Tetrahedron | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahe

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Related people

Related units

Related concepts

Related courses

Related lectures

Summary

Official source

About this result

Related publications (29)

Related publications

Related people

Related units

Related concepts (96)

Related concepts

Related courses (25)

Related courses

Related lectures (43)

Related lectures

Properties of Ir4 clusters in the gas phase and on oxide substrates

Vladan Stevanovic

Results of a theoretical study on the properties of Ir4 clusters in the gas–phase and on oxide surfaces are presented. The work is based on density functional theory (DFT) within the generalized gradient approximation (GGA) and ultrasoft pseudopotentials. Properties of a small particle such as Ir4 cluster are entirely determined by its geometry. The already known result that the most stable form of Ir4 in the gas–phase is the square structure which is significantly more stable than the butterfly and tetrahedron is confirmed. This result is in contradiction with experiments which indicate that the oxide supported Ir4 adopts a tetrahedral configuration. It is shown in this thesis that the chemical environment has a strong influence on the relative stability of Ir4 clusters. On MgO(100) surface, the square isomer remains the most stable Ir4 structure, well separated in energy from the other two. Moreover, the tetrahedron is heavily distorted by the interaction with the surface oxygen. Presence of point defects (neutral and charged O vacancies) affects the energy ordering making tetrahedron and square very close in energy, but the structural distortion of the tetrahedron is even bigger and the predicted data do not correspond to experiments. On TiO2(110) the tetrahedron and square structures become degenerate and the butterfly becomes the least stable isomer. Moreover, structural distortions are very small, in agreement with experimental data. It is shown that the TiO2 surface influences the relative stability of the three isomers through a particularly strong electrostatic field. Interactions of Ir4 with H, C and O atoms as well as with CO molecules have been studied. Adsorption of a single C atom strongly influences the relative stability of the three isomers. Upon C adsorption, the butterfly becomes the most stable gas–phase isomer while on both surfaces the tetrahedron is the most probable structure. Adsorption of a single H or O atom does not produce the same effect. The interaction with CO molecules is also important given the experimental procedure used for producing supported Ir4 clusters. It is shown that on MgO(100), CO dissociation is as probable as the competing process CO desorption justifying the presence of carbon adatoms on Ir4 clusters which brings theoretical predictions in better agreement with experimental data.

EPFL2009

Described are the synthesis and characterization of two, potentially tetradentate, N2S2 Schiff-base ligands, containing a disulfide bond, N,N'-bis(3-phenylprop-2-en-l-ylidene)-2,2'-disulfanediyldianiline (L-1) and N,N'-bis(3,3-diphenylprop-2-en-1-ylidene)-2,2'-disulfanediyldianiline (L-2), and their reaction with Zn2+. Surprisingly, both L-1 and L-2 undergo reductive disulfide bond scission upon reaction with Zn2+ in alcoholic media to give, under alcohol oxidation, the respective Zn(NS)2 complexes Zn(L-3)(2) (1) and Zn(L-4)(2) (2), where the L-3 and L-4 are the respective bidentate thiolate-imine anions. The ligands L-1 and L-2 and the complexes I and 2 have been characterized spectroscopically, and the crystal and molecular structures of the two complexes have been determined by single crystal X-ray diffraction. The coordination geometry around Zn(II) centers in both complexes is a distorted tetrahedron. In addition, DFT calculations (B3LYP/LANL2DZ/6-311++G(d,p)) support the structure of I. Cyclic voltammetric studies demonstrate that Zn(II) shifts the reduction potential of the disulfide ligands L-1 and L-2 to less negative values thus making them more susceptible to reductive cleavage of the disulfide bond. The results of semi-empirical PM6 calculations offer key insight into the nature of the transition state for this reaction. (C) 2014 Elsevier Ltd. All rights reserved.

Elsevier2014

Structural elucidation and antibacterial activity of new dialkylstannyl- and chlorodialkylstannyl(IV) 3,4-dihydroisoquinoline-2-(1H)-carbodithioate

In extension with growing concern to develop new antimicrobial organometallic drugs, an attempt has been made to synthesize five new antimicrobial organotin(IV) carbodithioates with general formula, Me2SnClL (1), Me2SnL2 (2), Et2SnClL (3), Et2SnL2 (4), Bu2SnClL (5), Bu2SnL2 (6), (where L = Sodium 3,4-dihydroisoquinoline-2-(1H)-carbodithioate). Compounds have been synthesized by refluxing organotin(IV) chlorides with ligand in dry toluene for 7-8 h. Compound 1, 3 and 5 exhibited Penta coordination, while 2, 4 and 6 showed hexacoordination in solid state, as evident from the difference obtained between symmetric and asymmetric CSS stretches in IR spectra. Solid-state structures of 1 was further attested by single crystal analysis as distorted trigonal bipyramidal geometry. H-1-, C-13- and Sn-119- NMR spectroscopy was used to assess the geometry of the compounds in solution state. It was observed that 1, 3, 4 and 5 completely dissociated in solution and showed tetrahedral geometry, whereas 2 and 6 exhibit pentacoordinate geometry after partial dissociation in solution. The pronounced activity was observed for all the compounds against five different strains of bacteria. (C) 2021 Elsevier B.V. All rights reserved.

ELSEVIER2021

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually d

Polyhedron

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron th

Platonic solid

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regula

CS-457: Geometric computing

This course will cover mathematical concepts and efficient numerical methods for geometric computing. We will develop and implement algorithms to simulate and optimize 2D and 3D geometric models with an emphasis towards computational design for digital fabrication.

CH-110: Advanced general chemistry I

Le cours comporte deux parties. Les bases de la thermodynamique des équilibres et de la cinétique des réactions sont introduites dans l'une d'elles. Les premières notions de chimie quantique sur les électrons et les liaisons chimiques, exemplifiées en chimie organique, sont présentées dans l'autre.

BIO-369: Randomness and information in biological data

Biology is becoming more and more a data science, as illustrated by the explosion of available genome sequences. This course aims to show how we can make sense of such data and harness it in order to understand biological processes in a quantitative way.