Intereting Posts

Probability distribution and their related distributions
Proof: Series converges $\implies $ the limit of the sequence is zero
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Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + … + r^{n-1} = 0$.
Symmetric matrix is always diagonalizable?
$m$ is a perfect square iff $m$ has an odd number of divisors?
Is the Bourbaki treatment of Set Theory outdated?
Jordan Canonical Form determined by characteristic and minimal polynomials in dimension $3$, but not beyond
Characterize finite dimensional algebras without nilpotent elements
A derivation of the Euler-Maclaurin formula?
How to prove that the collection of rank-k matrices forms a closed subset of the space of matrices?
Given $a_1,a_{100}, a_i=a_{i-1}a_{i+1}$, what's $a_1+a_2$?
Order of some quotient ring of Gaussian integers
Number of horse races to determine the top three out of 25 horses

What is the shortest path between two opposing corners of cube (two corners that have no edge in common)?

- Calculation mistake in variation of length functional?
- On continuously uniquely geodesic space II
- Geodesics are minimizing in a simply connected manifold without conjugate points?
- Hamiltonian for Geodesic Flow
- Distance between two points on the Clifford torus
- Example for conjugate points with only one connecting geodesic
- Is a uniquely geodesic space contractible? I
- Uniqueness of minimizing geodesic $\Rightarrow$ uniqueness of connecting geodesic?
- Why is the geometric locus of points equidistant to two other points in a two-dimensional Riemannian manifold a geodesic?
- Is a uniquely geodesic space contractible? II

Visual solution:

Length $= \sqrt{2^2 + 1^2} =\sqrt 5.$

*Here is a non-rigorous approach:*

The geodesic must clearly consist of two straight lines segments on the two faces connected the opposing corners, and it should be pretty obvious (due to symmetry) that they should have the same “slope” w.r.t. the face that they are on.

This results in the first line segment going from the bottom-right corner of the first face (we just start here arbitrarily) to the middle of its top edge. This last point is then the middle of the lower edge of the second face through which the geodesic runs and it connects to its top-left corner.

If we’re dealing with a cube of unit side length, the total distance becomes

$$2\sqrt{1^2+\left(\frac{1}{2}\right)^2}=2\sqrt{\frac{5}{4}}=\sqrt{5}$$

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- Using Squeeze Theorem to show the following limits