Search Results

You are looking at 1 - 10 of 344 items :

  • "Platonic solids" x
Clear All

5 Graph theory and platonic solids In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is con- structed by congruent (identical in shape and size) regular (all angles equal and all sides equal) polygonal faces with the same number of faces meeting at each vertex. There are only five solids thatmeet these criteria: the tetrahedronwhichhas four faces, the cubewhich has six faces, the octahedronwhich has eight faces, the dodecahedron which has 12 faces and the icosahedron which has 20 faces. The ancient Greek geometers extensively

How Old Are the Platonic Solids? David R. Lloyd Recently a belief has spread that the set of five Platonic solids has been known since prehistoric times, in the form of carved stone balls from Scotland, dating from the Neolithic period. A photograph of a group of these objects has even been claimed to show mathematical understand- ing of the regular solids a millennium or so before Plato. I argue that this is not so. The archaeological and statistical evidence do not sup- port this idea, and it has been shown that there are problems with the photograph. The

Chapter 8 Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes Mathematics is concerned only with the enumeration and comparison of relations. —Carl Friedrich Gauss1 Mathematicians do not study objects, but relations between objects. —Henri Poincaré2 “That’s great, but what’s it good for?” the skeptical student asks, sarcasm dripping from his voice. Beauty is a wonderful trait, but some say use- fulness is a more important measure of the worth of a theorem. What is Euler’s formula good for? That is a fair question to ask of any mathematical theorem. Euler

Chapter 8 Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes Mathematics is concerned only with the enumeration and comparison of relations. —Carl Friedrich Gauss1 Mathematicians do not study objects, but relations between objects. —Henri Poincaré2 “That’s great, but what’s it good for?” the skeptical student asks, sarcasm dripping from his voice. Beauty is a wonderful trait, but some say use- fulness is a more important measure of the worth of a theorem. What is Euler’s formula good for? That is a fair question to ask of any mathematical theorem. Euler

Plato’s mathematical skills made him equal to the task of calculating the correct number of combinations, as well as the possible reasons why he rejected the hypothesis of there being five worlds. Keywords: Plato; Timaeus; Platonic solids; Dodecahedron At Timaeus 55C–D, immediately after the account of the five ‘Platonic solids’ (pyramid, octahedron, icosahedron, cube and dodecahedron), there is a passage which reads as follows: ‘Now if anyone, taking all these things into account, should raise the pertinent question, whether the number of worlds should be called

the context of the Timaeus. As well as recalling the definitions and proper- ties of plane angles and solid angles in Euclid’s Elements, I offer an alternative interpretation, which in my opinion improves the comprehension of the pas- sage, and makes it consistent with both the immediate and wider context of the Timaeus. I suggest that the passage marks a transition from plane geometry to solid geometry within Plato’s account of the universe. Keywords: Plato’s Timaeus, Plane angles, Solid angles, Platonic solids Ernesto Paparazzo: Consiglio Nazionale delle Ricerche

. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g > 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combi- natorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g > 2. Key words. Platonic solids, regular polyhedra, regular maps, Riemann surfaces, polyhedral em- bedding

Zeitschrift für Kristallographie 210, 3-4 © by R. Oldenbourg Verlag, München 1995 3 On the shape of crystals S. Andersson*, M. Jacob and S. Lidin Chemical Center, Inorganic Chemistry 2, P.O. Box 124, S-22100 Lund, Sweden Received March 7, 1994; accepted March 26, 1994 Crystal shape / Polyhedra j Exponential scale / Elliptic geometry / Platonic solids Abstract. The so-called exponential scale analytical functions are used to describe the shape of crystals. The Platonic solids and also irregular polyhedra such as the pyritohedron have been calculated. The

details of mean field equations on a closed surface. In this paper, we will construct blow-up solutions to ( 1.1 ) with blow-up points forming regular configurations, i.e., the vertices of equilateral triangles on a great circle or inscribed platonic solids (tetrahedrons, cubes, octahedrons, icosahedrons and dodecahedrons). Moreover, these solutions possess the corresponding symmetries of the configuration. To make the construction easier to understand, we will consider ρ → 32 ⁢ π {\rho\rightarrow 32\pi} and focus on a configuration of tetrahedron. The solutions we

amounts a productive synthetic route has still to be found [ 27 ]. We focus in this paper on the improvement of the magnetic response of plasmonic nanoclusters that are accessible to the methods of nanochemistry. However, instead of improving the structural parameters of the plasmonic raspberry model based on a large number of metallic satellites randomly distributed around a dielectric core [ 21 ], we investigated an alternative model based on a determined number of metallic satellites precisely located at the center of the faces of a Platonic solid, i.e. a convex