5 ms·

Does anyone know of a good visualization of a hypercube? Specifically, I'd like to be able to rotate it around various axes to see what its shadow looks like in

Does anyone know of a good visualization of a hypercube? Specifically, I'd like to be able to rotate it around various axes to see what its shadow looks like in 3D. On that note, a VR rendering might help since a 2D projection overcomplicates the shadow. It's easy to construct, just build a point, extrude the point out to make a line, extrude the line out to make a square, extrude the square out to make a cube, then extrude the cube out to make a hypercube. It looks like two cubes with the corners connected by lines through a 4th dimension that we can't construct in 3D. One way to visualize it is to use time, so if you translate a cube from point A to point B, the "extrusion" would be along time and you can kind of visualize that extra "dimension". But what would scaling, rotation, etc along that invisible axis look like? To fully grok it, we'd need to be able to sculpt the hypercube's shadow in 3D as easily as drawing a cube on a piece of paper. Asking in the hopes of building a mental bridge to 4D and then possibly 5D in order to generalize to higher dimensions.

ogogmad 5y ago
You could use different projections, like gnomonic or stereographic projection. There won't be any time dimension to deal with, but it should make it easier to apply rotations and see the effects.
enriquto 5y ago
> rotate it around various axes Wait, doesn't it rotate around a plane?
mariusor 5y ago
4D Toys from Marc Ten Bosch: https://4dtoys.com/ https://4dtoys.com/ Based on the same 4D geometry concepts he's working on the game Miegakure: https://miegakure.com/ https://miegakure.com/
p1mrx 5y ago
Stella4D free demo: https://www.software3d.com/Stella.php#stella4D https://www.software3d.com/Stella.php#stella4D
corysama 5y ago
https://apps.apple.com/de/app/the-fourth-dimension/id504201783?l=en https://apps.apple.com/de/app/the-fourth-dimension/id5042017... https://www.theverge.com/2012/3/29/2912766/the-fourth-dimension-for-ios-hands-on https://www.theverge.com/2012/3/29/2912766/the-fourth-dimens...
JoeDaDude 5y ago
I created a Hypercube screen saver back in the day from instructions provided in a Scientific American article, which if memory serves was written by Martin Gardner. A casual search leads me to believe the article is in his book Mathematical Carnival, though I could be wrong. https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/dp/039472349X https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/... While you program it up, you can play this little Hypercube game. Do it in 3D, then turn on the 4D to get an intuitive feel for the 4D world. http://harmen.vanderwal.eu/hypercube/ http://harmen.vanderwal.eu/hypercube/
mikewave 5y ago
Back in the day this Java applet by the famous Ken Perlin was fun: https://mrl.cs.nyu.edu/~perlin/demox/Hyper.html https://mrl.cs.nyu.edu/~perlin/demox/Hyper.html Recommendation: Your brain actually has two depth perception systems: hardware (using your two eyes for parallax) and software (using visual cues to establish a 3D scene, the way you can with one eye, or when looking at 3d projections onto a 2d surface). Use each depth perception mechanism for a different axis! In this old applet you could enable stereo projection and also thick-lines-mode which really helped this, and gave me a strong intuitive feeling for the motions of the hypercube.
falsifian 5y ago
This demo is actually fairly close to showing you a hypercube. The six rooms in the house form six cells of the hypercube. You can make one more cube by joining up the tops of the walls: that cell is always above you. And finally, there would be one more cell below you. (To be a bit more precise: the six rooms of the house would have to be extended to be twice as tall. The tops of the walls are at coordinates (+-1, 1, +-1, +-1), but right now the bottoms of the walls are at (+-1, 0, +-1, +-1). You have to remove the floor and make the walls go all the way down to (+-1, -1, +-1, +-1). You can do this for any 4D polytope. Taking it down a dimension, the layout of the house shows how to put a cube on a 2-sphere. I could have done it for any other 3D polyhedron. Give the sphere and the polyhedron the some centre point, and then project all the lines of the polyhedron onto the sphere. Each face of the polyhedron becomes a 2-D region on the sphere (in the case of the house, each room is a square). Going a dimension up, you can centre the 3-sphere and the polytope on the same point, and do the same kind of projection. Every cell of the polytope turns into one 3-D region of the 3-sphere. I wish I had had time to include that.

ogogmad 5y ago

You could use different projections, like gnomonic or stereographic projection. There won't be any time dimension to deal with, but it should make it easier to apply rotations and see the effects.

enriquto 5y ago

> rotate it around various axes Wait, doesn't it rotate around a plane?

mariusor 5y ago

4D Toys from Marc Ten Bosch: https://4dtoys.com/ https://4dtoys.com/ Based on the same 4D geometry concepts he's working on the game Miegakure: https://miegakure.com/ https://miegakure.com/

p1mrx 5y ago

Stella4D free demo: https://www.software3d.com/Stella.php#stella4D https://www.software3d.com/Stella.php#stella4D

corysama 5y ago

https://apps.apple.com/de/app/the-fourth-dimension/id504201783?l=en https://apps.apple.com/de/app/the-fourth-dimension/id5042017... https://www.theverge.com/2012/3/29/2912766/the-fourth-dimension-for-ios-hands-on https://www.theverge.com/2012/3/29/2912766/the-fourth-dimens...

JoeDaDude 5y ago

I created a Hypercube screen saver back in the day from instructions provided in a Scientific American article, which if memory serves was written by Martin Gardner. A casual search leads me to believe the article is in his book Mathematical Carnival, though I could be wrong. https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/dp/039472349X https://www.amazon.com/Mathematical-Carnival-Martin-Gardner/... While you program it up, you can play this little Hypercube game. Do it in 3D, then turn on the 4D to get an intuitive feel for the 4D world. http://harmen.vanderwal.eu/hypercube/ http://harmen.vanderwal.eu/hypercube/

mikewave 5y ago

Back in the day this Java applet by the famous Ken Perlin was fun: https://mrl.cs.nyu.edu/~perlin/demox/Hyper.html https://mrl.cs.nyu.edu/~perlin/demox/Hyper.html Recommendation: Your brain actually has two depth perception systems: hardware (using your two eyes for parallax) and software (using visual cues to establish a 3D scene, the way you can with one eye, or when looking at 3d projections onto a 2d surface). Use each depth perception mechanism for a different axis! In this old applet you could enable stereo projection and also thick-lines-mode which really helped this, and gave me a strong intuitive feeling for the motions of the hypercube.

falsifian 5y ago

This demo is actually fairly close to showing you a hypercube. The six rooms in the house form six cells of the hypercube. You can make one more cube by joining up the tops of the walls: that cell is always above you. And finally, there would be one more cell below you. (To be a bit more precise: the six rooms of the house would have to be extended to be twice as tall. The tops of the walls are at coordinates (+-1, 1, +-1, +-1), but right now the bottoms of the walls are at (+-1, 0, +-1, +-1). You have to remove the floor and make the walls go all the way down to (+-1, -1, +-1, +-1). You can do this for any 4D polytope. Taking it down a dimension, the layout of the house shows how to put a cube on a 2-sphere. I could have done it for any other 3D polyhedron. Give the sphere and the polyhedron the some centre point, and then project all the lines of the polyhedron onto the sphere. Each face of the polyhedron becomes a 2-D region on the sphere (in the case of the house, each room is a square). Going a dimension up, you can centre the 3-sphere and the polytope on the same point, and do the same kind of projection. Every cell of the polytope turns into one 3-D region of the 3-sphere. I wish I had had time to include that.