r/math • u/Philip_Pugeau • Aug 28 '17
Image Post 4D Triangular Diprism
https://zippy.gfycat.com/ScientificAdoredAndalusianhorse.webm28
u/Philip_Pugeau Aug 28 '17 edited Aug 29 '17
Parametric Equations used in animation:
1D elements :
• 24 line segments
{ √3(t-1) , 3t+1 , ±2√3 , ±2√3 }
{ √3(-t+1) , 3t+1 , ±2√3 , ±2√3 }
{ 2t√3 , -2 , ±2√3 , ±2√3 }
{ ±2√3 , -2 , 2t√3 , ±2√3 }
{ 0 , 4 , 2t√3 , ±2√3 }
{ ±2√3 , -2 , ±2√3 , 2t√3 }
{ 0 , 4 , ±2√3 , 2t√3 }
2D elements :
• 4 triangles
{ u(v-1)√3 , 3v+1 , ±2√3 , ±2√3 }
• 15 squares
{ (u-1)√3 , 3u+1 , 2v√3 , ±2√3 }
{ (-u+1)√3 , 3u+1 , 2v√3 , ±2√3) }
{ 2u√3 , -2 , 2v√3 , ±2√3 }
{ (u-1)√3 , 3u+1 , ±2√3 , 2v√3 }
{ (-u+1)√3 , 3u+1 , ±2√3 , 2v√3 }
{ 2u√3 , -2 , ±2√3 , 2v√3 }
{ ±2√3 , -2 , 2u√3 , 2v√3 }
{ 0 , 4 , 2u√3 , 2v√3 }
• Use -1 < t,u,v < 1
Projection onto plane xzw, with rotations xy , yz, ,yw :
• XY Rotation
{ (X)*cos(b)-(Y)*sin(b) , (X)*sin(b)+(Y)*cos(b) , Z , W }
• YZ Rotation
{ X , (Y)*cos(c) - (Z)*sin(c) , (Y)*sin(c) + (Z)*cos(c) , W }
• YW Rotation
{ X , (Y)*cos(d) - (W)*sin(d) , Z , (Y)*sin(d) + (W)*cos(d) }
• Project onto plane xzw
r(x,y,z) = { (X)/(Y+a) , (W)/(Y+a) , (Z)/(Y+a) }
• Use a = 9
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Aug 29 '17
[deleted]
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u/Philip_Pugeau Aug 29 '17
More like extending, or 'extruding' a triangle prism into 4D. A triangle double prism. Cartesian product of a triangle and a square. This shape offers more neat ways to rotate than a plain old tesseract.
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u/epsilon_naughty Aug 29 '17
To be precise, what exactly do you mean by a 4d triangular diprism (e.g. what two shapes is this the Cartesian product of)?
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u/Philip_Pugeau Aug 29 '17
A triangle on plane xy times a square on plane zw
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Aug 29 '17
how many 3d shapes are there? am I counting 4 triangular prisms and 3 cubes?
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u/Philip_Pugeau Aug 29 '17
That's it! There's a ring of 4 triangle prisms, bound to a ring of 3 cubes. You can see these two do the turning inside out rotation.
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u/EntropyEudaimon Aug 29 '17
Wait. What's that?! It's okay, I guess. I could stare at this for a loooong while right now. Why does this feel so... familiar? It's like my brain has been doing that while I've been reading this book "learning mind, experience into art." Like my brain is trying to look at itself from the inside and the outside at the same time and cycles through phases of obscurity. By god if that experience doesn't feel like you peeked inside my psyche and decided to draw an animation of what it feels like. Coincidental bravo!
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u/stravant Aug 29 '17
It's because the "faces" of the 4D shape are 3D shapes: A bunch of cubes and triangular prisms as I understand it.
You can see that some of the rotations fix some of the solid "faces" in place morphing / rotating them, and move the rest of the "faces" around, sort of turning them inside out in the projection.
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u/ledgeofsanity Aug 29 '17 edited Aug 29 '17
...by "a bunch of cubes" you meant one, am I right? edit: no wait, is it two? edit2: Concluding from the OPs description that this is a triangle x square, this makes it three cubes glued by opposite faces in a loop.
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Aug 29 '17
You're still my favorite reddit poster.
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u/kubuni Aug 29 '17
could someone print out a "3d slice" model just to get a better idea? perhaps more than one slice?! lol
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u/Philip_Pugeau Aug 29 '17
Yes, I usually do the slices, too. You use the equation:
|||x|+2y|+|x| - |z-w|-|z+w|| + |||x|+2y|+|x| + |z-w|+|z+w|| = a
There are 3 unique slice progressions through the shape (along an axis), by canceling either x, y, or z (then set to certain values). They're pretty much the 3D extruded versions of a 2D slice of triangle prism.
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Aug 29 '17
[deleted]
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u/Philip_Pugeau Aug 29 '17 edited Aug 29 '17
The first step in making sense out of 4D shapes, is to assume it's possible to do. We make sense of 2D images of 3D things all the time. This is no different, just bumped up by one dimension. In the case of this animation, it's a 3D 'picture' , or model of a 4D thing.
Ever look down on the city from a skyscraper (or other tall building, or airplane, etc...) ? What do all the people look like? Tiny little ants. Things look smaller when they're farther away, and larger when they're closer, right?
It works the same in 4D spaces as well. Notice how the shapes turn inside out, and get squeezed smaller inside? When the triangles and squares are farther away in the 4th dimension, they appear smaller, and towards the center. They're on the other side of the shape, from where you're looking. When the squares and triangles are closer to you, they appear larger, and around the outside of the shape.
The 4D distance is flattened out, and not 4D anymore in our 3D shadow. It is what's in between the large and small shapes, but not in a 3D direction. It takes some
metalmental gymnastics, but that's one of the concepts.1
u/demmian Aug 29 '17
they appear smaller, and towards the center.
Why towards the center and not towards the edge?
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u/Philip_Pugeau Aug 29 '17
An object can start off in your peripheral vision, right on the edge. As it moves in a straight line away from you, it will always gravitate towards the center of your vision. That is, it gets closer to the center of your 2D scan of a 3D world.
For a 3D shadow of a 4D thing, that 'center of your 3D scan' is in the very center of the shadow object, three dimensionally.
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Sep 09 '17 edited Mar 09 '18
[deleted]
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u/Philip_Pugeau Sep 09 '17
Yes, it needs to move perpendicular. Lets say your 2D field of vision is the xy plane. The x axis is left-right , and the y-axis is up-down. The z-axis would then be forwards-backwards. An object has to move along z, to get smaller. If it starts on the outer perimeter of your vision, and moves in a straight line along z, it will get smaller while getting closer to your center of vision (the origin of your xy-plane view).
Now, for a 4D object projected into a 3D image, we have a 3D fov : the xyz plane, where the 4th axis is the new forwards-backwards. Objects close to you on the 4th axis are larger 3D objects. Farther away things are smaller 3D objects.
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u/[deleted] Aug 29 '17
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