Defining the process of twisting a prism : Twisting the top face of a prism with no walls.
The prism can show these two behaviors while getting twisted :
- An ideal prism(side length not constant) will simply have it's face twisted with no other change,
- A real scenario where the side length is constant and hence there is a slight compression perpendicular to the top face,
For this post I am concerned about the second point i.e when the side length is constant.
Some more examples I constructed-
I am providing the link of a google drive folder where I have uploaded the geogebra files so you guys can experiment with them.
As I was constructing these, I noticed that the length all the figures were getting compressed was equal (the polygons had equal radii, and the side length was also equal). I did it only till Pentagon
- so I hypothesize that it will be equal for every regular polygon given the radius and the side length are equal. Is my hypothesis correct? If yes how to prove it?
I noticed another thing- every 180° rotation resulted in the first intersection for every polygon prism not depending on the radius/side length. I tried thinking a lot about it but wasn't able to visualize it,
- why does the first intersection happens after rotating 180°?
My last but not the least question:
- how can we find the relation between the angle by which the top face gets twisted and the changing angle between the polygon side and the side length i.e
In the process of construction, I found out the locus of the vertices : taking the example of a square prism the vertex $\text{B}_1$ follows : $$x=\sqrt{l^2 - (r\cos (\phi + \pi /2)-h)^2 - (r\sin (\phi + \pi /2)-k)^2}-m \\ y=r\cos(\phi +\pi /2) \\ z=r\sin(\phi + \pi /2) \\ \text{the prism is along x axis}\\ \text{ $(m,h,k)$ are the $x,y,z$ coordinates of $ \text{A}_1 $respectively} \\ \text{ $\phi$ is the angle by which the top face is getting rotated.} \\ \text{ $r,l$ are the radius and length of the prism respectively.}$$ Note that I have added a + $\pi /2$ in the angle to denote the initial coordinate of the vertex.
Thank you for reading my long question, I will appreciate all answers/comments :)
from Hot Weekly Questions - Mathematics Stack Exchange
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