All Questions
60
questions with no upvoted or accepted answers
7
votes
0
answers
121
views
Shape of very long wire between two very tall posts (many km tall) which are attached to the earth
Consider for a moment a length of uniform wire or chain which goes between two posts of equal height. If we assume the earth to be flat then we can predict the shape of the curve using
$$y = a \cosh \...
- 181
7
votes
0
answers
812
views
What are the equations that describe a Gömböc?
By 'Gömböc', I am referring to the family of shapes pictured on the Wikipedia page S.V. 'Gömböc',
https://en.wikipedia.org/wiki/Gömböc
In their paper "Static equilibria of rigid bodies: dice, ...
6
votes
0
answers
198
views
A light beam enters a closed room. What is the maximal number of reflections?
I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
- 181
6
votes
0
answers
140
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Analytic caustics for 3D objects
Is it possible to efficiently calculate caustics for a given 3D object, like a torus, or a cube?
To be more precise: let's assume that we have a 3d torus, resting on a 2d plane and a single light ...
- 405
4
votes
0
answers
69
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Why don't more celestial bodies exhibit higher-order rotations?
It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
- 434
4
votes
0
answers
129
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How to integrate scalar field over quarter torus? Infinite series does not converge.
This seems to be physics question, but the problem just concerns math.
Preface
If one wants to calculate the permeance $P$ of a rectangular bar:
it is an easy task:
$$P = \frac{\mu a b}{L} ~~~~ \...
- 295
3
votes
0
answers
37
views
Smallest Polymer that can Pass through a Circular Orifice
In a microfluidic setting, I have encountered a puzzle of finding the minimum size (smallest total-length $L$) of a polymer (can be hyperbranched or loop, whatever shape that you can make from merging ...
- 81
3
votes
0
answers
57
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Approximating the intersection of a line and the iPhone X screen as well as its normals
I am creating a simulation where little, fast moving, particles need to intersect with the edges of the iPhone X screen.
Previously I have had no difficulty with my collisions. The particles have ...
- 485
3
votes
1
answer
1k
views
maximal principle uniform covering of a sphere with uniform geodesics
Summary
This question is about covering a sphere with great circles so that the resulting density is uniform and that if we for each circle associate a normal vector, then the magnitude of the sum of ...
- 889
3
votes
0
answers
110
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What proportions make a regular right prism a fair dice?
If the base of a right prism is a regular $n$-gon of side 1, what height makes it a fair dice? The $n=4$ case is obvious by symmetry. Assume constant density, constant downwards gravity, throwing on a ...
- 3,388
3
votes
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answers
1k
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What's so special about involute curves??
An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler.
Why is this? What special ...
- 43.7k
2
votes
0
answers
53
views
Optical path of a light ray reflected from two mirrors and into a pinhole camera
I have been staring at this problem for longer than I would like to admit.
I am trying to determine the path of a light ray from an object that is reflected from two plane mirrors and into the ...
- 21
2
votes
0
answers
46
views
How many stable equilibrium points are there for a floating body?
Given a convex solid with uniform density, which is less than the density of water, if it's thrown into water, it will float.
How many stable equilibrium points can it settle into?
To think about the ...
- 3,806
2
votes
0
answers
92
views
Criteria of a manifold
I am self studying Frankel's The Geometry of Physics, early in the text he poses the example:
The x axis of the xy plane can be described as the locus of the quadratic $F(x,y) := y^2=0$. Both ...
- 111
2
votes
1
answer
146
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What would be the calculations behind the skyscraper view of a sunset as a demonstration of a rotating earth?
This is my first post here. I have been challenged to proved proof that the earth is not flat and that we live on a spinning ball. This was a fairly easy task for me and I proved several observable ...
- 121