In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. We associate with this family of words a family of curves that are like the Fibonacci word fractal and reveal some fractal features. Finally, we describe an infinite family of polyominoes stems from the generalized Fibonacci words and we study some of their geometric properties, such as perimeter and area. These last polyominoes generalize the Fibonacci snowflake and they are double squares polyominoes, i.e., tile the plane by translation in exactly two distinct ways.

In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.

Abstract. In this paper we define the k-Fibonacci words in analogy with the definition of the k-Fibonacci numbers. We study their properties and we associate to this family of words a family of curves with interesting patterns.

Abstract. In the present article, we study a word-combinatorial interpretation of the biperiodic Fibonacci sequence of integer numbers (F (a,b) n ). This sequence is defined by the recurrence relation aF (a,b) n−1 + F (a,b) n−2 if n is even and bFn−2 if n is odd, where a and b are any real numbers. This sequence of integers is associated with a family of finite binary words, called finite biperiodic Fibonacci words. We study several properties, such as the number of occurrences of 0 and 1, and the concatenation of these words, among others. We also study the infinite biperiodic Fibonacci word, which is the limiting sequence of finite biperiodic Fibonacci words. It turns out that this family of infinite words are Sturmian words of the slope [0, a, b]. Finally, we associate to this family of words a family of curves with interesting patterns.

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