New answers tagged discrete-geometry
4
votes
Accepted
For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
Jarkko Kari and Markus Rissanen construct such (even a substitutive one), called Sub Rosa, for even $n$ in [1]. ArXiv is https://arxiv.org/abs/1512.01402
The second-named author is Markus Rissanen, ...
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22
votes
Accepted
Can you see through a cannonball packing?
Yes. View the FCC packing as a series of stacked square packings, with spheres of unit radii centered at the points $(2a,2b,2\sqrt{2}c)$ and $(2a+1,2b+1,(2c+1)\sqrt{2})$ for all $a,b,c,\in\mathbb Z$:
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- 3,068
3
votes
Accepted
References: rigorous algorithms for elementary computations in base-b with complexity estimates
In cryptography, one often needs to implement modular (and polynomial) arithmetic $\mathbb{Z}/N\mathbb{Z}$, but your hardware only natively supports computations in $\mathbb{Z}/n\mathbb{Z}$.
Typical ...
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1
vote
Packing problems where parts of objects are allowed to intersect
Sometimes you will be able to relate this to a usual packing problem. For the initial example with a region $X$ and a complementary region $Y$, let $Z$ be the set of points which are closer to $X$ ...
- 148k
10
votes
Accepted
6
votes
Accepted
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
For $d=2$ the maximal $n$ is $3=d+1$. Indeed, between 4 points $x_1,x_2,x_3,x_4$ either one of them, $x_k$, belongs to the convex hull of three others, then $\langle \theta,x_k\rangle$ can not be the ...
- 109k
4
votes
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
This is not a complete answer, but too large to fit reasonably as a comment. As shown by Fedor Petrov, it is not true that $n \leq d+1$, since for $d = 3$ we can construct $5$ such points and ...
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3
votes
Do triple-linked graphs exist?
I have since come across the following paper which seems to answer the question affirmatively in a very strong sense:
E. Flapan, B. Mellor, R. Naimi, "Intrinsic linking and knotting are ...
- 13.6k
2
votes
Accepted
Do triple-linked graphs exist?
Yes.
Theorem 1.
For every $k$ there exists $N=N(k)$ such that in every embedding of the complete tripartite graph $K_{N,N,N}$ into $\mathbb{R}^3$ there are $k$ disjoint pairwise linked triangles; in ...
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