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4D Toys: a box of four-dimensional toys

marctenbosch.com

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Re: 4D Toys: a box of four-dimensional toys

#71

Post by mabbo »

pbhjpbhj wrote: post #66
mabbo wrote: post #55

Earlier quoted context omitted.

Watch the part about the slicing of 2d world looking at the 3d objects. That slice is infinitely large in both dimensions, yet it can't see the 3D objects when they aren't in the same plane. Now apply the same analogy to our world: our 3D world is just an infinitely large 3D plane in a 4D world. When the objects aren't in our 'plane', we can't see them.

See also Flatland ( http://amzn.to/2qKFrOh , aff. link) which is a short novel in part about 2D shapes discovering the third dimension. Edit: Didn't notice it was mentioned at the end of the comments, https://news.ycombinator.com/item?id=14472395 , there's a free version linked from archive.org.

A favorite novel of mine actually! Hard to believe that it was written in the 1800s, yet still Abbott understood the 4th dimension better than most people today.

Re: 4D Toys: a box of four-dimensional toys

#72

Post by jacobolus »

eriknstr wrote: post #32
eriknstr wrote: post #24

Earlier quoted context omitted.

Same here. I understand perfectly everything that was said in the video but I still don't understand how the fourth dimension quite works. However, having only heard of hypercubes and not hyperspheres before I decided to see if there was anything useful about them online and I found this video that I just started watching and already 1 min 50 sec into the video something very interesting was said; > Everybody knows w…

Another quote from the video I linked in parent comment. 6:19 > As complex numbers are to real numbers, quaternions are to complex numbers. It's like a way to build up even further. [...] Real numbers are one-dimensional. Complex numbers are two-dimensional. [...] For three dimensions there is no natural number system, but for four dimensions there is and it looks like this.

That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.

In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)

Re: 4D Toys: a box of four-dimensional toys

#74

Post by jacobolus »

jerf wrote: post #49
indspenceable wrote: post #40

Earlier quoted context omitted.

Often when I see something about 4d it uses this same analogy: a 2d being seeing a crosssection of a 3d world, in 2d. However, the video is in 2d, and it's able to show 3d in a much better/more clear way, clearly, cause when it's showing that example it's got a cutout of the "3d" view (which is still 2d!). Why can't we do the same thing with 4d? Why does the object just disappear when it bounces into the 4th dimensio…

"Why does the object just disappear when it bounces into the 4th dimension, can't we maybe see a projection of it onto the 3rd dimension?" Yes, you can. This program just doesn't.

There’s a big problem with that, which is that we don’t actually see in 3 dimensions. Our eyes only get 2-dimensional projections of a 3-dimensional world.

Re: 4D Toys: a box of four-dimensional toys

#75

Post by jacobolus »

tomdre wrote: post #69
js8 wrote: post #23

If you make a Kickstarter to 4D print them, I would support it for my kids. I think for kids, it's more important to play with real-world physical objects rather than their virtual computer representation.

As we represent on a 2d paper a projection of a 3d cube, we can print a 3d object that represents a projection of a 4d one. I'm down for the kickstarter.

You might like http://www.3dprintmath.com/chapters/3-Four-dimensional-space

Re: 4D Toys: a box of four-dimensional toys

#76

Post by jacobolus »

pbhjpbhj wrote: post #70
sp332 wrote: post #50

Earlier quoted context omitted.

That's if you do a "projection", like light rays from a 3D object focused onto a screen. This is more like taking a 3D slice of a 4D object. If you think of taking a 2D slice of a 3D object, you can see how objects disappear when they move out of the plane that you're drawing.

If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear? Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in…

A 3-sphere (hypersphere embedded in 4 dimensional space) is a sphere that can be incrementally rotated in 6 orthogonal directions. :-)

Re: 4D Toys: a box of four-dimensional toys

#77

Post by sp332 »

pbhjpbhj wrote: post #70
sp332 wrote: post #50

Earlier quoted context omitted.

That's if you do a "projection", like light rays from a 3D object focused onto a screen. This is more like taking a 3D slice of a 4D object. If you think of taking a 2D slice of a 3D object, you can see how objects disappear when they move out of the plane that you're drawing.

If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear? Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in…

You could take a 2D slice of a tesseract, but what most visualizations do is take a 3D slice and then project that in a more familiar way to 2D. So you could get any of these shapes: https://en.wikipedia.org/wiki/Tesseract#/media/File:Orthogon... If you 2D-slice a regular cube you could get various shapes too. https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-g...

A hypersphere has a similar description to a regular sphere or a circle, which is that every point a fixed distance from the center is part of the sphere. So I'd say it's a little boring. No matter what 3D section you take, it's just a larger or smaller sphere. It gets smaller if you take a slice off to the side, just like taking a slice of a sphere gives you a smaller circle near the side.

Re: 4D Toys: a box of four-dimensional toys

#78

Post by lorenzosnap »

jacobolus wrote: post #74
jerf wrote: post #49

Earlier quoted context omitted.

"Why does the object just disappear when it bounces into the 4th dimension, can't we maybe see a projection of it onto the 3rd dimension?" Yes, you can. This program just doesn't.

There’s a big problem with that, which is that we don’t actually see in 3 dimensions. Our eyes only get 2-dimensional projections of a 3-dimensional world.

I am not sure you are correct. We have 2 eyes so that our brain can work out the depth of the image.

Re: 4D Toys: a box of four-dimensional toys

#79

Post by js8 »

placeybordeaux wrote: post #64
js8 wrote: post #23

If you make a Kickstarter to 4D print them, I would support it for my kids. I think for kids, it's more important to play with real-world physical objects rather than their virtual computer representation.

Is this satire?

It's a joke - I don't actually have kids. And I mean the 2nd sentence seriously. Also, quick googling reveals that 4D printers haven't been invented yet, so presumably one would need a KS for those first.

Re: 4D Toys: a box of four-dimensional toys

#80

Post by Crespyl »

pbhjpbhj wrote: post #70
sp332 wrote: post #50

Earlier quoted context omitted.

That's if you do a "projection", like light rays from a 3D object focused onto a screen. This is more like taking a 3D slice of a 4D object. If you think of taking a 2D slice of a 3D object, you can see how objects disappear when they move out of the plane that you're drawing.

If you sliced a tesseract then presumably you'd encase it in the knife and a square, with no depth (the depth is in the 4th dimension that we're not experiencing), would appear? Hypercubes have always been a difficult one for me to intuit, basically I have no 4th dimensional intuition. When I think of a 4D hypersphere all I can get is a simple sphere. I don't have any intuition as to whether that's wrong, it seems in…

It might help if you consider spheres as a surface.

3D spheres are a 2D surface wrapped into a 3D space, likewise, hyperspheres would be a 3D surface wrapped into a 4D space. There's no "infinite spheres on its surface", I think the "rotation" is a better analogy.

Take a line rotated around an orthogonal axis and you have a circle, a circle rotated around an axis orthogonal to the other two is a sphere, a sphere rotated around another orthogonal axis is a hypersphere.

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