Browsing by Author "Hilden, Hugh Michael"

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    Impresión de diseños simétricos en la obra de Escher
    (2012-11) Hilden, Hugh Michael; Montesinos, José María; Tejada Jiménez, Débora María; Toro Villegas, Margarita María
    In his need to find beautiful designs, forms and colors in the decoration of walls and the tiling of floors, humankind produced symmetric patterns that are examples of the concept of tessellation. In its own way, in order to solve its problems, nature had found wonderful tessellations. A tessellation or a surface tilling is the process of covering completely a surface with one type of tile that is repeated over and over without gaps or overlaps. Although it seems that there are infinitely many ways to produce symmetric plane designs, there are basically only 17 possible ways to produce a design. We will show that the execution of these tessellations follows some simple and concise rules, that we have used to construct 17 artifacts that can be used in the impression of any symmetric plane design. These artifacts are practical examples of Bill Thurston´s concept of Orbifold. As an illustration of our presentation, we will exhibit these artifacts by using some pictures and some of Escher´s designs. We will show that it is possible to teach, in an easy way, the concepts of rotation, translation and reflection by using these artifacts.
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    Publication
    On the classification of 3-bridge links
    (2012) Hilden, Hugh Michael; Montesinos, José María; Tejada Jiménez, Débora María; Toro Villegas, Margarita María
    Using a new way to represent links, that we call a butterfly representation, we assign to each 3-bridge link diagram a sequence of six integers, collected as a triple (p/n, q/m, s/l), such that p ≥ q ≥ ≥ s ≥ 2, 0 < n ≤ p, 0 < m ≤ q and 0 < l ≤ s. For each 3-bridge link there exists an infinite number of 3-bridge diagrams, so we define an order in the set (p/n, q/m, s/l) and assign to each 3-bridge link L the minimum among all the triples that correspond to a 3-butterfly of L, and call it the butterfly presentation of L. This presentation extends, in a natural way, the well-known Schubert classification of 2-bridge links. We obtain necessary and sufficient conditions for a triple (p/n, q/m, s/l) to correspond to a 3-butterfly and so, to a 3-bridge link diagram. Given a triple (p/n, q/m, s/l) we give an algorithm to draw a canonical 3-bridge diagram of the associated link. We present formulas for a 3-butterfly of the mirror image of a link, for the connected sum of two rational knots and for some important families of 3-bridge links. We present the open question: When do the triples (p/n, q/m, s/l) and (p’/n’, q’/m’, s’/l’) represent the same 3-bridge link?