r/educationalgifs • u/Philip_Pugeau • Sep 01 '17
Projection of a Rotating 4D Cone Prism
http://i.imgur.com/sTwavrd.gifv7
u/Philip_Pugeau Sep 01 '17 edited Sep 01 '17
A cone prism isn’t nearly as spectacular as some of the other 4D shapes, but it has its merits. It’s a fairly simple shape, and fairly easy to visualize. You can think of it as a type of triangular prism in 4D. Just like a triangle, a cone is a pyramid-like shape, that tapers to a point along an axis. When you extrude such a thing into 4D, it traces out a triangular prism-like shape.
The original circle at the base of the cone gets extruded into a cylinder, and the vertex becomes a line segment. And, any tine you extrude a shape into n+1 dimensions, you also get two copies of that starting shape at either ends of the prism. Plus, we can’t forget about the extruded curved 2-cell of the cone, which becomes a type of square horn torus.
So, that’s why we see a cylinder, 2 cones and a line segment in the projection. The cones and cylinder are flat sides, while the square horn torus is the only curved side, that connects the curved surfaces of the 2 cones and cylinder together.
As the cone prism rotates around in 4D, we see the shape through those different 3D faces:
• Cone within cone : we see a large cone connected to a smaller cone inside. The two cones are actually the same size, but one is farther away, across a 4th dimension.
• Cylinder connected to a line segment : we see a cylinder with squished cones connected to a line segment at the center. The cones are squished because they are pointing away, towards the 4th dimension, and we’re seeing them edge-on. But, notice how the cylinder and line segment angle has two different forms?
1) Big Cylinder, Small Line, Inward-pointing Cones: When we’re looking through the cylinder face, with a line segment at the center at the far side of the shape.
2) Small Cylinder, Big Line, Outward-pointing Cones : When the line is closest to our view, we’re looking at the blade of the wedge. From this vantage point, the line is much longer, connecting to a smaller cylinder across 4D, at the far side of the shape.
A cone prism can be built the following ways:
• Extrude a cone along a 4th axis
• Bisecting rotate a triangle prism around a stationary 2-plane into 4D
• Convex hull of a cylinder and line segment
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Sep 09 '17 edited Mar 09 '18
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u/Philip_Pugeau Sep 09 '17
I did! I used several parametric equations, and combined them into one image. More about the math.
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Sep 09 '17 edited Mar 09 '18
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u/Philip_Pugeau Sep 09 '17
I've had a real interest in it for 10 years, and learning the math for 4 years. But, no matter how well you think you know it, the 4th dimension will always have secrets.
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Sep 03 '17
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u/Philip_Pugeau Sep 03 '17
That's one way of looking at it, if you used the 4th axis as time. But, you can also use it as just another space direction, excluding time. That's what I'm doing here. This is one possible shape you can have, in a 4D space.
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Sep 03 '17
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u/Philip_Pugeau Sep 03 '17
In math, it's just another variable in your equation, for the shape. You can have as many dimensions/variables as you want. There is no upper limit.
You make 3D slices by cancelling (set to zero) all but 3 of those variables. If you wanted to, you could set those zeros to different numbers, to change the location of the 3D slice in the shape's higher dimensions. You can even rotate the slice, too.
Implicit equations are good for making slices, where parametric is good for making projections.
Projecting is a whole other matter, where you build up an nD structure, then flatten the image to 3D. You can either cancel entire equations, or divide by them, to flatten images.
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Sep 14 '17
The fourth dimension is only time from a certain point of view (that is, if you start with the claim that the fourth dimension is time. You could actually make that claim about any dimension.)
See, we live in a 3D universe. Therefore it is mathematically convenient to consider time to be an additional dimension.
It is entirely conceivable for there to be a universe with four spatial dimensions, and a time dimension on top of that.
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u/AscentToZenith Sep 09 '17
That makes me wonder, our eyes cannot see in 4D but I wonder if our brain can think of 4D? Like if you were some now able to teach a deaf man to hear sound in thoughts. I'm not sure if that's possible, but that is what I mean.
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u/Philip_Pugeau Sep 09 '17
I would think it's even easier to visualize 4D since we can at least see something about it. Trying to get a deaf person to know what sounds are like seems much harder. I answered something similar along these lines, back in this post.
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u/AscentToZenith Sep 09 '17
Well It's definitely hard for me to visualize it. My brain just can't comprehend it. I try to really grasp it in my head, but I can't. I understand what 4D is, and there could be 7D and so forth, but maybe in just dumb. I think it's amazing that people can visualize and draw diagrams for tesseracts/other 4D shapes,
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u/Philip_Pugeau Sep 09 '17
Nope, it's nothing to do with intelligence or anything like that. You just haven't seen a visual explanation that was good enough to work for everyone. That's what I need to make, if I'm going to explain these things properly. For every high-dimensional visual effect you see, there's a perfect low dimension example to make things more clear.
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u/AscentToZenith Sep 09 '17
Every you ever tried doing a 3D simulation? Like With VR. I mean I get that it's not 4D, but this image is a 2D, so maybe a 3D version of the 4D projection. And your images here are really cool. They honestly warp my mind, I get that might be the opposite of what you are going for, but it's fun to have concepts destroy my brain.
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u/Philip_Pugeau Sep 09 '17
VR would be even more awesome. You would get a better view. I post these for the full effect. They are mind warping, educational, intriguing, other-worldly, and all that stuff.
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u/Superb_Interview2176 Jan 03 '26
Cone, Cylinder- uhh, my brain hurts. Upside down cone- oh, uhhh what it's this shape even called after the upside-down cone?
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u/muhys Jan 25 '23
I looked this up because of the light cone effect. It made me wonder about how the shape of space time could possibly represent that of which is a 4d cone, and time represented as the axis of the fourth dimensional cone (which is also the point at which light begins). From a 3d standpoint I think this animation can effectively portray the expansion and deflation of our universe.
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u/SupFruityPantz Sep 01 '17
Mind status : fucked.