New algebraic and geometric constructs arising from Fibonacci numbers

Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$ C n , it is possible to obtain an infinite (right) word $$W_n$$ W n on the binary alphabet $$\{0,1\}$$ { 0 , 1 } , which we will call the nth Carboncettus word. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $$n=5$$ n = 5 . The fifth Carboncettus word $$W_5$$ W 5 is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000 100 100 010 010 01. Finally, we also define a further word $$W_{\infty }$$ W ∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\infty }$$ W ∞ itself.

Coordination shells and coordination numbers of the vertex graph of the Ammann–Beenker tiling

The vertex graph of the Ammann–Beenker tiling is a well-known quasiperiodic graph with an eightfold rotational symmetry. The coordination sequence and coordination shells of this graph are studied. It is proved that there exists a limit growth form for the vertex graph of the Ammann–Beenker tiling. This growth form is an explicitly calculated regular octagon. Moreover, an asymptotic formula for the coordination numbers of the vertex graph of the Ammann–Beenker tiling is also proved.

Generalized Scheme Based on Octagon-Shaped Shell for Data Hiding in Steganographic Applications

The exploiting modification direction scheme is a well-known data hiding method because of its high payload and low distortion. Its most criticized drawback is that the secret data must be converted to a non-binary numeral system before the embedding procedure. To overcome this drawback, one study proposed a turtle shell-based scheme for data hiding, in which a reference matrix was constructed based on a hexagon-shaped shell to embed three secret bits into each pixel group of a pair of pixels of the cover image. In a subsequent work, the embedding capacity was increased by using an octagon-shaped shell instead of a hexagon-shaped shell in the reference matrix. The above scheme was extended by utilizing a regular octagon-shaped shell in the reference matrix to reduce the distortion of the worst case. In another approach, the payload of the octagon-shaped-shell-based data hiding scheme was maximized by searching for the appropriate width and height of the octagon. In this study, we generalize the data hiding scheme based on an octagon-shaped shell so that the above-mentioned four schemes can be regarded as specific cases of the proposed method.