- Contact Details
- Interdisciplinary Laboratory for Computational Science
- Department of Physics
- University of Trento
- via Sommarive 14
- 38123 Trento, Italy
- tel +39 461 314175
- ruggero.gabbrielli unitn.it
- Ph.D. in Mechanical Engineering, University of Bath, UK
- Laurea in Mechanical Engineering, University of Florence, Italy
- Research interests
- Cell aggregates and soap films (require Java). The geometry of monodisperse foams. Ground state for cellular material and aggregates of equisized soft particles.
The Kelvin problem
(video 1, video 2, video 3, comics), the three-dimensional shape of bubbles in monodisperse foam, disorder in foams,
n-dimensional honeycombs, periodic tilings from point sets, low-dimensional non-lattice sphere packings and coverings, spatial distribution of points. Applications: foam modelling, cell aggregates, metal crystallites, grain structure and boundaries.
- Triply periodic surfaces (require Java). Trigonometric approximations by implicit functions of triply periodic minimal surfaces and their use in materials science. Optimization of low volume fraction porous materials (stress levelling). Modelling of structures using triply periodic surfaces. Stress-leveling analysis of porous materials.
Application: bone substitutes, architecture.
Modelling and three-dimensional printing of functionally graded materials.
- Pattern formation applied to optimization problems in mathematics (sphere coverings and quantizers), physics (structure
of condensed matter and metallurgy), chemistry (crystallography), engineering. Swift-Hohenberg equation, Brusselator reaction-diffusion system.
- Auxetics (require Java). Bifurcation and geometric instabilities in solid mechanics. The internal geometry of
negative Poisson's ratio structures.
- R. Gabbrielli, Y. Jiao and S. Torquato. Dense periodic packings of tori Phys. Rev. E 89, 022133 (2014).
- R. Gabbrielli, Y. Jiao and S. Torquato. Families of tessellations of space by elementary polyhedra via retessellations of face-centered-cubic and related tilings Phys. Rev. E 86, 041141 (2012).
- R. Gabbrielli, S. Hutzler, D. Weaire, A. Meagher, K. Brakke. An experimental realization of the Weaire-Phelan structure in monodisperse liquid foam
Phil. Mag. Lett. 92:1 (2012), pp. 1-6.
- F.P.W. Melchels, K. Bertoldi, R. Gabbrielli, A.H. Velders, J. Feijen and D.W. Grijpma. Mathematically defined tissue engineering scaffold architectures prepared by stereolithography
Biomaterials 31:27 (2010), pp. 6909-6916.
- R. Gabbrielli. A new counter-example to Kelvin's conjecture on minimal surfaces
Phil. Mag. Lett. 89:8 (2009), pp. 483-491.
- R. Gabbrielli and M. O'Keeffe. A new simple tiling, with unusual properties, by a polyhedron with 14 faces
Acta Cryst. A64 (2008), pp. 430-431.
- R. Gabbrielli, I.G. Turner and C.R. Bowen. Development of modelling methods for materials to be used as bone substitutes Key Eng. Mat. 361-363 (2008), pp. 903-906.
- 3D Gallery (require Java)
- 3D tilings by octahedra and tetrahedra: There seem to be only 4 side ratios that allow mixtures of identical octahedra and identical tetrahedra to fill space.
- Sodalite and related structures: A net by vertex-sharing tetrahedra and three of its configurations.
- P42 and P42a: Two periodic partitions of space that use polyhedra with 13 and 14 faces. P42 is quite possibly the simplest example of a hexagonal monodisperse micellar phase other than hcp (see lyotropic liquid crystals). P42a has less surface area than the partition by truncated octahedra. 3D also available as open and closed-cell models.
- rug: A simple tiling by a polyhedron with 14 faces. Better than Williams' polyhedron at approximating the actual distribution of polygons found in uniform mixtures of soap bubbles.
- P8: A polyhedron with 13 faces that fills space together with its mirror image in a racemic mixture. Halfway between the rhombic dodecahedron and the truncated octahedron. This is the Voronoi cell of alpha nitrogen (space group Pa3). Note that the tiling is made of congruent polyhedra. Also note that due to surface area minimization constraint and the symmetry of the tiling, this geometrical embedding contains flat quadrilaterals and hexagons, but also non-planar pentagons, thus making the polyhedron non-convex. A convex version is available upon request.
- Q12: A space-filling polyhedron with 12 faces. There are many known dodecahedral space-fillers. I haven't been able to find this in the literature, though.
- Comparison between Lord Kelvin's truncated octahedron, the Weaire-Phelan structure and the P42a partition.
- Classification of periodic models for foams based on symmetry.
- Affine nodal gyroid periodic surfaces: Obtained from an approximation by implicit functions of Schoen's G minimal surface. Related tiling.
- Weaire-Phelan foam: The foam associated with the partition of space into regions of equal volume having the lowest known surface area.
- All the pictures have been created with 3dt (Gavrog) + Sunflow renderings: P42a, rug, P8
- A few more pictures on P8 in a plain html page and in flash with more info
- The movies below show how pattern-forming equations (in this case the Swift-Hohenberg equation) are able to find low energy lattices and nonlattices.
- Here two isosurfaces of the BCC lattice are shown. The lattice points are at the centre of the green surfaces.
The white surface is 5% off the absolute minima.
- Same with A15, a nonlattice. Also here.
- Presentations and Posters
- The Kelvin Problem, Bioceramics 20, Periodic Triangulations from Pattern Forming Equations
- 2011-2014 € 149,380 Marie Curie Fellowship - FP7 COFUND; Construction sets provided by GEOMAG
- Javaview is a 3D geometry viewer and a mathematical visualization software. The 3d objects in this website use this format unless differently specified.
- The Surface Evolver is an interactive program for the modelling of liquid surfaces shaped by various forces and constraints.
- The Gavrog Project is a package for the analysis of periodic nets and tilings. It contains 3dt and Systre.
- K3DSurf is a program to visualize and manipulate mathematical models in three, four, five and six dimensions.
It supports parametric equations and isosurfaces.
- vcs computes Voronoi diagrams in three dimensions also with periodic boundary conditions.
- Voro++ is a software library for carrying out three-dimensional computations of the Voronoi tessellation.
Last updated on 18 April 2014