Branched Surfaces and Colored Patterns

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Session Title:

  • Computing and Aesthetics

Presentation Title:

  • Branched Surfaces and Colored Patterns

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Abstract:

  • Tiling problems have appeared in many branches of mathematics and physics, and during the last few decades there has been much progress in understanding their nature.  In the visual and sound arts, they have potential interest as a system of reference in constrictive preforming for channeling the expressive energies.

    Research on aperiodic tilings has been very intensive in connection with the field of mathematical quasicrystals and recently it has been suggested that aperiodic order already was present in the medieval islamic architecture. A cell complex is defined in the analysis of the cohomology of tiling spaces. It contains a copy of every kind of tile that is allowed, with some edges identified for the 2D case, and the result is a branched surface.  When the tiling does not force the border, collared tiles can be used. In this paper we discuss the use of cohomology for the generation of colored aperiodic tessellations which represent branched manifolds. The prototiles with the same shape, color and orientation appearing in the resulting patterns, represent the same tile in the complex. In the time domain substitution tilings and their appearance in the field of astronomy have been on the basis of several works where time harmonizations and sound synthesis play a central role.

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