Look into 4D Space – a Stereographic Simulation
Welcome everybody to something unusual. Today we'll be looking into four-dimensional space. Skip the talk and jump to the damned simulation.
You'll need functioning Java in order to see this.
If you happen to not know what the fourth dimension is, here's a great explanation:
OR you could read Robert Heinlein's hugely amusing short story "And He Built a Crooked House" - which describes the weird experience of a family that moved into a tesseract-shaped house. You can get it here and here. Or maybe here.
In our case, time will NOT be the forth dimension, but the fifth. Damn, I could have named this thing "Looking into 5D Space," but never mind.
My goal was to offer something better than the usual tesseract animations you can find on the internet. What is a tesseract?
I wanted to make it as real as it gets - and help you experience it as fully as possible.
See, we have a basic problem - our spacial awareness is three-dimensional. We have a limitation, that needs to be overcome.
Luckily however, our brains already have a means for overcoming a similar limitation: We can perceive the world in 3D, even though our eyes can only see flat, two-dimensional images.
The way our brains do it is to gather images of the same spatial object from many perspectives - and then put all this data together for us - in the form of a 3D perception.
One way to do this is to move while looking. As our point of view shifts, near objects shift a lot whereas distant ones hardly move. Thus, we have a means to gauge the distance.
As you might have guessed, the other way is through binocular vision - by looking with both eyes. Each eye gives a slightly different perspective, and when both are put together, spatial awareness can arise.
That's why here I'm offering you a stereoscopic simulation of a tesseract. We'll try to tap into the brain's capability to go one dimension further - and give it two perspectives of the 4D scene.
What you are about to see:
You'll be shown a simulation of a tesseract, rotating in 4D space. There'll be a separate projection for each of your eyes.
Normally, to do this, you'd have to first project the tesseract onto a 3D screen, and then make two 2D projections of that screen - one for each eyes. This didn't sound good enough for me though, so I went for something a bit more fancy:
The simulation below projects the tesseract onto two 3D screens - one for each eye - and then projects each of them onto a 2D screen.
Obviously, here we also have to deal with the limitations of current technology. 3D displays are not the standard. Not yet. So we'll have to use workarounds.
There are two widespread ways for looking at stereoscopic images. One requires some skill, the other one - a very cheap piece of equipment. I'm offering you implementations for both.
Anaglyphs combine two images into one, by using different colors for those images. The way each eye only gets its image is, by looking at the anaglyph with special glasses - with different color filters for both eyes.
Red and cyan are the most frequently used colors.
Through a red filter, white and red look the same, just as cyan and black look the same.
Through a cyan filter, cyan and white look the same, just as red and black look the same. Thus, each eye gets only what was meant for it.
The way I got to know anaglyphs was through a book I encountered as a kid: "Descriptive Geometry with Anaglyphic Illustrations." It came with a pair of glasses. I was instantly hooked. While the theory seemed too dry to me - I was just a kid after all - I couldn't get enough from the illustrations.
It was an amazing experience to look at the page through the glasses - and see all sorts of cool shapes jump into the air.
After a while, I even learned to look at the anaglyphs without glasses; and to draw my own.
And then I forgot all about it - never to remember it for many many years.
On a crossed-view stereogram, the two images reside next to each-other. In order to see the spatial figure, you have to be looking with you right eye at the left image, and with the left eye at the right image. If you imagine two straight lines connecting your eyes with their respective images, the two lines will intersect somewhere in mid-air, and that's where the spatial figure will appear.
I hope you read that well, because you'll have to do it yourself. It's gonna be fun ...
How to look at the crossed-view stereogram:
Here's a brief manual for how to look at the monochrome simulation below.
Look at the monitor straight. Don't look at an angle. Do NOT tilt your head.
It's easier if you look from some distance. An arm's length would be a good start.
Stretch your arm toward the screen and point your finger to the middle between the two images. Almost touch the screen ...
Look at your fingertip with both eyes.
While looking, slowly move your fingertip along a straight line toward the point between your eyes.
Somewhere in mid-air, you'll see the two images align, and turn into something REALLY COOL ...
Move the finger away, keep looking.
There's one tricky part. See, each time you look at something, two different things happen simultaneously - so you've been conditioned to always do both. Now you'll have to separate them. When you look at something:
- The lenses of your eyes focus at the object you are looking at - so that you can see it in high detail.
- Your eyeballs move - so that the images from both eyes align - and you see just one object and not two.
As you reach the point when both images align and it's time to remove your finger, you'll have to refocus the lenses of your eyes without moving your eyeballs. See, your the computer screen will be more distant than your finger, so you'll have to switch focus from the finger to the monitor. However, your eyes will already be looking at the point in mid-air, where the tesseract should appear.
It may take a bit of practice.
Making it interactive
Watching cool stuff is fun, but you won't learn much unless you can engage it. Our vision evolved to work in concert with movement. The best way to see is to also touch.
That's why I didn't just make a video, but wanted to make the simulation interactive. Admittedly, this part leaves a lot of room for improvement.
How do you touch a virtual 4D object? For now, we'll have to use the mouse.
You'll be able to control the rotation speed on two of the 6 possible planes. See, in 3D we can think of rotation around an axis. In 4D, this doesn't work anymore. There, you can have rotation on a plane; and there are six possible planes (rather than just three axes).
So many controls would have cluttered the display. Feel free to suggest better ways of doing it.
The simulations - at last!
Congratulations - you've arrived. Now it's time to start applying what you've learned.
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