Answer by efs for Proofs where higher dimension or cardinality actually...
Let $P$ be a convex polytope in $\mathbb{R}^d$ with vertices $v_1,\dots,v_n\in \mathbb{Z}^d$. A nice trick that helps to visualize, understand and prove that the number of lattice points in dilations...
View ArticleAnswer by coudy for Proofs where higher dimension or cardinality actually...
Hopefully, you will agree that infinite dimensionis higher dimension.A fruitful approach to solve a nonlinear problemon a finite dimensional space is to convert it intoa linear problem on an infinite...
View ArticleAnswer by Marco Golla for Proofs where higher dimension or cardinality...
The Toepliz problem, also known as "inscribed square problem" or "square peg problem", asks whether every Jordan curve in the plane contains the vertices of a square.Vaughan's proof of the rectangular...
View ArticleAnswer by Ivan Meir for Proofs where higher dimension or cardinality actually...
The aperiodic Penrose tiling can be generated as a cross section of a regular tiling in 5 dimensions, which is periodic! See this answer for more details.
View ArticleAnswer by Arun Debray for Proofs where higher dimension or cardinality...
This answer is quite different in spirit from my other answer, so I've factored it out.The $n$th bordism group $\Omega_n$ is the abelian monoid of diffeomorphism classes of closedsmooth $n$-manifolds...
View ArticleAnswer by Arun Debray for Proofs where higher dimension or cardinality...
There are several examples related to the mathematics of quantum field theory in which using higher-dimensionalthinking in physics led to mathematical theorems answering questions which might not have...
View ArticleAnswer by Chua KS for Proofs where higher dimension or cardinality actually...
For a graph $G$, its multi-dimensional characteristic polynomials $\Phi_G=\det(I_x-A)$ where $A$ is adjacency and $I_x=diag\{x_1,...,x_n\}$. It's definition depends on the labeling of the vertices but...
View ArticleAnswer by Noam D. Elkies for Proofs where higher dimension or cardinality...
The symmetry of the octahedron projects toSimpson's Rule.Recall that Simpson's Rule is the approximation$$\int_a^b f(x) \, dx \approx \frac{b-a}{6} \Bigl(f(a) + 4 f\bigl(\frac{a+b}{2}\bigr) + f(b)...
View ArticleAnswer by Thomas Lesgourgues for Proofs where higher dimension or cardinality...
Given that you asked about planar graph : In graph theory, there is the Heawood conjecture proven in 1968 by Ringel and Youngs: If a graph $G$ has genius $g>0$ then$$ \chi(G)\leq \left\lfloor...
View ArticleAnswer by Sandeep Silwal for Proofs where higher dimension or cardinality...
Not a theorem but a cool result regardless: Given a convex $n$ sided polygon in 2D, give an algorithm to find the largest circle that can fit inside it. I am not aware of any non messy or particularly...
View ArticleAnswer by Fedor Petrov for Proofs where higher dimension or cardinality...
The famous result by Bang is that if a convex compact set $K\subset \mathbb{R}^n$ is covered by a finite number of open planks, then the sum of their widths is greater than a width of $K$. [The closed,...
View ArticleAnswer by Abdelmalek Abdesselam for Proofs where higher dimension or...
An example in the spirit (in fact a generalization) of the answer by Sam T is the triviality of the $\phi^4$ quantum field theories from lattice approximations.In dimension 5 or more this was done a...
View ArticleAnswer by Mike Wise for Proofs where higher dimension or cardinality actually...
In 3D-graphics, 3D-points are translated into 4D-points using a technique refereed to as "homogeneous coordinates". Then 3D-perspective transformations and coordinate translations (which are non-linear...
View ArticleAnswer by Denis Serre for Proofs where higher dimension or cardinality...
The Cauchy problem for the wave equation$$\partial_t^2u=c^2\Delta_xu$$is not too difficult to solve explicitly in $3$ space dimensions, by the method of spherical means. This yields a close formula for...
View ArticleAnswer by Peter LeFanu Lumsdaine for Proofs where higher dimension or...
(Uri Bader points this out in comments, but it should really be an answer.)A classic example is the calculation of the one-dimensional integral $\int_{-\infty}^\infty e^{-x^2}dx$ by squaring it,...
View ArticleAnswer by Andrea Ferretti for Proofs where higher dimension or cardinality...
Differential equations of order $n$ in $\mathbb{R}$, like $\frac{d^n}{dt^n}x(t) = F\left(t, x(t), \frac{d}{dt}x(t), \dots, \frac{d^{n-1}}{dt^{n-1}}x(t)\right)$ can be transformed into first order...
View ArticleAnswer by Noam D. Elkies for Proofs where higher dimension or cardinality...
Tarski's plank theorem (1932).A plank of width$w$ in ${\bf R}^n$ is the closed region betweentwo parallel hyperplanes at distance $w$ from each other.Q:Can a unit disc in ${\bf R}^2$ be covered witha...
View ArticleAnswer by Max Xiong for Proofs where higher dimension or cardinality actually...
Here is an example from projective geometry. Desargues's theorem states that for two triangles, if the lines connecting their corresponding vertices are concurrent, then the intersection of each pair...
View ArticleAnswer by John Bentin for Proofs where higher dimension or cardinality...
The following is copied from an answer to another question on this site.Here's an example in planar euclidean geometry. Consider an equilateral triangle of side $a$ and a general point in the plane...
View ArticleAnswer by Sam OT for Proofs where higher dimension or cardinality actually...
Perhaps I am just partially blind and someone has already said this, but the thing that springs to mind for me is to show that there no percolation (ie no infinite component) at criticality for...
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