Wrap your mind around a Tesseract

Tunnelcat · Post by **Tunnelcat** » Wed Aug 13, 2008 4:24 pm

I was reading a sci-fi book the other night and stumbled upon the name of an object that I hadn't seen since high school. It's called a Tesseract, the analog of a cube in the fourth dimension. What's neat is although we technically can't see in the fourth dimension, we can see it's shadow in the third dimension. Hard to grasp even when you know what it is. Mind bending.

http://en.wikipedia.org/wiki/Tesseract

QuestionableChaos · Post by **QuestionableChaos** » Wed Aug 13, 2008 4:38 pm

okay i am quite confused now

haha

what a great thing to read about before heading off to my thermodynamics class

Lothar · Post by **Lothar** » Wed Aug 13, 2008 4:49 pm

mmmm, polychorons... yum

Duper · Post by **Duper** » Wed Aug 13, 2008 5:48 pm

Welcome to 1962; \"A Wrinkle in time\".

Bet51987 · Post by **Bet51987** » Wed Aug 13, 2008 6:28 pm

Mind bending it is...

Here is my favorite person explaining the hypercube like nobody ever could... Also, there is an awesome computer animated version of it somewhere but I don't have that anymore...

Bee

Dakatsu · Post by **Dakatsu** » Wed Aug 13, 2008 7:55 pm

I don't understand... at all... then again I still can't figure out what the difference is between Major and minor keys...

Can someone PLEASE tell me what analog means in the phrase \"analog of a cube\", I can't find anything on it...

QuestionableChaos · Post by **QuestionableChaos** » Wed Aug 13, 2008 8:40 pm

\"Fourth Dimension 101\"

*head explodes*

Lothar · Post by **Lothar** » Wed Aug 13, 2008 9:13 pm

Dakatsu wrote:Can someone PLEASE tell me what analog means in the phrase "analog of a cube", I can't find anything on it...

Think of "analog" as "analogy" -- it's something that's the same in some sense. The tesseract is the 4D analog, or 4D version, of a cube. It has many of the same properties of a cube, just in more dimensions.

Sedwick · Post by **Sedwick** » Wed Aug 13, 2008 10:47 pm

I always thought it was neat how the eighth outer cube is really turned inside-out and overlapping with the other 7 cubes in 3-D space. And how every cube in the hypercube touches all the others except for the one 2 cubes away. For my old website, I made this little animation of a hypercube rotating through 3-D configurations, almost like a torus...

Tunnelcat · Post by **Tunnelcat** » Thu Aug 14, 2008 3:59 pm

I heard this simplified explanation a long time ago in science class. Since we can't see into the 4th dimension, the tesseract is essentially the SHADOW of a 4D cube that we can see when it's projected into the 3rd dimension. Just as the square is the 2D shadow of a 3D cube. Still, it's a hard concept for our 3D brains to grasp. Here's another one that's kind of neat. Observe the motion of the darker colored panels and the way they shift around.

Alter-Fox · Post by **Alter-Fox** » Fri Aug 15, 2008 9:21 am

How many faces does a tesseract have?

Duper · Post by **Duper** » Fri Aug 15, 2008 9:53 am

It's not 3 dimensional, so technically it doesn't.

Alter-Fox · Post by **Alter-Fox** » Fri Aug 15, 2008 11:13 am

Ok. I meant that partially as a joke though.

heftig · Post by **heftig** » Fri Aug 15, 2008 11:23 am

A tesseract has got 24 square faces.

Sedwick · Post by **Sedwick** » Sat Aug 16, 2008 12:12 am

heftig wrote:A tesseract has got 24 square faces.

He's right. The 6 of the outer cube, the 6 of the inner cube, four others each from the top and bottom warped cubes (their other two are shared by the inner and outer cubes), and the four between the warped side cubes. 6 + 6 + 4 + 4 + 4 = 24.

Jon the Great · Post by **Jon the Great** » Sat Aug 16, 2008 7:55 pm

Bet51987 wrote:Here is my favorite person explaining the hypercube like nobody ever could...

(a little off topic but funny nonetheless)

Kilarin · Post by **Kilarin** » Wed Aug 20, 2008 5:41 am

Duper wrote:Welcome to 1962; "A Wrinkle in time"

My nine year old is just going through that series for the first time. Ah, I do LOVE L'Engle.

Bet51987 · Post by **Bet51987** » Wed Aug 20, 2008 6:37 am

Jon the Great wrote:
Bet51987 wrote:Here is my favorite person explaining the hypercube like nobody ever could...

(a little off topic but funny nonetheless)

Awesome

I loved it.

Bee

Foil · Post by **Foil** » Wed Aug 20, 2008 1:01 pm

Bet51987 wrote:Here is my favorite person explaining the hypercube like nobody ever could...

Nice! He's basically re-iterating the ideas presented in the book he mentioned ("Flatland" by Edwin Abbott), but I hadn't seen them visually done that way before. [Note: I'm assuming there must be some missing from the end of that video, because in that clip he never gets around to tying it back to his first comments about curved spacetime and unboundedness.]

I have a copy of Flatland somewhere; it was a good discussion trigger for an Advanced Geometry class I took during my undergraduate studies. (However, I don't recommend it as a casual read; it's really a political book more than anything, and has some very strange social elements, especially in regards to issues of gender, color, class, etc.).

-------------------------

Here's another interesting one for you guys: How about visualizing other 4-D shapes?

The 4-D hypersphere ("4-sphere"), for example...

That one may actually be slightly easier to try to visualize than the tesseract. If a 4-sphere with radius of 1 foot passed through our 3-D world, we would see it's "shadow"/"cross-section" as a sphere changing size (from nothing, to a point growing to a sphere of radius 1 foot, then shrinking again to a point and vanishing).

Note that things in 4-D don't work like they do in 3-D. For example, here's a game where you are tasked with taking two separate 3-D cubes and putting one inside the other. In our world, you can't do it without breaking one of the cubes, but we have an extra dimension to work with in 4-D... try it!

Tunnelcat · Post by **Tunnelcat** » Wed Aug 20, 2008 4:02 pm

Here's a thought to ponder. Is the 4th dimension 'time' or just a fourth axis in space?

Lothar · Post by **Lothar** » Wed Aug 20, 2008 4:16 pm

tunnelcat wrote:Here's a thought to ponder. Is the 4th dimension 'time' or just a fourth axis in space?

Extra dimensions are mathematical constructs. They can represent physical things like "space" and "time", or not, as you choose.

Mathematicians often work in infinite-dimensional spaces. If you've ever seen a Fourier series, you can think of that as a Hilbert space with basis functions cos(nx) and sin(nx). In other words, each "dimension" is a wave with a particular frequency. There are lots more types of spaces where your "dimensions" are functions.

Kilarin · Post by **Kilarin** » Wed Aug 20, 2008 4:25 pm

Foil wrote:I have a copy of Flatland somewhere;

Another great book!

Foil · Post by **Foil** » Wed Aug 20, 2008 4:28 pm

Lothar wrote:
tunnelcat wrote:Here's a thought to ponder. Is the 4th dimension 'time' or just a fourth axis in space?
Extra dimensions are mathematical constructs. They can represent physical things like "space" and "time", or not, as you choose.

Exactly! My current project at work actually involves the use of a certain algorithm which works in an n-dimensional space (where n varies depending on the scenario being solved). Those extra dimensions definitely come in handy.

Regarding "time as the 4th dimension", I don't remember all the details from my studies in Relativistic Geometry, but there are some subtle differences between the structure of relativistic spacetime and R4 (the 4-D space we've been talking about with tesseracts and such). Maybe one of our resident Physicists (Munk?) can jump in here and clarify.

Tunnelcat · Post by **Tunnelcat** » Wed Aug 20, 2008 4:34 pm

Now I'm not a mathematician or a physicist, but wouldn't that preclude that 'time', since it can be another axis in space, could move in more directions instead of just forward? If space is multi-dimensional or even curved around on itself, maybe time is in reality not linear or always going in what we perceive as forward.

Lothar · Post by **Lothar** » Wed Aug 20, 2008 4:49 pm

tunnelcat wrote:'time'... could move in more directions instead of just forward?

Mathematically speaking, yes. Physically speaking, there's no reason to think so.

Foil · Post by **Foil** » Wed Aug 20, 2008 4:53 pm

That's part of what I'm trying to remember from my undergrad Physics work. If time were exactly like R4 (the simple orthogonal w,x,y,z space), you'd be right and there wouldn't be any natural orientation/direction to time.

However, as I recall, one of the subtle differences between spacetime and R4 is that the time dimension has an orientation to it; I believe it has to do with the relativistic (yep, Einstein) structure.

Kilarin · Post by **Kilarin** » Wed Aug 20, 2008 5:48 pm

I love the explanation from H. G. Wells \"The Time Machine\"

You know of course that a mathematical line, a line of thickness _nil_, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.'

'That is all right,' said the Psychologist.

'Nor, having only length, breadth, and thickness, can a cube have a real existence.'

'There I object,' said Filby. 'Of course a solid body may exist. All real things--'

'So most people think. But wait a moment. Can an _instantaneous_ cube exist?'

'Don't follow you,' said Filby.

'Can a cube that does not last for any time at all, have a real existence?'

Filby became pensive. 'Clearly,' the Time Traveller proceeded, 'any real body must have extension in _four_ directions: it must have Length, Breadth, Thickness, and--Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.'

Not necessarily great science, but it DOES make intuitive sense.

Sirius · Post by **Sirius** » Wed Aug 20, 2008 7:28 pm

It is pretty much bogus. Objects don't \"start\" and \"stop\" in the real world; at best their component parts change from one form into another. Time cannot be used in the same way as the other three dimensions, therefore it is questionable why it should be considered a dimension at all.

Drakona · Post by **Drakona** » Wed Aug 20, 2008 9:23 pm

Well, you can consider it a dimension if it's convenient or appropriate for the problem you're working on. If you want to chart, say, the path of an airplane, it's four-dimensional with time as one of the dimensions. If you just want a chart of a building's structure, then not so much.

Time can be thought of as a dimension in the sense of \"another degree of freedom\" if it's relevant to the situation at hand. That just a model, though; it isn't anything particularly deep. It doesn't make time any more or less special than it already is, or particularly like a spatial dimension.

Back on topic, I'm delighted to see so many geometers frequenting the forums.

QuestionableChaos · Post by **QuestionableChaos** » Wed Aug 20, 2008 9:30 pm

time is only considered to be the 4th dimension in spacetime theory. in terms of spatial dimensionality, the 4th dimension is this complicated mess of tesseracts and such

So in this argument of what is the 4th dimension, everyone is right!

Kilarin · Post by **Kilarin** » Thu Aug 21, 2008 5:34 am

Sirius wrote:It is pretty much bogus.

Yep. I agree. I LIKE the argument, it's interesting and intuitive. But it's not great science.

Foil · Post by **Foil** » Thu Aug 21, 2008 8:11 am

No, no, it's not bogus at all.

For local comparisons (non-relativistic speeds), the structure of spacetime is nearly identical to the structure of R4, your basic 4-dimension space. Obviously there are limitations due to things like the orientation of time (we can only move one direction along the t-axis), but the mathematical similarities are definitely there.

And again, back to what Lothar mentioned earlier: the term \"dimension\" is a mathematical abstraction. It's not always a spatial thing, it can refer to all kinds of mathematical structures.

Kilarin · Post by **Kilarin** » Thu Aug 21, 2008 8:28 am

foil wrote:No, no, it's not bogus at all.

It's not bogus, but it's not complete. It's overly simplistic.

foil wrote:the mathematical similarities are definitely there.

Yes, but I think it is a BETTER description to say that time seems to involve movement along a 4th directional axis. When we say "Time is the 4th dimension", I'm afraid we give the impression that the 4th dimension IS time, when it would be better to say that time is 4 dimensional. Right? Wrong? Not even wrong?

Lothar · Post by **Lothar** » Thu Aug 21, 2008 11:25 am

Kilarin wrote:When we say "Time is the 4th dimension", I'm afraid we give the impression that the 4th dimension IS time, when it would be better to say that time is 4 dimensional.

It would be better to say that you can model time as another dimension if it's convenient. Sometimes that makes it the 4th dimension, sometimes the 3rd (if you're working on a spatially 2D problem), sometimes the 10th or 20th, and sometimes you already have an infinite number of other dimensions in play.

Referring to time as "THE 4th dimension" demonstrates a fundamental misunderstanding of dimensions as fixed, necessary physical quantities, rather than as mathematical abstractions that can be assigned to whatever you need to assign them to. (In Fourier series, "length", "width", and "depth" are not dimensions at all, and "time" is certainly not the fourth one.)

Wrap your mind around a Tesseract

Wrap your mind around a Tesseract

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