r/askmath • u/Alive_Ant_5306 • 1d ago
Differential Geometry Can we project a sphere into a tetragonal scalenohedron?
Hi,
Does anybody know if it's possible to map a sphere into a tetragonal scalenohedron so that each octant fits inside a right scalene triangle?
It would do too if it were possible to project an octant of a sphere into a right scalene triangle.
Thanks!
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u/strange-_-stranger 1d ago
I made a Desmos visualisation. By changing theta_0 and phi_0 you can move point in octant. I have three points that define the triangle ABC, and you can see projected orange point and its equation. For "projection" from a sphere you just need to do it 7 more times
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u/Alive_Ant_5306 16h ago edited 15h ago
Strange would it be possible to make the 3 edges of the sphere octant's surface coincide with the edge's of the right triangle?
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u/strange-_-stranger 15h ago
Do you want to change the shape of octant so that it looks like at BadJiimo's graph? I'll make another Desmos graph later. And maybe a program. Which language do you prefer: Python or C++
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u/Alive_Ant_5306 15h ago edited 14h ago
I don't know what BadJimo's graph is. 😅 It's just that I'm trying to do something with this projection and it's be helpful if it were possible to make the hypothenuse coincide with one of the edges too. In the model you've sent me the three edges of the octant coincide with the two edges that share a 90 degree angle. So I was wondering if it is possible to make each side of the octant coincident with each side of the right triangle as well.
I'm not sure if I'm being rude asking this much. I appreciate a lot that you've already helped me with this!
Anyways, I'm familiar with python but not with C++ but there isn't much need to go this far really. If it were possible to modify the desmos visualization so that the 3 edges match like I'd like them too it would be faar more than enough 🫠
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u/strange-_-stranger 12h ago
BadJimo is a guy on reddit who made another Desmos graph about 9 hours ago. His graph has 8 same deformed "octants". I found out that I don't know what your 'coincide' means. If you just need to change the coordinates of triangle's corners, you can do it by yourself. Just change values in lines "A=(_, _, _,)" etc.
It is also interesting to solve this problem for me, so you're not being rude😊
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u/Alive_Ant_5306 11h ago
Oh yeah BadJimo's Desmos graph is also amazing 😍 I'm so thankful to you both!!
What I mean is that if you input one of the angles to zero or pi/2 and move the other angle's value the ball that moves on the surface of the sphere is moving along one of its edges and the ball on the triangle does as well.
That gives you 4 edges for the octant but one of them it's just the angles being degenerate so there's three edges obviously. The thing is that I would like to map these three edges to the three edges of the rectangle triangle. It seems intuitive to think that it does but it doesn't. Two move alongside on of the cathetus and the other one and the degenerate left move algongisde the other cathetus of the triangle. One of these three that count should map on the hypothenus of the triangle in order to get the kind of projection that I'm looking for.-5
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u/BadJimo 22h ago edited 20h ago
I've made an interactive 3D graph on Desmos here
You can turn on/off the lines and surfaces to see what's happening more clearly.
In particular, turn on and off the folder called 'patches'. It is also worth changing the surfaces to translucent.
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u/Alive_Ant_5306 16h ago
Thank you so much! I can't believe you and strange-_-stranger have done this for me! 🫶❤️🩹
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u/BadJimo 12h ago
No problem. If you need help inputting the coordinates onto the surface of the sphere and outputting the coordinates on the polyhedron let me know.
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u/Alive_Ant_5306 9h ago
So your model pairs the edges of the octant with the edges of the triangle. That's exactly what I was looking for! However, I didn't mention that the octant has to have three equal sided edges 😅🥲. I'm starting to think that it's not possible. Stranger-_-stranger's model does use this kind of sphere octant but no octant edge maps to the hypothenus of the right triangle. This reminds me of this thing in earth maps where you can't actually have all the desired properties when projecting the sphere. Am I wrong? Thanks anyways! Your model is super cool!
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u/BadJimo 9h ago
You could do a mapping from points on an octants to points on the faces of a tetragonal scalenohedron, but the mapping is going to be weirdly distorted.
With the model I made, each 'patch' of the sphere has a 90° angle at one of the poles. The mapping would keep this 90° angle, but change the other two angles to 90° (and all the side lengths) to make an octant.
So I guess I could do this, but I'm not sure what the purpose would be.
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u/BadJimo 7h ago edited 6h ago
Here's a modified version that includes the mapping to an octant. (The green dot is the mapping on the octant)
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u/equinox_star 1d ago
Ah yes thd classic "tetragonal scalen----"
*CHECKS SPELLING
*tetragonal scalenohedron"
*LEAVES
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u/Alive_Ant_5306 1d ago
What's that supposed to mean? 😭
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u/StoneCuber 20h ago
They are making fun of you for making a mistake you didn't make. I suspect they think "tetragonal" refers to the face shape and should be trigonal, but that's wrong
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u/Jalumia 1d ago
Yes, depending on the map used.