52
votes

Accepted

### Why do bees create hexagonal cells ? (Mathematical reasons)

There are two principles at play here: a mathematical principle that favors hexagonal networks, and a physical principle that favors a network with straight walls.
The mathematical principle that ...

43
votes

Accepted

### Understanding sphere packing in higher dimensions

There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition ...

26
votes

### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

There are many ways to interpret this question depending on how you "randomly" choose your vectors. Here's one example. Take the set of vectors $v\in\{-1,1\}^N$, where the coordinates of $v$ are ...

25
votes

### Why do bees create hexagonal cells ? (Mathematical reasons)

There is this theorem of Thomas Hales from 1999, which proves the Honeycomb Conjecture:
Theorem.
Let $\Gamma$ be a locally finite graph in $\mathbb{R}^2$, consisting of smooth curves, and such that $\...

24
votes

Accepted

### Does every positive-definite integral lattice admit an angle-preserving homomorphism into $\Bbb Z^n$ for some $n$?

Yes, this is true. There's some fancy number theory that one can apply (the Hasse-Minkowski invariant and embedding of quadratic forms), but one can see this directly without number-theoretic ...

17
votes

### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

The results from a large number of repeated trials can be viewed as a random point in a high-dimensional space, so we can use our intuition about probability in the long run as a substitute for our ...

16
votes

Accepted

### On (a generalization of) the Gauss Circle Problem

You can write a similar expression to the usual formula for the remainder term in the circle problem by using the Poisson summation formula. The shift of the center of the circle simply means that ...

16
votes

Accepted

### 24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise

Yes, such a configuration exists, even with all inner products positive
(as the title of the question requires, even though the text allows
either sign).
We shall use the standard normalization of ...

16
votes

Accepted

### Counting primitive lattice points

No, there is no result in this form because in dimension 3 or higher it is allowed to have some non-first minima relatively small even when the first minimum is very small.
For example, for any $N>...

15
votes

Accepted

### Is there a contractible hyperbolic 3-orbifold of finite volume?

Yes. For example, let $M$ be the figure-eight knot complement. So $M$ is a hyperbolic manifold with volume a bit more than 2. The manifold $M$ has a two-fold symmetry $\tau$ that fixes, pointwise, a ...

14
votes

### Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?

Pick two random unit vectors. After picking the first vector, switch to a coordinate system in which this is the first basis vector. The probability distribution for the second vector is assumed to be ...

14
votes

Accepted

### Is it possible to completely embed complete Heyting Algebras into upsets of a poset?

No, not in general: for instance, the real interval $([0,1],{\le})$, or any non-atomic complete Boolean algebra, do not have such an embedding. This follows from the following characterization:
...

13
votes

Accepted

### Polynomials leaving invariant the Gaussian integers

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \...

13
votes

Accepted

### Simple conjecture about rational orthogonal matrices and lattices

Proof
Let $R$ be any matrix. We have the obvious exact sequence
$$ 0 \longrightarrow\mathbb{R}^N \xrightarrow[\left(\begin{matrix} I \\ R \end{matrix}\right)]{} \mathbb{R}^N \oplus \mathbb{R}^N \...

12
votes

### An interesting sum over lattice points in a large disk centered at the origin

The limit equals $-\pi\log 2$, in accordance with Henri Cohen's remark above.
For the proof, we combine the formula
$$r_2(k):=\#\{(m,n)\in\mathbb{Z}^2:m^2+n^2=k\}=4\sum_{d\mid k}\chi_4(d)$$
with ...

12
votes

Accepted

### what is the number of paths returning to 0 on the hexagonal lattice

This is answered by Ian Agol here, with the reference "All Roads Lead to Rome-Even in the Honeycomb World", Brani Vidakovic, Amer. Statist. 48 (1994) no. 3, 234-236.
An exact formula is
$$ p(n) = \...

12
votes

### A note on orders in quaternion algebras

Two orders need not be isomorphic.
First of all, in number fields $K$ other than $\mathbf Q$ not all orders are isomorphic rings (even if they are isomorphic abelian groups): the full ring of integers ...

11
votes

Accepted

### An interesting sum over lattice points in a large disk centered at the origin

It is problem number 10 of IMC 2018, you may find the solution on the official site.

11
votes

### Which even lattices have a theta series with this property?

I don't have a full answer to your question, and have not tried to work out details, but I can offer some thoughts, at least in even dimension:
First, to satisfy that property, the theta series needs ...

11
votes

Accepted

### Higman's lemma and a manuscript of Erdős and Rado

I didn't have much time when I wrote my initial answer, so here's an update.
It occurred to me that I ought to recommend Kruskal's classic paper "The theory of well-quasi-ordering: a frequently ...

11
votes

### Why do bees create hexagonal cells ? (Mathematical reasons)

Isn't it just the 2d sphere packing? If one assumes that the larvae needs a disc of fixed radius to grow up to an adult form and that the bees want to have as many cells as possible then the hexagonal ...

11
votes

Accepted

### Outer automorphisms of finitely generated linear groups

Yes. Furthermore, every recursively presented countable group embeds in such a group. Indeed, first Higman's embedding reduces to proving that every finitely presented group embeds into such a group (...

10
votes

Accepted

### Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths?

I was able to find the result at Ex. 6.39 s in Stanley's EC Vol.2. The reference given there is to
Ciucu, M. "Perfect Matchings of Cellular Graphs" Journal of Algebraic Combinatorics 5 (1996), 87-...

10
votes

Accepted

### The smallest volume possible for a lattice with integer distances?

After scaling your lattice by $\sqrt{2}$, the Gram matrix has integral entries, so the absolute value of its determinant, being a nonzero integer, is at least $1$. This gives a lower bound of $2^{-n/2}...

10
votes

Accepted

### Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard?

The NP-hardness of the shortest vector problem in $L_2$ norm is discussed in this 2015 lecture by Vinod Vaikuntanathan. An algorithm for this problem would give a randomised algorithm for any problem ...

10
votes

### Kissing number lower bound vs. upper bound - precise meanings?

Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...

9
votes

Accepted

### Is this obfuscation scheme unbreakable?

"Is this obfuscation scheme unbreakable?"
"Well.. no." said people a couple of years later.
On GGHRSW13 specifically: Cryptanalyses of Candidate Branching Program Obfuscators
See also (concurrent, ...

9
votes

### Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?

Let $k$ be a global field, and $S$ a non-empty finite set of places of $k$ containing the archimedean places. It is certain that you meant to assume $G$ and $H$ are smooth and affine (hence the ...

9
votes

### in search of a transformation between determinants

This doesn't answer the original question but answers the later SNF
question for the matrix $B_n$. Let $C_n$ be the $n\times n$ matrix
whose $(i,j)$-entry ($1\leq i,j\leq n$) is $\binom{x+1}{2j-i}$. ...

9
votes

Accepted

### in search of a transformation between determinants

There is such a transformation, of the form predicted in Linear transformation that preserves the determinant.
Denoting $R$ the involution matrix $e_i\mapsto e_{n+1-i}$, it turns out that the matrix ...

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