Modeling of the Effective Permittivity of Open-Cell Ceramic Foams Inspired by Platonic Solids

Open-cell solid foams are rigid skeletons that are permeable to fluids, and they are used as direct heaters or thermal dissipaters in many industrial applications. Using susceptors, such as dielectric materials, for the skeleton and exposing them to microwaves is an efficient way of heating them. The heating performance depends on the permittivity of the skeleton. However, generating a rigorous description of the effective permittivity is challenging and requires an appropriate consideration of the complex skeletal foam morphology. In this study, we propose that Platonic solids act as building elements of the open-cell skeletal structures, which explains their effective permittivity. The new, simplistic geometrical relation thus derived is used along with electromagnetic wave propagation calculations of models that represent real foams to obtain a geometrical, parameter-free relation, which is based only on foam porosity and the material’s permittivity. The derived relation facilitates an efficient and reliable estimation of the effective permittivity of open-cell foams over a large range of porosity.

Groups, Platonic solids and Bell inequalities

The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.

Platonic entanglement

We present a construction of highly entangled states defined on the topology of a platonic solid using tensor networks based on ancillary Absolute Maximally Entangled (AME) states. We illustrate the idea using the example of a quantum state based on AME(5,2) over a dodecahedron. We analyze the entropy of such states on many different partitions, and observe that they come on integer numbers and are almost maximal. We also observe that all platonic solids accept the construction of AME states based on Reed-Solomon codes since their number of facets, vertices and edges are always a prime number plus one.

Path planning for the Platonic solids on prescribed grids by edge-rolling

The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—have found many applications in mathematics, science, and art. Path planning for the Platonic solids had been suggested, but not validated, except for solving the rolling-cube puzzles for a cubic dice. We developed a path-planning algorithm based on the breadth-first-search algorithm that generates a shortest path for each Platonic solid to reach a desired pose, including position and orientation, from an initial one on prescribed grids by edge-rolling. While it is straightforward to generate triangular and square grids, various methods exist for regular-pentagon tiling. We chose the Penrose tiling because it has five-fold symmetry. We discovered that a tetrahedron could achieve only one orientation for a particular position.

Where’s Number Four?: The Place of Structure in Plato’s Timaeus

Through a detailed reading of the Timaeus, I show that the concept of structure plays an implicitly central but untheorized role in Plato’s philosophy. Plato holds apart form and matter, while theorizing the participation of particulars in universal ideas. I argue that “structure” is the concept necessary to understand the doctrine of participation, and that it mediates between the ideal and the material, the formal and the physical, in Platonic theory. This argument is developed through an engagement with the theory of the so-called Platonic solids in the Timaeus: Plato’s account of the geometrical structure of the elements. The chapter concludes by positioning this theory of structure in relation to Badiou’s “Mark and Lack” and Derrida’s Introduction to Husserl’s Origin of Geometry.

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.

Design of Flat Vaults with Topological Interlocking Solids

This paper investigates the principles that regulate complex stereotomic constructions as a starting point for the design of a new two-dimensional floor structure based on the principles of TIM (Topological Interlocking Materials). These interlocking systems use an assembly of identical Platonic solids which, due to the mutual bearing between adjacent units and the presence of a global peripheral constraint, lock together to form pure geometric shapes. This type of structure offers several advantages such as a high energy dissipation capacity and tolerance towards localised failure, which has made it a popular research topic over the last 30 years. The current research project includes a case study of an assembly of interlocking cubes to create a “flat vault”. The resulting vault design features a striking appearance and its geometry may be manipulated to achieve different two-dimensional solutions, provided certain geometric conditions necessary for the stability of the system are followed.