Exploring the Possibilities of Drawing 4D Shapes in 3D Space

[||spoon||]

this may seem quite stupid to many of you but i was wondering how it is possible to draw 4D objects. I have seen pictures of 4D objects such as the hyper-cube etc, but they are drawn in three dimensions. The picture of a hyper cube that i looked at seemed to me to just look like a cube with a smaller cube attached inside it.

How is it possible to draw a 4D shape with only three spatial dimensions avialble to draw within?

also is it possible to make a model of a 4D shape because wouldn't you come across the same problem?

right now i don't have the picture but if it is needed i am sure i could find one to link in another comment.

||spoon||

 

[robphy]

To view higher dimensional objects, you basically have to project down into the viewing plane or space. Think about how you draw a cube on a piece of paper... one can do this with perspective to get a "square in a square, with more edges joining corresponding vertices"... or one can do this with an "orthogonal projection" where you have a "square displaced from another square, with more edges joining corresponding vertices".

Another way is to do a slice of the higher-dimensional object, like a plane cuting through a cube.

In all of these cases, one might have to vary the viewpoint or cutting plane to get a "feel" for the object... just like turning a cube in your outstretched hand to get a feel for it. Stereoscopic glasses might help a little.

 

[||spoon||]

thanks but i still find it kinda hard to see how u can draw a 4d shape becasue are you not only drawing in three dimensions? i know that you can use 4 vectors to describe the shape... i am currentlky studying vectors in specialist math and am familiar with the i-j-k system, but i can't see where the next vector would go.. how do u add another vector without it falling into the other three dimensions??

 

[||spoon||]

i mean how can u have a 4 dimensional axes?

 

[radou]

thanks but i still find it kinda hard to see how u can draw a 4d shape becasue are you not only drawing in three dimensions? i know that you can use 4 vectors to describe the shape... i am currentlky studying vectors in specialist math and am familiar with the i-j-k system, but i can't see where the next vector would go.. how do u add another vector without it falling into the other three dimensions??

When using the term 'dimension' near the term 'vector' you should be careful. Further on, not everything has a geometrical representation, in general.

 

[planish]

robphy said:

Another way is to do a slice of the higher-dimensional object, like a plane cuting through a cube.

That reminded me of a phrase that triggered a lengthy dinner-table discussion between me and my two sons: "Four-dimensional cheese slicer". We think we figured it out. We could not guess at how to fabricate one, but we devised a test to see if it worked.

It is convenient to use time as the fourth dimension, since we have a sense of notions such as "long time", "short time" or "distant past" in the same way we understand long and short distances and "far away". For this thought experiment, choose a time unit of arbitrary duration and say that it is equivalent to a given distance. Let's say one hour is equivalent to a length of one foot.

When you take a block of cheese that is, say, 1 foot by 1 foot by 1 foot, and cut it in half with the three-dimensional cheese slicer, you end up with two smaller blocks of cheese that are 6" x 1' x 1'. It might be difficult to tell it's been sliced until you separate them by some distance, say 2 feet. There is still the same volume of cheese, but it exists as two smaller pieces.

The 4-D cheese block not only has to have spatial dimensions, but also some duration. Make that a cubic foot of cheese that exists for one hour. A hypercube of cheese, which has a volume-duration of one cubic foot-hour.

Our 4-D cheese slicer would slice it in half perpendicular to the time dimension. It also (conveniently) automatically separates the two halves in Time (say, by 30 minutes), otherwise we would not know whether or not it worked.

What we would observe then, is a cubic foot of cheese that pops into existence at 1:00 pm, poofs out of existence at 1:30, reappears (all 1 cubic foot of it) at 2:00, and finally disappears (forever) at 2:30.

The total volume-duration of cheese is still 1 cubic foot-hour of cheese, but it has been sliced into two pieces that are only 1/2 hour long.

If you want to make sandwiches, then you have to adjust the 4D cheese slicer so that it appears for 59 minutes, disappears, and then reappears for 1 minute.

If you slice it parallel to the Time dimension, then you simply end up with two 1' x 6" x 1' pieces of cheese simultaneously. (Or so our think-tank believes.) Read the instructions, and try again.

If you slice it at an angle, you might end up with a 1' cube that shrinks gradually out of existence, then grows from nothing back up to the original 1' cube, maybe 3 hours later. It's still a total volume-duration of 1 cubic foot-hour of cheese, and is the 4D equivalent of slicing a 3D cube at an angle - it's tapered like two wedges. It's also recalls to the representation of a hypercube that looks like a smaller cube within a larger cube, although it might be more correct to say that's like a block that shrinks down to a smaller block over a given length of time (or expands).

As I said, we could not figure out how to fabricate the 4D Cheese Cutter. We also had difficulty figuring out how to make the cheese appear out of non-existence in the first place. That is left as an exercise for the student to solve over the weekend.

There's a book "Flatland: A Romance of Many Dimensions ", written in 1884, which has also been made into a short animated film called "Flatland" - see http://en.wikipedia.org/wiki/Flatland - about a society of 2D entities - polygons of varying numbers of sides. The hero of the story (a square)encounters a 3D being, a sphere which passes through his universe. The sphere first appears to him as a dot appearing out of (seemingly) nowhere, growing into large circle, and then smaller again as it passed through the 2D plane. Depending on how it approached, a cube passing through Flatland might appear as a square that pops into existence for a while, or as a small triangle that morphs into a large hexagon, and then back to a shrinking triangle again. Just imagine feeding a styrofoam cube into a belt sander at various angles, and consider what the sanded face would look like.

||spoon|| said

i am currentlky studying vectors in specialist math and am familiar with the i-j-k system, but i can't see where the next vector would go.. how do u add another vector without it falling into the other three dimensions??

Forgetting about the vector/dimension thing for a moment, you don't.
You either use time (animation, anyone?) or you start moving 3D objects some arbitrary direction in 3D space, much like adding a Z axis to an XY Cartesian diagram, it can be 30 degrees (or whatever you like) off the X axis. If you can arrange for the objects to change shapes as they move, so much the better. (Where's my Holo-tank?)

At any rate - moving, say, a cube some distance through 3D space is the 4D equivalent of sliding a square along a plane, and joining the corners to represent a cube. Each face of the moving cube could also be said to represent the 6 "cubes" that join the start and end cubes of the hypercube.

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BTW - Another name for the hypercube is "tesseract" - http://en.wikipedia.org/wiki/Tesseract
More n-dimensional cube thingies - http://en.wikipedia.org/wiki/Hypercube

Less-serious short story involving a tessaract-shaped house: Robert Heinlein's "-And He Built A Crooked House". - http://www.scifi.com/scifiction/classics/classics_archive/heinlein/heinlein1.html

 

[neurocomp2003]

how do you draw 3D in 2D? use tha tsame concept...probably tooth picks or a chemistry set would be bets used...oh wait the magnetix toys...those are awesome.