Spatial Geometric Cells — Quasipolyhedra

Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.

Polyhedrons the Faces of which are Special Quadric Patches

We seize an idea of Oswald Giering (see [1] and [2]), who replaced pairs of faces of a polyhedron by patches of hyperbolic paraboloids and link up edge-quadrilaterals of a polyhedron with the pencil of quadrics determined by that quadrilateral. Obviously only ruled quadrics can occur. There is a simple criterion for the existence of a ruled hyperboloid of revolution through an arbitrarily given quadrilateral. Especially, if a (not planar) quadrilateral allows one symmetry, there exist two such hyperboloids of revolution through it, and if the quadrilateral happens to be equilateral, the pencil of quadrics through it contains even three hyperboloids of revolution with pairwise orthogonal axes. To mention an example, for right double pyramids, as for example the octahedron, the axes of the hyperboloids of revolution are, on one hand, located in the plane of the regular guiding polygon, and on the other, they are parallel to the symmetry axis of the double pyramid. Not only for platonic solids, but for all polyhedrons, where one can define edge-quadrilaterals, their pairs of face-triangles can be replaced by quadric patches, and by this one could generate roofing of architectural relevance. Especially patches of hyperbolic paraboloids or, as we present here, patches of hyperboloids of revolution deliver versions of such roofing, which are also practically simple to realize.

Parametric methods for virtual reconstruction. The hyperbolic paraboloids designed by Félix Candela

<p>The paper explains the process used in a research project whose objective was the reconstruction of the hyperbolic-parabolic surfaces of a selection of non-built projects designed by Félix Candela.</p><p>The workflow was based on an automation process. After setting up the variables and define a procedural design, it has been possible to obtain the variation of the projects designed by Candela.</p><p>The virtual reconstruction was done using models parameterized with Grasshopper and experimentally with Houdini FX.</p><p>The results are valid not only for their scientific debate but also can have an educational role in the fields of geometry and architectural communication.</p>