However, the dodecahedron and the icosahedron can be realized with rational coordinates if we do not require them to be precisely regular: If you perturb the vertices of a regular icosahedron “just a bit” into rational position, then taking the convex hull will yield a rational polytope that is combinatorially equivalent to the regular icosahedron. Similarly, by perturbing the facet planes of a...

A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces that is related to such fundamental music theoretical constructs as the Circle of ...

The article is devoted to modeling and visualization of the formation flat-nosed (snub-nosed) dodecahedron (snub dodecahedron). purpose research model snub (flat-nosed dodecahedron) visualize process its formation. faces consists in truncation edges vertices Platonic with subsequent rotation new around their centers. values edges, angle length edge are parameters three equations composed as dis...

This paper presents an interactive modeling system of uniform polyhedra using the isomorphic graphs. Especially, Kepler-Poinsot solids are formed by modifications of dodecahedron and icosahedron. Keywords—Kepler-Poinsot solid, Shape modeling, Polyhedral graph, Graph drawing. I. INTRUDUCTION EPLER-POINSOT solids are subset of uniform polyhedra, which include 5 regular polyhedra (Platonic soilids...

In this paper we consider geodesic triangulations of the surface of the regular dodecahedron. We are especially interested in triangulations with angles not larger than π/2, with as few triangles as possible. The obvious triangulation obtained by taking the centres of all faces consists of 20 acute triangles. We show that there exists a geodesic triangulation with only 10 non-obtuse triangles, ...

This paper gives an elementary account of the ‘‘Penrose dodecahedron,’’ a set of 40 states of a spin3 2 particle used by Zimba and Penrose @Stud. Hist. Phil. Sci. 24, 697–720 ~1993!# to give a proof of Bell’s nonlocality theorem. The Penrose rays are constructed here from the rotation operator of a spin2 particle and the geometry of a dodecahedron, and their orthogonality properties are derived...

The Neurohedron is a multi-modal interface for a nonlinear sequencer software model, embodied physically in a dodecahedron. The faces of the dodecahedron are both inputs and outputs, allowing the device to visualize the activity of the software model as well as convey input to it. The software model maps MIDI notes to the faces of the device, and defines and controls the behavior of the sequenc...

Is every edge unfolding of every Platonic solid overlapfree? The answer is yes. In other words, if we develop a Platonic solid by cutting along its edges, we always obtain a flat nonoverlapping simple polygon. We also give self-overlapping general unfoldings of Platonic solids other than the tetrahedron (i.e., a cube, an octahedron, a dodecahedron, and an icosahedron), and edge unfoldings of so...

The antiferromagnetic Heisenberg model on the dodecahedron possesses a number of ground state magnetization discontinuities in a field at the classical and quantum level, even though it lacks magnetic anisotropy. Here the model is considered for two dodecahedra coupled antiferromagnetically along one of their faces, as a first step to determine the magnetic response of collections of fullerene ...