Related papers: Golden lattices
A main goal in lattice theory is the construction of dense lattices. Most of the remarkable dense lattices in small dimensions have an additional symmetry, they are modular, i.e. similar to their dual lattice. Extremal lattices are densest…
It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic…
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and…
Let $\alpha=(1+\sqrt 5)/2$, the golden ratio, and $\beta=-1/\alpha=(1 - \sqrt 5)/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=(\alpha^n -\beta^n)/\sqrt 5$ and $L_n=\alpha^n + \beta^n$, for all non-negative…
The theta series of a lattice is a power series that characterizes the number of lattice vectors at certain norms. It is closely related to a critical quantity widely used in physical layer security and cryptography, known as the flatness…
On each nonorientable surface of odd genus $g \geq 5$, we give a mapping class whose dilatation on an invariant subsurface is the golden ratio.
It is known that the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$ has many applications in geometry. In this paper we consider some geometric properties of finite Blaschke products related to the golden ratio.
It is a well known result that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ and $x\in(0,\frac{1}{\beta-1})$ there exists uncountably many $(\epsilon_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that…
We present a multi-edge-length aperiodic tiling which exhibits 6--fold rotational symmetry. The edge lengths of the tiling are proportional to 1:$\tau$, where $\tau$ is the golden mean $\frac{1+\sqrt{5}}{2}$. We show how the tiling can be…
Mimicking the idea of the generalized Hamming weight of linear codes, we introduce a new lattice invariant, the generalized theta series. Applications range from identifying stable lattices to the lattice isomorphism problem. Moreover, we…
A golden-ratio-based rectangular tiling of the first quadrant of the Euclidean plane is constructed by drawing vertical and horizontal grid lines which are located at all even powers of $\phi$ along one axis, and at all odd powers of $\phi$…
For any natural number $\ell $ and any prime $p\equiv 1 \pmod{4}$ not dividing $\ell $ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\Q [\sqrt{-\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series…
There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the…
Fractals and quasiperiodic structures share self-similarity as a structural property. Motivated by the link between Fibonacci fractals and quasicrystals which are scaled by the golden mean ratio $\frac{1+\sqrt{5}}{2}$, we introduce and…
We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus…
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent…
Given an integral lattice $\Lambda$ of rank $n$ and a finite sequence $m_1 \leq m_2 \leq ... \leq m_k$ of natural numbers we construct a modular form $\Theta_{m_1,m_2,...,m_k,\Lambda}$ of level $N=N(\Lambda)$. The weight of this modular…
An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some…
It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could…
We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using Kneser neighbours to diagonalise spaces of…