You're probably trying to think about it in spatial terms. But it's impossible to understand a 4D object with 3D spatial reasoning. I would say ALL humans are too "dumb" to grasp this concept. Of course the math is fairly simple, but there's a difference between understanding the math and truly understanding what a world with 4 spatial dimensions would be like.
My first CompSci professor told me that he had trouble at MIT grasping some of his early math where they were using n-dimensions, and when he asked his professor what the 'n-ball' was, he drew a circle on the board and said 'that, when n is equal to two'. Fun times.
It's tricky because there is just no way to show 4D phenomenon without a lot of sacrifice. I ended up using a combination of "perspective" for size and differing transparency and a fair amount of movement. All of these are artificial of course.
The explanation on the page takes another approach: all black and white line drawings and a lot of brevity.
But it's really a hell of a lot of fun tying to figure out how to present impossible to visualize phenomenon.
And you are not remotely too dumb: if my story didn't come through it's my fault.
Displaying a 3D image in a 2D space also requires sacrifices. Obviously to a lesser extent, because you only lose one dimension, but the principles are the same. Reconstructing a 3D scene from photographs is an insanely tricky problem, and it only looks simple to us because our brains have specialized circuits for that task.
I think of it this way: if you try to draw a 3D cube on a 2D screen, what you really have is a 2D image. If oriented correctly, and nobody told you anything about that cube, you might mistake it for a 2D hexagon. If instead that cube were rotating, you would see the change in the shape of the hexagon and realize that the image is supposed to be a cube.
But the reality is, it's still a 2D image. It's just a representation, and context and animation help you get back to the notion that it's a 3D image.
Now, think for a minute about how the screen works. There are rows and rows of pixels. Each of those rows is actually a 1-dimensional image, a slice of the whole. If you were to only see one row of pixels at a time, you'd not be able to figure out what the image is. Maybe if you knew certain parameters about the image, that it was of very low fidelity, simplistic shapes, you'd be able to analyze one row after another and figure it out. You'd have extreme difficulty just looking at a 1-dimensional image, or a serious of such, and figuring out it would be of a 3-dimensional object. To really see the image, you again need it animated. The rows are scanned out very quickly, and your brain blurs it all together.
So, when you're dropping from N to N-1 dimensions, you lose something, something that you can sort of get back if you add some motion in. When you drop from N to N-2 or more, you lose so much that you have a really hard time figuring it out.
Now we can finally come back to the tesseract. It's a 4D object. We're trying to display it on a 2D display. It's not going to work out too well. You've really got to stare at it, think about what is going on, understand that you're dealing only with a very low fidelity, simplistic shape. Analyze it, rotation after rotation, and you might be able to figure it out.
Or, in other words, it would be a lot easier if you had a real 3D display.
A 3D display doesn't help much; being able to take advantage of stereo effects helps a little, since you can see it from multiple slightly different angles at a time (one per eye). But your retina is still 2D- you can't perceive 3D objects directly through the visual system, just 2D projections of them.
A native 4D creature would presumably have a 3D retina that can directly perceive 3D projections of 4D objects and infer 4D structure from them like we infer 3D structure from 2D projections. Maybe a tactile display could improve things somewhat (though you'd still be lacking access to 3D-internal structure), but as long as you're going through the human visual system, it takes some significant effort to reconstruct the 3D projections and then go the extra step to inferring 4D structure, no matter how good your display technology is.
One can use parallel coordinates to visualize N-dimensional objects in the 2d plane. The hypercube's pretty easy to understand in the orthogonal projection, but for anything more complicated than a 24-cell, parallel coordinates really helps.
I struggle with picturing 4D rotating objects, I find it easier to deal with slicing the object up and picturing that. To clarify if you think of a single picture from an MRI it's a slice of a 3D object and you can move up and down the object to see each slice. If you move through all the slices you can get an idea of the 3D shape. With the 4th dimension it's similar except each of the slices is a 3D shape. So a 4D “sphere” sliced up this way would look like a 3d sphere that gradually got bigger until halfway through the 4D “sphere” then gets smaller again. You can also somewhat consider it a 3D object that changes shape with time but since time is a dimension you can go back and forward along it. That's how I picture 4D objects which is admittedly not as well as some people do.
For those trying to wrap their heads around this ... I find this segment by Carl Sagan from the original Cosmos to be pretty well presented on the topic: http://www.youtube.com/watch?v=UnURElCzGc0
I've always thought a 4d first person game where you could toggle pairs of dimensions could be really interesting. At any point you could swap between wxy/wxz/xyz. The rendering would be standard 2d projections of the 3 dimensions you have toggled on...Add guns and you have the trippiest FPS I've ever heard of.
It's an amazing experience to think of visualizations. I ended up with a grid of grids, like you see in the readme. Each smaller grid is a 2D slice of the world, each column a 3D stack, and the set of all columns the full 4D space.
If you think that us 3D people will never be able to wrap their heads around 4D, you have to read Charles Hinton's (http://en.wikipedia.org/wiki/Charles_Howard_Hinton) The Fourth Dimension (https://archive.org/details/fourthdimension00hintarch). This is the guy who coined the terms ana and kata (from Greek) for the two additional 4D directions, analogous to up/down, etc. (he also invented a gunpowder baseball pitching machine that, some say, led to his dismissal from Princeton, due to player injuries).
Now that's interesting on its own, but his sister-in-law, Alicia Boole Scott (http://en.wikipedia.org/wiki/Alicia_Boole_Stott) is also amazing. She was the daughter of the famous Boole, and made big contributions to higher dimensional mathematics, especially 4D, e.g. she proved there are exactly six regular polytopes in 4D. She "made beautiful cardboard models" of 3D cross sections of these. This, while she was working as a secretary in Liverpool (sad reality of early women scientists/mathematicians).
They don't make them like that anymore! It would be interesting to have documentary on 4D based on these interesting people.
EDIT: Also, in catastrophe theory, the Swallowtail catastrophe, which has 3 control and behavioral dimension was named by the French mathematician Bernard Morin (http://en.wikipedia.org/wiki/Bernard_Morin), who is blind since he was 6!
I love every blog post you write and every image i see released of the game.
It's just that i've come to terms with the high possibility that i simply might not get to play this game at any point.
I'm simply trying to say that the path to the realization should be made easier for other people before they become enamored only to face the harsh reality of the last paragraph on the game's website. ;)
Reminds me of the time that Sting sang "Message in a klein bottle". Of course, as soon as the bottle was thrown into the sea, the water seeped in destroying the message on the paper.
"Bollocks", said Sting. "There goes my proof of the Riemann Hypothesis."
For a more in-depth discussion on 4 dimensional geometry, I highly recommend the Dimensions films[1], which are available for free download. It's a little bit mind-bending, so to speak.
43 comments
[ 2.7 ms ] story [ 90.6 ms ] threadThe explanation on the page takes another approach: all black and white line drawings and a lot of brevity.
But it's really a hell of a lot of fun tying to figure out how to present impossible to visualize phenomenon.
And you are not remotely too dumb: if my story didn't come through it's my fault.
But the reality is, it's still a 2D image. It's just a representation, and context and animation help you get back to the notion that it's a 3D image.
Now, think for a minute about how the screen works. There are rows and rows of pixels. Each of those rows is actually a 1-dimensional image, a slice of the whole. If you were to only see one row of pixels at a time, you'd not be able to figure out what the image is. Maybe if you knew certain parameters about the image, that it was of very low fidelity, simplistic shapes, you'd be able to analyze one row after another and figure it out. You'd have extreme difficulty just looking at a 1-dimensional image, or a serious of such, and figuring out it would be of a 3-dimensional object. To really see the image, you again need it animated. The rows are scanned out very quickly, and your brain blurs it all together.
So, when you're dropping from N to N-1 dimensions, you lose something, something that you can sort of get back if you add some motion in. When you drop from N to N-2 or more, you lose so much that you have a really hard time figuring it out.
Now we can finally come back to the tesseract. It's a 4D object. We're trying to display it on a 2D display. It's not going to work out too well. You've really got to stare at it, think about what is going on, understand that you're dealing only with a very low fidelity, simplistic shape. Analyze it, rotation after rotation, and you might be able to figure it out.
Or, in other words, it would be a lot easier if you had a real 3D display.
Adding the motion restores the subtracted dimension as time. That's why that works.
A native 4D creature would presumably have a 3D retina that can directly perceive 3D projections of 4D objects and infer 4D structure from them like we infer 3D structure from 2D projections. Maybe a tactile display could improve things somewhat (though you'd still be lacking access to 3D-internal structure), but as long as you're going through the human visual system, it takes some significant effort to reconstruct the 3D projections and then go the extra step to inferring 4D structure, no matter how good your display technology is.
Adanaxis[1] is a 4 dimensional space shooter- only sort of playable but still fun.
http://www.mushware.com/
http://www.amazon.com/Flatland-Dover-Thrift-Editions-Abbott/...
Touching all surfaces != fun.
Maybe collecting an item(s) on some surfaces could be ok.
Or a "finish line" box to go through on some surface.
It's more fun, IMO, to have specific goals.
Uninstalled for now but am excited to see where you might take it.
It's an amazing experience to think of visualizations. I ended up with a grid of grids, like you see in the readme. Each smaller grid is a 2D slice of the world, each column a 3D stack, and the set of all columns the full 4D space.
Now that's interesting on its own, but his sister-in-law, Alicia Boole Scott (http://en.wikipedia.org/wiki/Alicia_Boole_Stott) is also amazing. She was the daughter of the famous Boole, and made big contributions to higher dimensional mathematics, especially 4D, e.g. she proved there are exactly six regular polytopes in 4D. She "made beautiful cardboard models" of 3D cross sections of these. This, while she was working as a secretary in Liverpool (sad reality of early women scientists/mathematicians).
They don't make them like that anymore! It would be interesting to have documentary on 4D based on these interesting people.
EDIT: Also, in catastrophe theory, the Swallowtail catastrophe, which has 3 control and behavioral dimension was named by the French mathematician Bernard Morin (http://en.wikipedia.org/wiki/Bernard_Morin), who is blind since he was 6!
As referenced in a XKCD http://xkcd.com/721/
I love every blog post you write and every image i see released of the game.
It's just that i've come to terms with the high possibility that i simply might not get to play this game at any point.
I'm simply trying to say that the path to the realization should be made easier for other people before they become enamored only to face the harsh reality of the last paragraph on the game's website. ;)
"Bollocks", said Sting. "There goes my proof of the Riemann Hypothesis."
[1] http://www.dimensions-math.org/Dim_E.htm