Maximum and Minimum Stable Random Packings of Platonic Solids

Motivated by the relation between particle shape and packing, we measure the volume fraction $\phi$ occupied by the Platonic solids which are a class of polyhedron with congruent sides, vertices and dihedral angles. Tetrahedron, cube, octahedron, dodecahedron, and icosahedron shaped plastic dice were fluidized or mechanically vibrated to find stable random loose packing $\phi_{rlp} = 0.51, 0.54, 0.52, 0.51, 0.50$ and densest packing $\phi_{rcp} = 0.64, 0.67, 0.64, 0.63, 0.59$, respectively with standard deviation $\simeq \pm 0.01$. We find that $\phi$ obtained by all protocols peak at the cube, which is the only Platonic solid that can tessellate space, and then monotonically decrease with number of sides. This overall trend is similar but systematically lower than the maximum $\phi$ reported for frictionless Platonic solids, and below $\phi_{rlp}$ of spheres for the loose packings. Experiments with ceramic tetrahedron were also conducted, and higher friction was observed to lead to lower $\phi$

Desain Pembelajaran Volume Kubus Dan Balok Menggunakan Filling Dan Packing Di Kelas V

This study was aimed at producing a learning design that can help the students to understand the concept of cube and cuboid volume using the filling and packing method in the fifth grade of the primary school. The approach used was PMRI. The subjects were the students of the fifth grade of MI Ma\u27had Islamy Palembang, South Sumatera. The study used validation study research design. The results show that the learning design is able to help students in understanding the concept of cube and cuboid volume; the contents of cube and cuboid through the activity of filling, the beams have more volume than the cube through comparing, the concept of the cubes and cuboid volume, the volume of a cube through the activity of packing, a formula of cube volume, the volume of the cuboid through the activity of packing, cuboid volume formula, and the conclusing out of cubes and cuboid formulas

A Cubical Flat Torus Theorem and the Bounded Packing Property

We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.Comment: 14 pages, 2 figures, submitted May 2015 Minor corrections and swapped sections 2 and 3 Corrected an unfortunate typo in Theorem 2.1 - the hypothesis that the cube complex be finite dimensional has now been adde

Packing subgroups in relatively hyperbolic groups

We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case.Comment: 45 pages, 2 figures. To appear in Geom. Topol. v2: Updated to address concerns of the referee. Added theorem that an infinite, nonparabolic relatively quasiconvex subgroup H of a relatively hyperbolic group has finite index in its commensurator. Added several new geometric results to Section 7. Theorem 8.9 on packing relative to peripheral subgroups is ne

Heterogeneous packing and hydraulic stability of cube and cubipod armor units

This paper describes the heterogeneous packing (HEP) failure mode of breakwater armor. HEP reduces packing density in the armor layer near and above the mean water level and increases packing density below it. With HEP, armor units may move in the armor layer, although they are not actually extracted from it. Thus, when HEP occurs, armor-layer porosity is not constant, and measurements obtained with conventional methods may underestimate armor damage. In this paper, the Virtual Net method is proposed to calculate armor damage considering both armor-unit extraction and HEP. The Cubipod concrete armor unit is then described as a solution to the effects of HEP on conventional cubic block armor. The hydraulic stability of cube and Cubipod armor units was compared in two-dimensional laboratory experiments. Cube and Cubipod armor layers were tested in two wave flumes under nonbreaking and non-overtopping conditions. The hydraulic stability was higher for double-layer Cubipod armor than for single-layer Cubipod armor, which had a higher hydraulic stability tan conventional double-layer cube armor.The authors are grateful for the financial support of CDTI (CUBIPOD Project), SATO Construction Co. (OHL Group), and Puertos del Estado (Convenio de Diques). The authors also thank Debra Westall for revising the manuscript.Gómez-Martín, ME.; Medina, JR. (2014). Heterogeneous packing and hydraulic stability of cube and cubipod armor units. Journal of Waterway, Port, Coastal, and Ocean Engineering. 140(1):100-108. doi:10.1061/(ASCE)WW.1943-5460.0000223S100108140

Hyperball packings related to truncated cube and octahedron tilings in hyperbolic space

In this paper, we study congruent and noncongruent hyperball (hypersphere) packings to the truncated regular cube and octahedron tilings. These are derived from the Coxeter truncated orthoscheme tilings $\{4,3,p\}$ (6< p \in \mathbb{N}) and $\{3,4,p\}$ (4< p \in \mathbb{N}), respectively, by their Coxeter reflection groups in hyperbolic space $\mathbb{H}^{3}$. We determine the densest hyperball packing arrangement and its density with congruent and noncongruent hyperballs. &nbsp; We prove that the locally densest (noncongruent half) hyperball configuration belongs to the truncated cube with a density of approximately $0.86145$ if we allow 6< p \in \mathbb{R} for the dihedral angle $2\pi/p$. This local density is larger than the B\"or\"oczky--Florian density upper bound for balls and horoballs. But our locally optimal noncongruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$. We determine the extendable densest noncongruent hyperball packing arrangement to the truncated cube tiling $\{4,3,p=7\}$ with a density of approximately $0.84931$