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Visualizing the Fourth Dimension Using Color

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Re: Visualizing the Fourth Dimension Using Color

#11

Post by mirosam »

elnate wrote: post #10

Can someone give a two dimensional analogy to the unknotting example?

Nontrivial knots don't exist in 2D, so not really.

Something almost analogous is if you imagine two circles: one smaller and one larger in 2D. Let's say the smaller circle is inside the larger. It's impossible to move the smaller one outside smoothly without intersecting the larger.

In 3D of course this is trivial, just move the smaller circle up away from the 2D plane, then over the larger one and down on the plane again.

Re: Visualizing the Fourth Dimension Using Color

#12

Post by heeton »

Houshalter wrote: post #8
jakebian wrote: post #3

OR you can simply project the fourth dimension into 2D, much like how we project 3D pictures onto 2D screens. Here's an example http://jakebian.github.io/hypercanvas.js/

It's a cool idea but I don't know how to comprehend what I'm looking at.

What would your response be to a child who said something similar when you drew a cube?

It does take practise to be able to visualise 4d objects, and I agree that 2d projection example isn't helpful. I find it interesting to think of the 3d -> 2d problem that some people have, and extend it to 4d.

Re: Visualizing the Fourth Dimension Using Color

#13

Post by ragle »

mirosam wrote: post #11
elnate wrote: post #10

Can someone give a two dimensional analogy to the unknotting example?

Nontrivial knots don't exist in 2D, so not really. Something almost analogous is if you imagine two circles: one smaller and one larger in 2D. Let's say the smaller circle is inside the larger. It's impossible to move the smaller one outside smoothly without intersecting the larger. In 3D of course this is trivial, just move the smaller circle up away from the 2D plane, then over the larger one and down on the plane…

I think your circles analogy is good.

To tie it in a bit tighter with OPs article, I would just add that if we could only perceive two dimensions, the inner circle would disappear as we "move the circle up and away from the 2d plane".

It would only reappear to us once we moved it back down into our plane of perception, safely outside of the other circle.

The relevant bit from OPs article that this is analogous to is:

> "What would we see if you watched this happen in real life? Since we can't see anything outside our 3D slice of 4D space, from our perspective the moving (green) loop would disappear, to reappear later in the unknotted position."

Re: Visualizing the Fourth Dimension Using Color

#14

Post by j2kun »

swift wrote: post #7
j2kun wrote: post #5

Earlier quoted context omitted.

> no matter how many senses you incorporate, you will always navigate a particular 3D projection. How do you mean? I can think of visualizations that do not.

Could you describe the visualizations you're thinking of?

An animated visualization, for example.

Re: Visualizing the Fourth Dimension Using Color

#15

Post by Houshalter »

heeton wrote: post #12
Houshalter wrote: post #8

Earlier quoted context omitted.

It's a cool idea but I don't know how to comprehend what I'm looking at.

What would your response be to a child who said something similar when you drew a cube? It does take practise to be able to visualise 4d objects, and I agree that 2d projection example isn't helpful. I find it interesting to think of the 3d -> 2d problem that some people have, and extend it to 4d.

I mean that there is no description on how these projections are even created.

A child would recognize a cube since we naturally see 2d projections of 3d objects as it is. Likewise a 2 dimensional being could recognize a 1d projection of a 2d object (looking at the 2d object on it's side.) But would a 3d object projected to 1 dimension make any sense?

Re: Visualizing the Fourth Dimension Using Color

#16

Post by Grue3 »

mirosam wrote: post #11
elnate wrote: post #10

Can someone give a two dimensional analogy to the unknotting example?

Nontrivial knots don't exist in 2D, so not really. Something almost analogous is if you imagine two circles: one smaller and one larger in 2D. Let's say the smaller circle is inside the larger. It's impossible to move the smaller one outside smoothly without intersecting the larger. In 3D of course this is trivial, just move the smaller circle up away from the 2D plane, then over the larger one and down on the plane…

The perfect analogy to knots would be planar and nonplanar graphs. It is always possible to draw a graph without intersections in 3D, but in 2D some graphs are impossible to draw without intersections (nonplanar graphs), such as 5-vertices complete graph.
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