Uvedieme históriu a prehľad výsledkov o ukladaní kociek do kvádra s minimálnym objemom a pridáme aj hlavné myšlienky niektorých dôkazov. V závere sa veľmi stručne zmienime o iných ukladacích problémoch.

In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68,84,85,86$, which are denser than the...

The set $𝒰{𝒟}_r$ of point sets of ${ℝ}^n,n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\>0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $𝒰{𝒟}_{r,f}\subset 𝒰{𝒟}_r$ of the finite point sets is compatible with the restriction of this topology to $𝒰{𝒟}_{r,f}$. We show that its subsets of Delone sets of given constants in ${ℝ}^n,n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature,...

The behavior of special classes of isometric foldings of the Riemannian sphere $S^2$ under the action of angular conformal deformations is considered. It is shown that within these classes any isometric folding is continuously deformable into the standard spherical isometric folding $f_s$ defined by $f_s(x,y,z)=(x,y,|z\left|\right)$.

Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes called Q...

Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{\>0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.