Gustavo Rubiano

A generalization of the Fibonacci word fractal and the Fibonacci snowflake

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.

GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION

In this paper, we introduce the p-circle inversion which generalizes the classical inversion with respect to a circle (p = 2) and the taxicab inversion (p = 1). 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 p-circle inversion.

A geometrical construction of inverse points with respect to an ellipse

In this note, we introduce the inverse point respect to an ellipse, which generalizes the classical inversion in a circle. Specifically, we show a geometrical construction of inverse points with respect to an ellipse. For this, we extend the classical method to construct the inverse of a point respect to a circle.

A generalization of the spherical inversion

Abstract In the present article, we introduce a generalization of the spherical inversion. In particular, we define an inversion with respect to an ellipsoid, and prove several properties of this new transformation. The inversion in an ellipsoid is the generalization of the elliptic inversion to the three-dimensional space. We also study the inverse images of planes, spheres, ellipsoids and other curves. We use the software Mathematica to generate these graphics. Finally, we generalize the Pappus chain theorem to ellipsoids.

On the k-Fibonacci words

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.