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 Post subject: Representing 4D
PostPosted: Tue Nov 12, 2013 4:13 pm 
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Maybe it's a stupid question but there you go:
If we can represent 3D on 2D, could we represent 4D on 3D?

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 4:24 pm 
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Eliasthecollector93 wrote:
Maybe it's a stupid question but there you go:
If we can represent 3D on 2D, could we represent 4D on 3D?


In simulations yes but not very well in a physical form.

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 4:26 pm 
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Eliasthecollector93 wrote:
Maybe it's a stupid question but there you go:
If we can represent 3D on 2D, could we represent 4D on 3D?

I'll assume you mean spatial dimensions...
Yes of course. This is actually what happens when you have the shadow of a 4D object projected into 3D:
Attachment:
4d-Hypercube.gif
4d-Hypercube.gif [ 276.73 KiB | Viewed 3379 times ]

It's the shadow of a hypercube, rotating along the YW plane (I think).
However this is very misleading, as we're only looking at this from a 3D perspective (pretend it's a 3D hologram).

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 4:28 pm 
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Yes.
Image

And this is a hypercube unfolded into 3D space:
Image

The same way a cube can be unfolded into 2D space like this:
Image

The problem with projecting it into 3D space, as you can see, is that the "cells" become distorted, the same way a cube's square faces become distorted when depicted in 2D space. In normal 4D space, the cells are connected to all the other cells and remain proportional.

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 4:32 pm 
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In a way, we can represent the fourth dimension with the 3 dimensional coordinate plane. Let's consider a fly in a box, for example. Now, let's say that this fly is flying around the box, and its location is given by the coordinates of x, y, and z, (3,4,5) for example. This represents the point that the fly is at within the three dimensional space. To add the fourth dimension, which is time, we'd incorporate another factor into that coordinate. Let's say, at 4 seconds, the fly is at coordinate (3,4,5). We could rewrite that to say (3,4,5) @ t=4 seconds where t represents time.

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 4:43 pm 
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cubeguy314 wrote:
In a way, we can represent the fourth dimension with the 3 dimensional coordinate plane. Let's consider a fly in a box, for example. Now, let's say that this fly is flying around the box, and its location is given by the coordinates of x, y, and z, (3,4,5) for example. This represents the point that the fly is at within the three dimensional space. To add the fourth dimension, which is time, we'd incorporate another factor into that coordinate. Let's say, at 4 seconds, the fly is at coordinate (3,4,5). We could rewrite that to say (3,4,5) @ t=4 seconds where t represents time.


The fourth dimension is not a dimension of time, although that seems to be a common misconception. It's a theoretical additional spatial dimension, of which there are many (5th, 6th, etc...). What you've described is a three-dimensional space with a single dimension of time.

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 5:19 pm 
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Obligatory 4D Rubik's Cube screenshot:
Attachment:
3x3x3x3.png
3x3x3x3.png [ 322.48 KiB | Viewed 3309 times ]


That's Magic Cube 4D (MC4D).

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 7:02 pm 
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cubeguy314 wrote:
In a way, we can represent the fourth dimension with the 3 dimensional coordinate plane. Let's consider a fly in a box, for example. Now, let's say that this fly is flying around the box, and its location is given by the coordinates of x, y, and z, (3,4,5) for example. This represents the point that the fly is at within the three dimensional space. To add the fourth dimension, which is time, we'd incorporate another factor into that coordinate. Let's say, at 4 seconds, the fly is at coordinate (3,4,5). We could rewrite that to say (3,4,5) @ t=4 seconds where t represents time.


Let's stress that 4d=time is only a theory :) I tend to disagree because we are talking about spatial dimensions, and in my limited, teenage view of metaphysics, time is just a linear expression of spatial existence. (I actually don't believe time exists but that's a philosophical discussion for another thread!)

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 12, 2013 7:24 pm 
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Also here's a fun video from Minute Physics. It's not exactly about visualizing a 4th dimension, but it's cool nonetheless.
http://youtu.be/M9sbdrPVfOQ

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 4:00 am 
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Eliasthecollector93 wrote:
Maybe it's a stupid question but there you go:
If we can represent 3D on 2D, could we represent 4D on 3D?


Oh yes! Perhaps we cannot present a 4D object in our 3D world (that is, until we unlock our brains),
but we may surely use the 4D symmetry to make some 4Ds puzzles. Here are some of my videos:

http://www.youtube.com/watch?v=YL-vlONT2QM
http://www.youtube.com/watch?v=zj1BZCwxf3k
http://www.youtube.com/watch?v=Drs98RH1sic

;)


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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 4:06 am 
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Brandon Enright wrote:
Obligatory 4D Rubik's Cube screenshot:]

That's Magic Cube 4D (MC4D).


Semi OT from the topic at hand, but has anyone here ever successfully solved this puzzle or any of its higher order variants (like the 4x4x4x4 or 5x5x5x5, or even the 3x3x3x3x3)? It's such a fun puzzle to play around with, but solving it seems so much more difficult. :shock:

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 8:53 am 
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Ferd Berfel wrote:
Semi OT from the topic at hand, but has anyone here ever successfully solved this puzzle or any of its higher order variants (like the 4x4x4x4 or 5x5x5x5, or even the 3x3x3x3x3)? It's such a fun puzzle to play around with, but solving it seems so much more difficult. :shock:

Yes. Several people have even solved the 3x3x3x3x3x3x3:

:shock:

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 10:18 am 
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One metaphor I like for conceptualizing n-dimensional objects from a n-1 dimensional organism's prospective is to view the n-1 dimensional features of the n dimensional whole as rooms connect by walls representing the n-2 dimensional features of the whole.

For example:
You have a 2-dimensional organism on the face of a cube.
This 2 dimensional being precieves the face as a square room with doors in each wall(the walls being 1 dimensional line segments) connecting to other rooms with the same layout.
As the 2 dimensional being explores the rooms, laying down markers to show where its been, it discovers that , by going in a straight line, he returns to where he started after the fourth room, an impossible loop from a 2 dimensional perspective.
Likewise, when the 2 dimensional being tries turning left at each new room, he returns to the start after 3 rooms instead of the expected four, another impossible loop from a 2D perspective.
An infinite square grid makes sense to the 2d being, but the cube presents a closed space of 6 square rooms connected in a manner that should be impossible, but makes perfect sense to us 3d beings.

Expanding to the Tesseract, a 3D being would see the individual cells as cubic rooms, each with a door in every face leading to other cubic rooms.
Again, the 3D being would travel in a straight line and find themselves back where they started after 4 rooms instead of an infinite lines of rooms that makes sense to the 3D perspective.
They would likewise discover that only three rooms surround an edge as opposed to the four of a cubic grid.
Just as the 2D being finds the squares of the cube connected in what they reason to be impossible, the 3d being would likewise see the cubes of the tesseract connected in an impossible fashion.

Using this metaphor, I have a concept of what a 3d being would see while travelling inside of a pentachoron, tesseract, polyhedral pyramid, polyhedral prism, or douprism, that I still have no grasp of the 16-, 24-, 120-, or 600-cells.

As for 3d shadows of 4d objects, my first hand experience with Zome Tool tells me it is quite excellent for building such shadows in a wireframe style, though best I can tell, a shadow of the tesseract is impossible and the Pentachoron and tetrahedral and Octahedral prism/ require green struts, which are not part of the basic Zome sets.

Speaking of shadows, does anyone know if the Cuboctahedral Pyramid is unique in that its shadow can be constructed in 3d with all edges of equal length?

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 11:25 pm 
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An interesting youtube video on imagining dimensions up to the 10th

http://www.youtube.com/watch?v=gg85IH3v ... qnQvKmVCIg

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 11:38 pm 
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TheCubingKyle wrote:
An interesting youtube video on imagining dimensions up to the 10th

http://www.youtube.com/watch?v=gg85IH3v ... qnQvKmVCIg


I've watched most of this one before. Emphasis on "imagining."

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 Post subject: Re: Representing 4D
PostPosted: Wed Nov 13, 2013 11:41 pm 
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TheCubingKyle wrote:
An interesting youtube video on imagining dimensions up to the 10th

http://www.youtube.com/watch?v=gg85IH3v ... qnQvKmVCIg

I just now remembered there's a really cool channel called wildstar2002, which has a bunch of higher-dimensional animations. They're really animated very well!

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 Post subject: Re: Representing 4D
PostPosted: Thu Nov 14, 2013 1:08 am 
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Ferd Berfel wrote:
Brandon Enright wrote:
Obligatory 4D Rubik's Cube screenshot:
That's Magic Cube 4D (MC4D).

Semi OT from the topic at hand, but has anyone here ever successfully solved this puzzle or any of its higher order variants (like the 4x4x4x4 or 5x5x5x5, or even the 3x3x3x3x3)? It's such a fun puzzle to play around with, but solving it seems so much more difficult. :shock:

The higher-dimensional twisty puzzles fall to standard commutators pretty much like every other twisty puzzle. Solving them is about breaking them down into small steps and tackling these baby problems one at a time.

One of the trickier things about higher dimensional twisty puzzles is that you end up with pieces with lots of different stickers. For example the "corners" on the 3^4 Rubik's Cube have 4 stickers rather than 3. Finding the piece your looking for becomes cumbersome and handling all of the orientations pieces can be in is somewhat tricky too. Everyone should give the 3x3x3x3 a try :D .

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 Post subject: Re: Representing 4D
PostPosted: Thu Nov 14, 2013 5:17 am 
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I was thinking it would be nice to show a series of 2D "nets" for increasing number of dimensions...

For example, the first 3D image below could be unfolded further into 2D. Similarly, a 5D hypercube could be unfolded into 4D, and then unfolded further into 3D, and then 2D. And so on with 6D, 7D, etc. You would end up with a series of 2D nets.

Thus in theory, I think/propose we can represent any higher dimension in 2D! :D

Modern-Day Warrior wrote:
And this is a hypercube unfolded into 3D space:
Image

The same way a cube can be unfolded into 2D space like this:
Image

One could even prepeat this series for triangles/tetrahedra, or any other simple shape...

PS. On a side note, I am used to representing multiple dimensions on a single 2D chart. For example, bubble charts are essentially 3D scatter charts, where the Z dimension is represented by bubble size. And you can represent further dimensions with bubble shape, fill colour, shading intensity, line colour, etc. But that quickly gets VERY complicated to interpret!

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 Post subject: Re: Representing 4D
PostPosted: Thu Nov 14, 2013 8:28 am 
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I once made a tesseract to play with.

Trying to comprehend a 4D object by looking at a 2D projection of its 3D projection is silly. Makes me wish we had 3D hologram projectors already.

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 Post subject: Re: Representing 4D
PostPosted: Thu Nov 14, 2013 8:32 am 
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Coaster1235 wrote:
Trying to comprehend a 4D object by looking at a 2D projection of its 3D projection is silly.

Why? Even if it does not give an intuitive picture, all the 4D information is kept in the 2D net/projection, isn't it?

Incidentally, isn't that precisely what you're looking at here (on your 2D screen):

Image

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Last edited by KelvinS on Fri Nov 15, 2013 7:10 am, edited 3 times in total.

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 Post subject: Re: Representing 4D
PostPosted: Thu Nov 14, 2013 8:38 am 
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KelvinS wrote:
Why? Even if it does not give an intuitive picture, all the 4D information is kept in the 2D net/projection, isn't it?
It is, but for me it would be easier to comprehend with less projections. For example, I have no idea what the tesseract is doing when it's rotated in the zw plane. I guess you're better at thinking spatially than me :D

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 Post subject: Re: Representing 4D
PostPosted: Fri Nov 15, 2013 7:29 am 
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KelvinS wrote:
I was thinking it would be nice to show a series of 2D "nets" for increasing number of dimensions...

For example, the first 3D image below could be unfolded further into 2D.

Likle this - a 4D hypercube unfolded into 3D, and then unfolded further into 2D:
Attachment:
4Dcube2D.png
4Dcube2D.png [ 14.86 KiB | Viewed 3017 times ]

I wonder if this would give a fractal with increasing number of dimensions?

Also interesting to count the number of points in this series:

2D: 1
3D: 8
4D: 16
5D: 32
6D: 64

How many square faces in each dimension?

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 Post subject: Re: Representing 4D
PostPosted: Sat Nov 16, 2013 8:16 am 
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TheCubingKyle wrote:
cubeguy314 wrote:
In a way, we can represent the fourth dimension with the 3 dimensional coordinate plane. Let's consider a fly in a box, for example. Now, let's say that this fly is flying around the box, and its location is given by the coordinates of x, y, and z, (3,4,5) for example. This represents the point that the fly is at within the three dimensional space. To add the fourth dimension, which is time, we'd incorporate another factor into that coordinate. Let's say, at 4 seconds, the fly is at coordinate (3,4,5). We could rewrite that to say (3,4,5) @ t=4 seconds where t represents time.


Let's stress that 4d=time is only a theory :)


It's not a theory, it's a definition. You can define time as a dimension to help model reality or whatever. In this context time is not the fourth dimension since we are talking about spatial dimensions. A simple but important distinction!


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 Post subject: Re: Representing 4D
PostPosted: Sat Nov 16, 2013 9:15 am 
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To be complete, any single independent continuous property or variable can define, or be defined by, a dimension (in the broadest sense).

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 Post subject: Re: Representing 4D
PostPosted: Sat Nov 16, 2013 9:49 am 
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KelvinS wrote:
I was thinking it would be nice to show a series of 2D "nets" for increasing number of dimensions...

For example, the first 3D image below could be unfolded further into 2D.

Your unfolded tesseract net isn't correct. A tesseract only has 24 squares. When it was unfolded into 8 cubes in 3D some of the 4D relationships between the cubes aren't easy to understand anymore. Those cubes are touching each other in a lot more ways than is shown in 3D and when you unfolded the 3D shape into 2D you included squares on some of the cubes that are actually shared between cubes.

I have colored squares that are actually the same square the same color:
Attachment:
tesseract_unfolded.png
tesseract_unfolded.png [ 8.58 KiB | Viewed 2952 times ]


The bottom face of the bottom cube is the same square as the top face of the top cube.

On the cube, there are three "interlocking rings" of 4 faces. For example, one of them is F R B L and another is U B D F and the last is U R D L. On the tesseract there are 6 interlocking rings of 4 cubes. One of the rings is the 4 cubes vertically stacked. Another is the 4 attached to the side of the vertical stack. When you picture the tesseract in this way it's easier to see how interconnected these cubes are and why there are only 24 square faces.

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Last edited by Brandon Enright on Sun Nov 17, 2013 2:17 pm, edited 1 time in total.

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 Post subject: Re: Representing 4D
PostPosted: Sat Nov 16, 2013 12:11 pm 
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^Speaking of face counts on polychora, isn't there a rule analogous to counting edges on a polyhedron?

i.e.
The number of edges of a polyhedron is half the sum of the edges of its faces because every edge is shared by two faces.
So is the following similar statment true?:
The number of faces of a polychoron is half the sum of the faces of the cells because each face is shared by two cells?

Also, is it correct to view 3D Honeycombs as degenerate polychora in the same manner that 2D tilings can be viewed as degenerate polyhedra.

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 Post subject: Re: Representing 4D
PostPosted: Sat Nov 16, 2013 10:08 pm 
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Jeffery Mewtamer wrote:
^Speaking of face counts on polychora, isn't there a rule analogous to counting edges on a polyhedron?

There is a branch of mathematics that deals with this sort of thing: https://en.wikipedia.org/wiki/Polyhedral_combinatorics

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 Post subject: Re: Representing 4D
PostPosted: Sun Nov 17, 2013 4:45 am 
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Brandon Enright wrote:
KelvinS wrote:
I was thinking it would be nice to show a series of 2D "nets" for increasing number of dimensions...

For example, the first 3D image below could be unfolded further into 2D.

Your unfolded tesseract net isn't correct. A tesseract only has 24 squares. When it was unfolded into 8 cubes in 3D some of the 4D relationships between the cubes isn't easy to understand anymore. Those cubes are touching each other in a lot more ways that in shown in 3D and when you unfolded the 3D shape into 2D you included squares on some of the cubes that are actually shared between cubes.

I have colored squares that are actually the same square the same color:
Attachment:
tesseract_unfolded.png


The bottom face of the bottom cube is the same square as the top face of the top cube.

On the cube, there are three "interlocking rings" of 4 faces. For example, one of them is F R B L and another is U B D F and the last is U R D L. On the tesseract there are 6 interlocking rings of 4 cubes. One of the rings is the 4 cubes vertically stacked. Another is the 4 attached to the side of the vertical stack. When you picture the tesseract in this way it's easier to see how interconnected these cubes are and why there are only 24 square faces.

Wow, thanks! Are there *any* external faces, or are they *all* folded in on each other???

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 Post subject: Re: Representing 4D
PostPosted: Sun Nov 17, 2013 1:23 pm 
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Every 2D face is between 2 3D faces. It's just like how every edge on a cube touches two faces. :)


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 Post subject: Re: Representing 4D
PostPosted: Sun Nov 17, 2013 4:31 pm 
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KelvinS wrote:
Wow, thanks! Are there *any* external faces, or are they *all* folded in on each other???

Nope! Only 3D "faces" (called cells). 4D is awesome. Our brains can almost but not quite understand it which makes thinking about it and manipulating it always fresh and novel feeling.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 4:15 am 
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Brandon Enright wrote:
KelvinS wrote:
Wow, thanks! Are there *any* external faces, or are they *all* folded in on each other???

Nope! Only 3D "faces" (called cells). 4D is awesome. Our brains can almost but not quite understand it which makes thinking about it and manipulating it always fresh and novel feeling.

Thanks. Is that "Nope, there are *no* external faces", or "Nope, they are *not* all folded in on each other"?

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 10:48 am 
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KelvinS wrote:
Thanks. Is that "Nope, there are *no* external faces", or "Nope, they are *not* all folded in on each other"?
I believe that is nope as in no external 2D faces. See Jared's post above. I think that answered this nicely.
Brandon Enright wrote:
Our brains can almost but not quite understand it which makes thinking about it and manipulating it always fresh and novel feeling.
4D always makes my brain hurt. I struggle with 3D at times and 4D is really tricky to get a handle on for me. Just when you think you have it you realize it is just a bit more complex then you thought.

By the way, does the 4th spacial dimension have a name? The first 3 are commonly called lenght, width, and depth. A table in 3D has all 3 of these. But a 4D object should have length, width, depth, and ???? I've often wondered about this but have never seen a good answer.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 11:31 am 
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^That is a good question. Also makes me wonder if there is a standard set of directions for the fourth special dimension akin to left/right, forward/backward, and up/down.

As for the 6 loops of four cubes contained within the tesseract, are the following statements true:
-Each cube is part of exactly 3 of these loops.
-each loop consists of two pairs of non-adjacent cubes.
-Each pair of non-adjacent cubes share the same three loops.
-Each loop would appear "straight" from the perspective of a 3d being moving within the cells of the tesseract.
-From the standard net of the tesseract, denoting the cubes as follows:
--h: the cube hidden within the cross octocube's crossing.
--f, b, l, r, u, d: the cubes ajcent to h, analogous to the notation for Rubik's cube faces.
--n: the cube not adjacent to h.
--the six loops are:
---u, h, d, n
---l, h, r, n
---f, h, b, n
---l, f, r, b
---l, u, r, d
---f, u, b, d

Also, while the "cross octocube" is the most common net of the tesseract, I would assume, that just as several hexaminoes are nets of the cube, that several octocubes are nets of the tesseract. My question is how many nets does the tesseract have, and is there one analogous to the "zig-zag" hexamino net of the cube?

Also, considering that at least three faces meet at every edge of a polychoron, is it even possible to unfold convex polychora into non-overlapping 2d nets?

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 12:14 pm 
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Jeffery Mewtamer wrote:
^That is a good question. Also makes me wonder if there is a standard set of directions for the fourth special dimension akin to left/right, forward/backward, and up/down?
Don't stop there. An aircraft has 6 degrees of freedom:

(1) It can go left/right.
(2) It can go forward/backward.
(3) It can go up/down.

But it also has yaw, pitch, and roll that it can also vary.
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How many degrees of freedom does an aircraft flying in 4D space have? I'm going to guess 8. If can be displaced positively or negatively along the 4th axis and it can also be rotated relative to this axis. Is that it? Or do some other wierd interactions com into play here? So at a minimum you not only need new terms for the displacement but you also need a term for the rotation which is now possible.

Carl

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 1:53 pm 
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Aren't there ten degrees of freedom? Movement along the four axes (x, y, z and w) and rotation in the six planes (xy, xz, xw, yz, yw and zw).

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 2:05 pm 
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Coaster1235 wrote:
Aren't there ten degrees of freedom? Movement along the four axes (x, y, z and w) and rotation in the six planes (xy, xz, xw, yz, yw and zw).
Quite likely... yes. My brain really struggles with 4D and often makes over simplifications when I try to play with 4D objects so I think you are probably correct. I'm eager to hear what the experts say... and I'd be surprised if these haven't all been named before. I've just never come across those names.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 2:16 pm 
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Okay, I made a very simplistic and not-perfect representation of exactly how dimensional scaling works.
http://imgur.com/a/4r2wq

You see here first a line, or a 1d object. A square is made by makinga second 1d object and connecting them (red lines) this creates a 2d plane. We follow suit with this method of extension, by making a second square,and connecting them by their vertices by planes into a cube. So logically, to make a 4d object, we duplicate a projected 3d object and connect all their correlated vertices, making the final shape shown, a tesseract in some stage of rotation.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 6:10 pm 
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wwwmwww wrote:
Coaster1235 wrote:
Aren't there ten degrees of freedom? Movement along the four axes (x, y, z and w) and rotation in the six planes (xy, xz, xw, yz, yw and zw).
Quite likely... yes. My brain really struggles with 4D and often makes over simplifications when I try to play with 4D objects so I think you are probably correct. I'm eager to hear what the experts say... and I'd be surprised if these haven't all been named before. I've just never come across those names.

Carl


SHouldn't it be 14 because of double rotations? I'm not sure if those apply here.


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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 7:38 pm 
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wwwmwww wrote:
KelvinS wrote:
Thanks. Is that "Nope, there are *no* external faces", or "Nope, they are *not* all folded in on each other"?
I believe that is nope as in no external 2D faces. See Jared's post above. I think that answered this nicely.l

I was on a plane for 6 hours today so that gave me lots of time to think about this. It turns out that just like how every feature of a cube (vertex, edge, face) is external, every feature of a tesseract is too.

I used some geometric reasoning to determine this but I couldn't figure out how to check / prove if I was correct so I switched to thinking abstractly about it.

First, lets take a cube centered on the origin. A cube is the set of points {x, y, z} such that x, y, and z are in the interval [-1, 1]. A point is considered external if there is some other arbitrarily close point that isn't in the cube. So take an edge on the cube as an example. For the edge to be external, for every point on the edge, there must be another point not contained in the cube such that the distance between the two points is <= epsilon where epsilon is arbitrarily small but > 0.

Here is a concrete example for and edge on the cube. Take the edge {1, 1, [-1, 1]} -- that is, the set of all points along the edge. For any point on the edge {1, 1, p} the point {1, 1 + epsilon, p} is epsilon distance away but not contained in the cube. Therefor the edge {1, 1, [-1, 1]} is an external feature.

Using this definition, it's also easy to count how many vertices, edges, and faces there are for a cube. There are (3 choose 3) * 2^3 = 8 vertices. There are (3 choose 2) * 2^2 = 12 edges. There are (3 choose 1) * 2^1 = 6 faces.


Now let me expand this to the tesseract. I'll define it as the set of points {x, y, z, w} such that x, y, z, and w are in the interval [-1, 1].

One of the 2D faces of the tesseract would be {1, 1, [-1, 1], [-1, 1]} which is the set off all points that are on that face. For any point, {1, 1, p, q} on that face, there exists a point {1, 1 + epsilon, p, q} that is arbitrary close to the point but not contained in the tesseract.

Also, as you can see it's easy to count the number of each type of feature.

vertices: ncr(4,4) * 2^4 = 16
edges: ncr(4,3) * 2^3 = 32
faces: ncr(4,2) * 2^2 = 24
cells: ncr(4,1) * 2^1 = 8

That is, just like on a cube, you can touch the vertices, edges, and faces, on any higher-dimensional cube you can still touch all of the features. There are no internal features. A face shared between two cubes in the tesseract may be "surrounded" in 3D but it isn't in 4D.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 8:02 pm 
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Brandon Enright wrote:
That is, just like on a cube, you can touch the vertices, edges, and faces, on any higher-dimensional cube you can still touch all of the features. There are no internal features. A face shared between two cubes in the tesseract may be "surrounded" in 3D but it isn't in 4D.
Nice... this makes it very easy to see that any 3D object has no "inside" in 4D as every point in the volume is at the surface in that extra dimension.

On the naming side...
In 2D we have perimeter and area.
In 3D we have surface area and volume.
In 4D we have surface volume and ???

I've seen this discussed before and here is one discussion that I found but I think the answer is simply "volume". I've always found that odd as I would tend to think another term sound be used to to make it clear one isn't talking about surface volume. To me it would seem the English language is rather deficient in the naming of 4D terms. Part of me tells me that is understandable as English has been around alot longer then these 4D discussions. However I would have thought the field of math would have been more then happy to pin down these terms. Maybe it has but if so they don't seem to be very easy to find.

Carl

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 8:04 pm 
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Brandon Enright wrote:
A face shared between two cubes in the tesseract may be "surrounded" in 3D but it isn't in 4D.

OK, that sounds *really* wierd, as if 3D space is "bigger than" - and fully inclusive of - 4D space!

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 8:14 pm 
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KelvinS wrote:
Brandon Enright wrote:
A face shared between two cubes in the tesseract may be "surrounded" in 3D but it isn't in 4D.

OK, that sounds *really* wierd, as if 3D space is "bigger than" - and fully inclusive of - 4D space!

Yeah it is very strange. I can only understand it via analogy. Take a sheet of paper (2D) and draw a square. Each edge of the square is a 1D line. Take two squares and push them together so they share an edge. There is no way to touch that edge -- it's completely surrounded by squares. In 3D though it's trivial to touch that edge because you come at it from the 3rd dimension. A face surrounded by two cubes can be reached via the 4th dimension.

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 Post subject: Re: Representing 4D
PostPosted: Mon Nov 18, 2013 10:52 pm 
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KelvinS wrote:
OK, that sounds *really* wierd, as if 3D space is "bigger than" - and fully inclusive of - 4D space!

Regarding "bigger" and dimensions, your comment reminded me of something Melinda Green showed Nan and me once. She was thinking about the amount of volume (in n-dimensional units) and surface area (in n-1 dimensional units) of hyperspheres and how it may change as the dimensions change. Unfortunately I couldn't find any post of hers on the Internet showing this so I've reproduced it.

The volume of a hypersphere is well-defined for real-number dimension. Don't ask me what negative or fractional dimensions "are". I can't explain them.

The volume of a n-dimensional unit-sphere is:
nvol(n) = (Pi^(n / 2))/gamma((n / 2) + 1)

The surface area of a n-dimensional unit-sphere is:
nsur(n)=(2 * Pi^(n / 2))/gamma(n / 2)

See https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area for details.

If you plot these you get:
Attachment:
hyphersphere_dimensions_graph.png
hyphersphere_dimensions_graph.png [ 33.13 KiB | Viewed 2622 times ]


The blue line is the ratio between the surface area and volume and it's always equal to the dimension. The surface area of a 3D sphere is 3x the volume. Of course the units of such a ratio are rather non-physical.

If it looks like the surface area curve is basically the same as the volume curve, it is. The volume curve nvol(x) is the surface curve nsur(x + 2) / (2*pi).

If you're wondering which dimension has the "most volume", that curve reaches its maximum around 5.2569464 (and so the surface area reaches its maximum at around 7.2569464).

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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 19, 2013 1:58 am 
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Hoping to help even more to visualize this wonderful world of 4D, here is an excerpt
from my 24cell.net site (which I also used for the CFF magazine article):

"So how can someone claim that we can create puzzles which are four (or higher) dimensional? Well,
no one can claim that a puzzle is four dimensional in our 3D world. But we may certainly use the symmetry
that comes from such structures.

The fourth dimension can be claimed to be related to either time or space (and there is a debate about it).
Here is my personal attempt to unify the time and space notions. 4D shapes exist in four spatial dimensions,
where the fourth dimension is not time, but a dimension of space. As we live in three spatial dimensions,
we cannot see in four spatial dimensions. Now, 4D shapes consist of 3D slices, just like a 3D cube can be
thought of as having 2D square slices. So viewing a sequence of these 3D slices is equivalent to a 4D being
seeing the depth of the four spatial dimensions. Therefore, by watching the 3D projections in a sequence,
it is possible to substitute the fourth spatial dimension by time, which is assumed to be zero.

But can we assume that time is zero? As agreed by top mathematicians and physicists, time can certainly be
assumed to be constant for a while, because as stated here, "time is a tiny window of a much greater underlying
fabric, which ultimately encompasses the multiverse of all possible universes and quantum indeterminacy".
And this explains why we are allowed to use this way to experience the symmetry of the four dimensions.

The Legend of the Pyramid, a very entertaining mini-story is highly recommended and it provides
a more "comparable" approach from our 3D world point of view."



What we need to understand is that the 3D distance is not perceived in 4D the way we are used to.
Infinite is zero, and zero is infinite. For example, a Hypercube is a structure that in 3D it is supposed
to have all of its edges straight and equal in length at all times. Mathematically and realistically it is impossible,
but if we use our imagination, we can try and imagine it as something whose extra dimension is causing it
to bend and move in a struggle to satisfy the above rules. It is when time also tends to zero, so that
the Hypercube will never be caught of violating those rules. And this is also where the "inside-out" effect
comes into the picture. As correctly stated above, there is no such thing as internal or external in 4D
(well, the way we think of it in 3D), the same way there is no external or internal of 2D when seen in 3D.

If we lived in 4D for a while, we could probably watch (in awe) all internal organs of everyone...!!! (yuk!)


:lol:


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 Post subject: Re: Representing 4D
PostPosted: Tue Nov 19, 2013 8:52 am 
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^And, if memory serves me well, The Legend of the Pyramid is what first introduced me to the room metaphor I mentioned in a previous post, though that story expalins it better than I could.

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