alex_Dubinsky View Post Is there stereoscopic software that lets you manipulate the hypercube, rotate it, etc, to train your brain? (I'm already good at cross-eye 3D btw.) There are several apps out there for visualizing and manipulating 4D Rubik's Cubes.

* Ponder Last edited by Ponderoid; 11-07-2009 at 06:08 PM. Reply With Quote

http://www.google.com/search?q=4d+rubik+cube

Old 11-07-2009, 10:50 PM snailboy snailboy is offline Guest

Join Date: Apr 2004 The reason it's impossible to visualize is because we see in polar coordinates. Our eyes see two angles (up/down and left/right) and (indirectly) distance. Pictures just remove the distance. Four-dimensional beings would see three angles and distance. Again, their pictures would just take away distance. They would look at this picture with the three angles in their eyes. We can't possibly visualize that. By talking about making holograms and such, you're swapping an angle for distance which won't work.

For one thing, we can't see the internal parts of a 3D object. A 4D being could see all points of it, including the interior. And they wouldn't simply see some parts as farther away than others, they would see them at a different angle, like we see all the points on a painting at different angles.

That is, unless they were looking at it from the same 3D realm, which would be like looking at a painting edge on. That's basically what you're trying to do by creating a hologram. Like I said, we only see distance indirectly, due to our stereoscopic vision and visual cues. You might as well say we see in two dimensions. That's why pictures look close to real life.

And I don't care what anyone says; I contest that no one can visualize a four-dimensional object. Imagine it and what properties it would have, possibly. But truly visualizing it would require a sense that we don't have. It would be like someone who was born blind imagining color. Reply With Quote

(irrelevant but interesting post from that page:

"One of the most interesting talks I've been to was given by Dmitri Tymoczko, who gave a presentation on his modern geometric view of music theory. As part of this talk, he displayed a mapping from the circle of fifths to a four-dimensional structure in which consonant chords are represented by adjacent points. To demonstrate the utility of this notion, he played an animation that showed the path taken by the chords in a piece by (IIRC) Schubert that has defied analysis by traditional theory. It was immediately obvious that the composer had an intuitive understanding of this structure. I have no idea whether he could visualize it, but it's not clear that that's important. "

)

http://www.urticator.net/maze/download.html

http://www.sciencenews.org/view/generic/id/35740/description/Seeing_in_four_dimensions

http://science.howstuffworks.com/science-vs-myth/everyday-myths/see-the-fourth-dimension.htm

http://eusebeia.dyndns.org/4d/vis/01-intro

http://pbr.psychonomic-journals.org/content/16/5/818.full.pdf

Postby thigle » Sun Feb 26, 2006 1:15 pm moonlord wrote:

    Try to impose yourself that those things further away in the 4th dimension are 'elsewhere', so these strange things are normal afterall.

and he got the point. :D

another (simpler) exercise for me was 4-simplex (aka pentatope or 5-cell or whathaveyou), from sci.math forums, by Donald Davis, march1999:

btw, moonlord or anyone, what is meant by "properly imagine a tesseract" ? does it allow one to rotate it freely in 4space and project it freely form any position into 3space ?

Postby thigle » Sun Feb 26, 2006 3:09 pm idea = unity of presences

to get an idea of a tesseract, one needs to be able to hold in one's awareness all the 8 3-cubes forming the hypersurface of tesseract. between these is a 4d chunk bounded by the tesseract.

what i don't understand, is the way the projection works from 4d to 3d ...

lets consider these analogies between seeing 3-cube(or looking at its 2-shadows in 2-plane, orthogonal to viewing axis) and 4-cube(or looking at its 3-shadows in 3-space):

in orthographic projection (from infinitely far):

3-cube, seen vertex first, which amounts to looking from viewpoint on 3-cube's diagonal, looks like hexagon with diagonals. 3-cube, seen face(2-cube = square) first, which amounts to looking at it from viewpoint at axis that passes through centre of the face and is orthogonal to it, looks like square.

between these 2 views of the 3-cube, one 'maximal' and one 'minimal', are all the other viewpoints with cube's image taking area between the hexagon and the square case. it is like seeing 3 axies of 3-space: one extreme it looks like 3 axes crossing, with 2,3,6-fold rotational symetry, other extreme is just a cross, one axis becomes a point of intersection.

for 4d case, projecting the 4-cube into 3-space, one extreme is the rhombic cuboctahedron with 4 diagonals from vertices where 3 edges meet. another extreme is a cube with its 4 diagonals. (one is cell first, other is vertex first, which is which ?)

now between these 2 extremes are the other projections, like the familiar one where one 3-cube is within another 3-cube and their vertices are connected, which gives other 6 3-cubes.

out of the blue, i gotta go now (my girlfriend needs me :oops:) but'll be back in the evening and completize my thoughts on this as well as pose some questions that might clarify what i don't get.

thigle Tetronian

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Top Re: How can I visualize 4D?

Postby Rkyeun » Wed Jun 03, 2009 7:18 pm When I first started imagining tetraspace, I used such diverse things as color, numerical value, and time.

Imagine a circle in a 2D graph. We will color it as follows: Starting from the center of the circle place a red dot. Draw concentric rings of color outwards towards the circumference. As you do this, change the color you use to draw so that the color approaches green. Do this more quickly as you progress, so there is quite a lot of green and not much red. This is the bottom half of a sphere in a 3D graph, where color corresponds to the Z-axis. To draw the upper part of the sphere we trace the circles back in towards the center on "top" of that, staying green for quite some time but then rapidly turning blue near the end. All points except the outer circumference have two different colors. Two different Z-values. Were drawn twice at different times.

Imagine a sphere in a 3D graph. We will color it as follows: Starting from the center of the circle place a red dot. Draw concentric shells of color outwards towards the surface. As you do this, change the color you use to draw so that the color approaches green. Do this more quickly as you progress, so there is quite a lot of green and not much red. This is the bottom half of a shape in a 4D graph, where color corresponds to the W-axis. To draw the "upper" part of the shape we trace the shells back in towards the center on "top" of that, staying green for quite some time but then rapidly turning blue near the end. All points except the outer surface have two different colors. Two different W-values. Were drawn twice at different times.

That shape is a kind of hypersphere. The glome perhaps? I'm not sure I know the right word for the 4D shape in which every point on its surface volume is the same fixed distance from its center.

I don't have to use color or transparency or ghostliness anymore. I can just directly in my head visualize four mutually perpendicular axis and the associated gridlines for graph paper. Five is tricky. For one brief instant I managed to visualize an infinite-dimensional space, but that gave me a headache for a week and I'm not eager to try again.

Rkyeun Dionian

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Top Re: How can I visualize 4D?

Postby Keiji » Sun Jun 14, 2009 11:26 am Yes, it's called the glome.

http://teamikaria.com/hddb/wiki/Glome

http://boards.straightdope.com/sdmb/showthread.php?t=223902