visualizing a hypercube

Discussions about how to visualize 4D and higher, whether through crosseyedness, dreaming, or connecting one's nerves directly to a computer sci-fi style.

visualizing a hypercube

Postby jrtmx » Thu Apr 15, 2004 12:06 am

When looking at this java applet game of a hypercube:
http://www1.tip.nl/~t515027/hypercube2.html
I see two different things. First there is a cube with another cube inside of it with their vertices connected, and when I rotate by right clicking the inner cube moves away from my mouse while the outer goes with my mouse. But then I can also see just a solid shape of lines that is just rotating when I click it with the right and with the left it looks like an inverted shape of the lines(right clicking makes the closer lines of when I left click become the farthest lines). Is it supposed to be a cube within a cube like my first explanation? or the other?
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Postby Keiji » Thu Apr 15, 2004 7:03 pm

Well first you must understand that a hypercube is NOT a cube within a cube, even if it looks like it. A hypercube is bordered by eight cubes just as a cube is bordered by six squares. Look for these in the image - these are the "cells" (3d faces) of the hypercube - if you have trouble finding them, print screen the image and paste it into paint, then try marking them out with the line tool. :D
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Postby PWrong » Sat Apr 24, 2004 11:07 am

I've seen that website before. Can anyone here actually get a high score on that game?
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Postby Keiji » Sun Apr 25, 2004 7:01 pm

Um, I didn't know it was a game.
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Postby PWrong » Wed Apr 28, 2004 3:32 pm

Click on the button on the right, Start HyperCube applet, and it brings up the ordinary hypercube. Uncheck the Basics box, and check the Game box. It's best to play in 3D mode first.

You control the little cross, and you have to hit the ball so that it bounces around and hits the large cross on the wall. Then the big cross disappears, so you have to leave the cube through the square opening, and go back in again before the little ball catches you. It's less complicated than it sounds. It's easy to play in 3D, but I can hardly do anything in the 4D version.
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Re:

Postby Secret » Mon Jul 04, 2011 11:31 pm

PWrong wrote:I've seen that website before. Can anyone here actually get a high score on that game?


Sorry for reviving this very old topic
But after 2-3 years of immersing into 4D cube applets, 4D insights etc.
I manage to get a score of 110 something

I can find myself arond the cube freely, even outside. However I'm still having trouble at the corners , where I always hit walls that seemed invisible no matter how I rotate the cube

The insights I used to resolve the unenhanced line drawing of the tesseract are:
1. The thing is not cube within a cube
2. How a normal cube in 4D will become as 'flat' as a piece of paper
3. The slice of cube you're actually at

Using these, the whoe thing will break down into two sets of 3Ds, you'll have to line up the view of your cross and the thing you want to hit/enter at both 3D views (which in reality is just positioning you ball in front or behind the object of interest), then head towards it
The above, however does not work at the corners or the cube slices that are closest to the cube cells wall themselves, where I always bump into the wall with no idea how
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Re: visualizing a hypercube

Postby Mrrl » Tue Jul 05, 2011 3:28 pm

My first result is -3618. But 4/5 of the game I thought that alternative rotation is on the middle button, and the right botton is for something else... Next time shold give better results.

UPD. Fifth attempt: -6.

Do you really need 4D imagination for this game? I'm not sure. It's just "check-turn picture-align-check again..." sequence of actions...
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Re: visualizing a hypercube

Postby Secret » Wed Jul 06, 2011 12:02 am

Mrrl wrote:My first result is -3618. But 4/5 of the game I thought that alternative rotation is on the middle button, and the right botton is for something else... Next time shold give better results.

UPD. Fifth attempt: -6.

Do you really need 4D imagination for this game? I'm not sure. It's just "check-turn picture-align-check again..." sequence of actions...


Without 4D insights, it will become trial and error (i.e. Hit wall often without any idea where the wall is)
Especially when near the corners
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Re: visualizing a hypercube

Postby Mrrl » Wed Jul 06, 2011 1:59 am

Secret wrote:
Without 4D insights, it will become trial and error (i.e. Hit wall often without any idea where the wall is)
Especially when near the corners


You don't hit walls when you find your way from the room: just find the door and go to it. Same is for the way from outside: all you need is to move cursor to the point from which the door should be visible. On the first stage (when you are looking for the strike direction) there may be problems. I try to keep close to the ball that is inside the room. But if I hit walls couple of times, I say that position of the ball is not convenient and push it ib any direction (without hitting walls). And all this is in 2D picture - the room may be 3D, 4D or 6D - algorithm will be the same (but in 6D you have to align cursor in space with more degrees of freedom).
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Re: visualizing a hypercube

Postby Mrrl » Wed Jul 06, 2011 9:27 am

First positive result: +73. But I was lucky - two times target had been destroyed not by direct hit, but after two bounces of sphere.
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Re: visualizing a hypercube

Postby Secret » Wed Jul 06, 2011 9:42 am

Mrrl wrote:...just find the door and go to it. Same is for the way from outside: all you need is to move cursor to the point from which the door should be visible...

Image
Same applies when you try to go back inside (using just one rotation)
Mrrl wrote:And all this is in 2D picture - the room may be 3D, 4D or 6D - algorithm will be the same (but in 6D you have to align cursor in space with more degrees of freedom)

Yup the rules are similar
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Re: visualizing a hypercube

Postby Mrrl » Wed Jul 06, 2011 9:53 am

169 :) That's all for now. Return to Magic 120-cell :)
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Re: visualizing a hypercube

Postby Secret » Thu Jul 14, 2011 1:58 pm

Image

(Label the figures from top left to bottom right as 1,2,3 etc.)

An analysis:
The first thing we know is the cross is located at the center of the tesseract (i.e. if the tesseract is 1x1x1x1, then the coordinates of the cross is (0.5,0.5,0.5,0.5))
According to fig 1, this is what the view is like when you first start the game (The red cube which is invisible in the game, is highlighted)

In fig 2, the tesseract is rotated ana and to fig 3
Note how the red cube is like a piece of 2D cardboard and become a line when the tesseract is rotated 90 ana
You are now looking at the tesseract from the ana direction

If you first move the cross so that it looks as it is within the ball and then rotate the tesseract 90 ana, you will understand that although you are in the same xyz position as the ball, you are actually far from the ball in the w direction. This operation also reveals that you are in the center cubic slice of the tesseract.

Fig 6 is a scenario where your cross seemed to have hit an invisible wall. (Assuming you have not gone anawards or katawards, i.e. staying at the center cube slice) the red cube reveals the wall that you have actually hit. As you are near the corner of the cube in xyz dimensions, you will hit the wall even if you walk for a short distance only, backward or forward. This is not obvious if the red cube is invisible.

The above analysis also revealed the controls of the game: Your cross can only move in directions a combination of 3 dimensions each time. (i.e. you cannot move in the dimension that is pointing out of your screen). The rotations allows you to switch your view of the tesseract hence swapping the pointing direction of the w axis with either the x y or z axes, which allows you to access the remaining dimension. Thus what you really need to handle are just two set of 3D views, where 2 of the axes were shared in each view.

If the cross is moving in 4D, what you'll see is the cross getting smaller (i.e. further) in the view but remains in the same position on the diagram when you rotate it in xyz dimensions (i.e. the 3 set of planar axes which causes the front cube cell to revolve around itself instead of swinging it to another position, which when projected onto the cube within cube diagram, result in the cube within cube thing rotating in a familar 3 dimensional way)

Using the above analysis, I manage to get a solid 361 points with 1-2 times bumping into walls

However this is not my goal: My goal is to visualize this thing so that I can get to any location (including the corners) of the tesseract without bumping into a single wall without trail and error

And now the main point is so far the above analysis DOES NOT WORK at the boundary cubic slices, where two cubes met at one plane, four cubes met at one point and 3 cubes met at one side. The current unexplained phenomenon is your cross got stuck between two walls shown below (moving the cross at one of the boundary cube slice and then rotate the view point so that it is looking at the front cube slice)
Image

Yet by rotating slightly to the left or right and then you'll be free to move forward to the opposite side (which gives an impression that the walls are arranged in a narrow corridor and your cross need to zig zag through)

So my question is, is there something wrong in my analysis? If so how to perceive the boundaries correctly?
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