Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat

Panoramic optical and near-infrared SETI instrument: optical and structural design concepts

We propose a novel instrument design to greatly expand the current optical and near-infrared SETI search parameter space by monitoring the entire observable sky during all observable time. This instrument is aimed to search for technosignatures by means of detecting nano- to micro-second light pulses that could have been emitted, for instance, for the purpose of interstellar communications or energy transfer. We present an instrument conceptual design based upon an assembly of 198 refracting 0.5-m telescopes tessellating two geodesic domes. This design produces a regular layout of hexagonal collecting apertures that optimizes the instrument footprint, aperture diameter, instrument sensitivity and total field-of-view coverage. We also present the optical performance of some Fresnel lenses envisaged to develop a dedicated panoramic SETI (PANOSETI) observatory that will dramatically increase sky-area searched (pi steradians per dome), wavelength range covered, number of stellar systems observed, interstellar space examined and duration of time monitored with respect to previous optical and near-infrared technosignature finders.Comment: 14 pages, 5 figures, 3 table

Exploiting lattice structures in shape grammar implementations

The ability to work with ambiguity and compute new designs based on both defined and emergent shapes are unique advantages of shape grammars. Realizing these benefits in design practice requires the implementation of general purpose shape grammar interpreters that support: (a) the detection of arbitrary subshapes in arbitrary shapes and (b) the application of shape rules that use these subshapes to create new shapes. The complexity of currently available interpreters results from their combination of shape computation (for subshape detection and the application of rules) with computational geometry (for the geometric operations need to generate new shapes). This paper proposes a shape grammar implementation method for three-dimensional circular arcs represented as rational quadratic Bézier curves based on lattice theory that reduces this complexity by separating steps in a shape computation process from the geometrical operations associated with specific grammars and shapes. The method is demonstrated through application to two well-known shape grammars: Stiny's triangles grammar and Jowers and Earl's trefoil grammar. A prototype computer implementation of an interpreter kernel has been built and its application to both grammars is presented. The use of Bézier curves in three dimensions opens the possibility to extend shape grammar implementations to cover the wider range of applications that are needed before practical implementations for use in real life product design and development processes become feasible