for the academic journal Tetrahedron journal Reg polyhedra db Reg polyhedron stat table T In geometry , a tetrahedron plural tetrahedra is a polyhedron composed of four triangular Face geometry faces , three .... The tetrahedron is the only convex polytope convex polyhedron that has four faces. ref name MW MathWorld urlname Tetrahedron title Tetrahedron ref The tetrahedron is the three dimensional case of the more general concept of a Euclidean geometry Euclidean simplex . The tetrahedron is one kind of pyramid ... 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 nets. ref name MW For any tetrahedron there exists a sphere the circumsphere such that the tetrahedron s vertices lie on the sphere s surface. Special cases A regular tetrahedron is one in which all four faces are equilateral triangle s, and is one of the Platonic solid s. An isosceles tetrahedron , also called a disphenoid , is a tetrahedron where all four faces are Congruence geometry congruent triangles. In a trirectangular tetrahedron the three face angles at one vertex are right angle s. If all three pairs of opposite edges of a tetrahedron are perpendicular , then it is called an orthocentric tetrahedron ... tetrahedron . An isodynamic tetrahedron is one in which the cevian s that join the vertices to the Incircle ... , and an isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron. Formulas for a regular tetrahedron Tetrahedral angle redirects here For a regular tetrahedron of edge length a class wikitable Base ... Between 2 Legs of a Tetrahedron Maze5.net ref also known as the tetrahedral angle align center math ... 2 1 proof . Volume The volume of a tetrahedron is given by the pyramid volume formula math V frac 1 ... more details
Infobox Journal title Tetrahedron cover Image Tetrahedron cover.gif 150 px discipline Organic chemistry abbreviation Tetrahedron website http www.elsevier.com wps find journaldescription.cws home 942 description?navopenmenu 1 publisher Elsevier country United Kingdom history 1957 to present ISSN 0040 4020 Tetrahedron is a scientific journal publishing full original research papers in the field of organic chemistry . The impact factor of this journal is 2.817 2007 . ref Journal Citation Reports, 2007 ref It has published a number of highly cited papers, seven of which having 1000 citation each according to Web of Science , 2008 Related conference http www.tetrahedron symposium.elsevier.com 13th Tetrahedron Symposiusm Challenges in Bioorganic & Medicinal Chemistry References references See also Tetrahedron Asymmetry Tetrahedron Letters External links http www.sciencedirect.com science journal 00404020 Tetrahedron online via ScienceDirect http www.elsevier.com wps find journaldescription.cws home 942 description?navopenmenu 1 Tetrahedron homepage Reed Elsevier chem journal stub Category Chemistry journals Category Elsevier academic journals de Tetrahedron fr Tetrahedron journal nl Tetrahedron tijdschrift pl Tetrahedron sk Tetrahedron asopis zh ... more details
A Heronian tetrahedron is a tetrahedron whose side lengths, face areas and volume are all rational number s. The faces must therefore all be Heronian triangle s. A regular tetrahedron with rational sides is not a Heronian tetrahedron because its face areas and volume are not rational numbers. A Heronian tetrahedron is sometimes called a perfect tetrahedron . 117 is the smallest possible length of the longest side of a perfect tetrahedron. Its other sidelengths are 51, 52, 53, 80 and 84. See also Euler brick External links mathworld HeronianTetrahedron http web.archive.org web 20091027105752 http geocities.com teufel pi papers perfectpyramids.pdf Perfect Pyramids Polyhedron stub Category Polyhedra Category Arithmetic problems of solid geometry fr T tra dre de H ron ... more details
In geometry , an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular . It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Antoine Jean L Huilier Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890. ref name Court Court, N. A., Notes on the orthocentric tetrahedron , American Mathematical Monthly , Oct 1934, pp. 499 502. ref In an orthocentric tetrahedron the four altitudes are Concurrent lines concurrent . This common point is called the orthocenter , and it has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid . ref name Court Hence the orthocenter coincides with the Tetrahedron Properties of a general tetrahedron Monge point of the tetrahedron. Characterizations All tetrahedron can be inscribed in a parallelepiped . A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron . From this it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair thus it is an orthocentric tetrahedron. ref name Court A tetrahedron ABCD is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges ref Reiman, Istv n, International Mathematical Olympiad 1976 1990 , Anthem Press, 2005, pp. 175 176. ref ref name Hazewinkel math displaystyle ... this condition for the tetrahedron to be orthocentric. Another necessary and sufficient condition for a tetrahedron to be orthocentric is that its three Tetrahedron Properties of a general tetrahedron ... regarding the edges imply that if only four of the six edges of an orthocentric tetrahedron are known ... the volume of an orthocentric tetrahedron can be expressed in terms of four edges a , b , c , d . The formula ... Trirectangular tetrahedron References reflist Category Polyhedra ... more details
Infobox journal title Tetrahedron Letters cover Image Tetrahedron Letters cover.gif 150 px editor discipline Organic chemistry peer reviewed language English language English formernames abbreviation Tetrahedron Lett. publisher Elsevier country United Kingdom frequency 52 year history 1959 present openaccess license impact 2.618 impact year 2010 website http www.elsevier.com wps find journaldescription.cws home 233 description description link1 http www.sciencedirect.com science journal 00404039 link1 name Archive link2 link2 name RSS atom JSTOR OCLC LCCN CODEN ISSN 0040 4039 eISSN boxwidth Tetrahedron Letters is a weekly international journal for rapid publication of full original research papers in the field of organic chemistry . Its 2010 impact factor was 2.618. Indexing Tetrahedron Letters is indexed in columns list 2 AGRICOLA BIOSIS Beilstein database CAB Abstracts Chemical Abstracts Chemical Engineering Biotechnology Abstracts Current Biotechnology Abstracts Current Contents Search Current Contents Life Sciences Current Contents Physics, Chemical, & Earth Sciences Derwent Drug File El Compendex Plus Excerpta Medica MEDLINE PASCAL database Pascal Research Alert Science Citation Index Scisearch Scopus See also Tetrahedron journal TetrahedronTetrahedron Asymmetry Reed Elsevier Category Chemistry journals Category Journal established in 1959 chem journal stub de Tetrahedron Letters es Tetrahedron Letters fr Tetrahedron Letters nl Tetrahedron Letters pl Tetrahedron Letters zh ... more details
File 2D simplex.svg thumb A trirectangular tetrahedron can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin, like BR x 0 BR y 0 BR z 0 BR and x a y b z c 1 In geometry , a trirectangular tetrahedron is a tetrahedron where all three face angles at one Vertex geometry vertex are right angles . That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base . The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron. Metric formulas If the legs have lengths a, b, c , then the trirectangular tetrahedron has the volume math V frac abc 6 . math The altitude h satisfiy ref Eves, Howard Whitley, Great moments in mathematics before 1650 , Mathematical Association of America , 1983, p. 41. ref math frac 1 h 2 frac 1 a 2 frac 1 b 2 frac 1 c 2 . math The area math T 0 math of the base is given by ref Gutierrez, Antonio, Right Triangle Formulas , http gogeometry.com pythagoras right triangle formulas facts.htm ref math T 0 frac abc h . math De Gua s theorem main De Gua s theorem If the area of the base is math T 0 math and the areas of the three other right angled faces are math T 1 math , math T 2 math and math T 3 math , then math T 0 2 T 1 2 T 2 2 T 3 2. math This is a generalization of the Pythagorean theorem to a tetrahedron. See also Disphenoid Goursat tetrahedron Orthocentric tetrahedron Schl fli orthoscheme References reflist External links MathWorld title Trirectangular tetrahedron urlname TrirectangularTetrahedron Category Polyhedra ... more details
Semireg dual polyhedra db Semireg dual polyhedron stat table dtT In geometry , a triakis tetrahedron is an Archimedean solid Archimedean dual solid, or a Catalan solid . Its dual is the truncated tetrahedron . It can be seen as a tetrahedron with triangular pyramid s added to each face that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name. If the triakis tetrahedron has shorter edge lengths 1, it has area math tfrac 5 3 scriptstyle sqrt 11 math and volume math tfrac 25 36 scriptstyle sqrt 2 math . Variations A triakis tetrahedron with equilateral triangle faces represents a polyhedral net net of the four dimensional regular polytope known as the 5 cell . Stellations File Stellation of triakis tetrahedron.png 160px This chiral figure is one of thirteen stellation s allowed by The Fifty Nine Icosahedra Miller s rules Miller s rules . See also Truncated triakis tetrahedron References The Geometrical Foundation of Natural Structure book Section 3 9 Citation last1 Wenninger first1 Magnus author1 link Magnus Wenninger title Dual Models publisher Cambridge University Press isbn 978 0 521 54325 5 id MathSciNet id 730208 year 1983 The thirteen semiregular convex polyhedra and their duals, Page 14, Triakistetrahedron The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman Strass, ISBN 978 1 56881 220 5 http www.akpeters.com product.asp?ProdCode 2205 Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron External links Mathworld2 urlname TriakisTetrahedron title Triakis tetrahedron urlname2 CatalanSolid title2 Catalan solid Polyhedron navigator Polyhedron stub Category Catalan solids ca Tetr edre triakis de Triakistetraeder es Triaquistetraedro eo Trilateropiramidigita kvaredro fr Triakit tra dre it Triacistetraedro ja no Triakistetraeder pl Czworo cian potr jny pt Tetraedro triakis ... more details
Image ReuleauxTetrahedron Animation.gif frame right Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed. Image Reuleaux tetrahedron intersection.png thumb Four spheres intersect to form a Reuleaux tetrahedron. The Reuleaux tetrahedron is the intersection of four sphere spheres of radius s centered at the Vertex geometry vertices of a regular tetrahedron with side length s . The sphere through each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces four vertices, and four curved faces, connected by six circular arc edges. This shape is defined and named by analogy to the Reuleaux triangle , a two dimensional curve of constant width . One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width , but it is not true the two midpoints of opposite edge arcs are separated by a larger distance, math sqrt3 sqrt2 2 s approx 1.0249s. math The volume of a Reuleaux tetrahedron is ref name Weisstein citation author Weisstein, Eric W authorlink Eric W. Weisstein title Reuleaux Tetrahedron publisher MathWorld A Wolfram Web Resource year 2008 url http mathworld.wolfram.com ReuleauxTetrahedron.html ref math frac s 3 12 3 sqrt2 49 pi 162 tan 1 sqrt2 approx 0.422s 3 math Meissner bodies linked from Meissner body , etc. Meissner and Schilling ref name Meissner citation last1 Meissner first1 Ernst last2 Schilling first2 Friedrich title Drei Gipsmodelle von Fl chen konstanter Breite journal Z. Math. Phys. volume 60 year 1912 pages 92 94 ref showed how to modify the Reuleaux tetrahedron to form a surface of constant width , by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced three that have a common vertex or three that form a triangle ... more details
The Logos Tetrahedron is a concert hall in Ghent, Belgium adjacent to the Logos Foundation recording studio and offices. It has seating for 150 people and is equipped with sound and light infrastructure. Since the concert hall is in the shape of a tetrahedron , it has no straight angles and, as a result, no standing waves can occur. Acoustic waves can never amplify one another in phase and are reflected by the walls under ever changing angles. The hall is exclusively used for contemporary and experimental music, such as electronic music and computer music and holds approximately 65 concerts a year, mostly organized by the Logos Foundation. The concert hall was built in 1990 by Godfried Willem Raes and the Logos workgroup using steel, concrete and plate metal. Construction took one year. The hall was opened in 1991 with a three day festival. External links http www.logosfoundation.org architecture tetrhall.html Logos Tetrahedron Concert Hall http www.logosfoundation.org tetrhall techspec.html Technical notes and infrastructure survey for musicians Category Buildings and structures in Ghent Category Buildings and structures completed in 1991 Category Event venues established in 1991 Category Buildings and structures completed in 1990 culture stub ... more details
Semireg polyhedra db Semireg polyhedron stat table tT In geometry , the truncated tetrahedron also known as perfect truncated tetrahedron or Archimedean truncated tetrahedron is an Archimedean solid . It has 4 regular hexagon al faces, 4 regular triangle geometry triangular faces, 12 vertices and 18 edges. Area and volume The area A and the volume V of a truncated tetrahedron of edge length a are math A 7 sqrt 3 a 2 approx 12.12435565a 2 math math V frac 23 12 sqrt 2 a 3 approx 2.710575995a 3. math Densest Packing The densest packing of the Archimedean Truncated Tetrahedron is believed to be 207 208, as reported by two independent groups using Monte Carlo methods ref name Damasceno cite web url http arxiv.org pdf 1109.1323v2.pdf title Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces date Dec 2011 ref ref name Jiao cite web url http arxiv.org pdf 1107.2300v4.pdf title A Packing of Truncated Tetrahedra that Nearly Fills All of Space date Sep 2011 ref . Although no mathematical proof exists that this is the best possible packing for those shapes, the high proximity to the unity and independency of the findings ... is slightly smaller than that of an Archimedean Truncated Tetrahedron, this new shape can be used ... vertices of a Truncation geometry truncated tetrahedron centered at the origin, with edge length ... tetrahedron, and combined they form a Compound polyhedron Uniform compounds uniform compound polyhedron ... Edge Face vertex align center Image File tetrahedron t01 ae.png 100px File tetrahedron t01 af36.png ... 1 2 3 4 Related polyhedra Tetrahedron family Use in architecture Giant truncated tetrahedra were ... Truncated tetrahedron urlname2 ArchimedeanSolid title2 Archimedean solid KlitzingPolytopes polyhedra.htm ...?net 3cqTmfu7gdEZ8I7kRUVvji6qxBATVSp2WpmIWGx7l7pWe7bveylFxv3piHnPNZN&name Truncated Tetrahedron applet Editable printable net of a truncated tetrahedron with interactive 3D view http www.mathconsult.ch ... more details
In geometry , the Reeve tetrahedron is a polyhedron , named after John Reeve mathematician John Reeve , in R sup 3 sup with vertices at 0, 0, 0 , 1, 0, 0 , 0, 1, 0 and 1, 1, r where r is a positive integer. Each vertex lies on a fundamental lattice point a point in Z sup 3 sup . No other fundamental lattice points lie on the surface or in the interior of the tetrahedron . In 1957 Reeve used this tetrahedron as a counterexample to show that there is no simple equivalent of Pick s theorem in R sup 3 sup or higher dimensional spaces. ref J. E. Reeve, On the Volume of Lattice Polyhedra , Proceedings of the London Mathematical Society , s3&ndash 7 1 378&ndash 395 ref This is seen by noticing that Reeve tetrahedra have the same number of interior and boundary points for any value of r , but different volumes. Notes and references reflist Ko odziejczyk, Krzysztof 1996 . An Odd Formula for the Volume of Three Dimensional Lattice Polyhedra , Geometriae Dedicata 61 271&ndash 278. Category Digital geometry Category Lattice points ... more details
In geometry , the Hill tetrahedra are a family of Space filling polyhedron space filling tetrahedron tetrahedra . They were discovered in 1896 by M.J.M. Hill, a professor of mathematics at the University College London , who showed that they are Hilbert s third problem scissor congruent to a cube . Construction For every math alpha in 0,2 pi 3 math , let math v 1,v 2,v 3 in Bbb R 3 math be three unit vectors with angle math alpha math between every two of them. Define the Hill tetrahedron math Q alpha math as follows math Q alpha , , c 1 v 1 c 2 v 2 c 3 v 3 mid 0 le c 1 le c 2 le c 3 le 1 . math A special case math Q Q pi 2 math is the tetrahedron having all sides right triangles with sides 1, math sqrt 2 math and math sqrt 3 math . Ludwig Schl fli studied math Q math as a special case of the Schl fli orthoscheme orthoscheme , and H.S.M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling. Properties A cube can be tiled with 6 copies of math Q math . Every math Q alpha math can be Dissection geometry dissected into three polytopes which can be reassembled into a prism geometry prism . Generalizations In 1951 Hugo Hadwiger found the following n dimensional generalization of Hill tetrahedra math Q w , , c 1 v 1 cdots c n v n mid 0 le c 1 le cdots le c n le 1 , math where vectors math v 1, ldots,v n math satisfy math v i,v j w math for all math 1 le i j le n math , and where math 1 n 1 w 1 math . Hadwiger showed that all such simplex simplices are scissor congruent to a hypercube . References M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits, Proc. London Math. Soc. , 27 1895 1896 , 39 53. Hugo Hadwiger H. Hadwiger , Hillsche Hypertetraeder, Gazeta Matem tica Lisboa , 12 No. 50, 1951 , 47 48. H.S.M. Coxeter , http matwbn.icm.edu.pl ksiazki aa aa18 aa18132.pdf Frieze patterns , Acta ... of a Hill tetrahedron into a triangular prism Category Polyhedra Category Space filling polyhedra ... more details
In geometry , tetrahedron packing is the problem of arranging identical regular tetrahedron tetrahedra throughout three dimensional space so as to fill the maximum possible fraction of space. File 120px Tetrahedron slowturn.gif thumb right A regular tetrahedron. Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63 . ref name chen2010 Cite arxiv first1 Elizabeth R. last1 Chen first2 Michael last2 Engel first3 Sharon C. last3 Glotzer title Dense crystalline dimer packings of regular tetrahedra date January 6, 2010 class arxiv 1001.0586 ref It has been known for hundreds of years that the tetrahedron does not tessellation tile space, but an upper bound below 100 namely math 1 2.6 ldots times 10 25 math has only recently been reported. ref cite journal author1 Simon Gravel author2 Veit Elser author3 Yoav Kallus doi 10.1007 s00454 010 9304 x title Upper bound on the packing density of regular tetrahedra and octahedra year 2010 journal Discrete and Computational Geometry arxiv 1008.2830 ref Historical results Aristotle claimed that tetrahedra could fill space completely. ref cite news title Do women have fewer teeth than men? url http education.theage.com.au cmspage.php?intid 147&intversion 80 newspaper The Age date 2011 03 14 author Burkard Polster and Marty Ross ref In 2006, John Horton Conway Conway and Torquato showed that a packing fraction about 72 can be obtained by constructing a non Bravais lattice packing of tetrahedra with multiple ... that the best tetrahedron packing cannot be a lattice packing with one particle per repeating unit ... and coworkers experimentally showed that tetrahedron like dice can randomly pack in a finite container .....876T ref Later these same authors obtained a denser random tetrahedron packing with a packing ... with an 82 tetrahedron unit cell, they obtained a packing density as high as 85.03 . ref cite ... of a Family of Dense Tetrahedron Packings and the Role of Symmetry eprint 0912.4210 class cond mat.stat ... more details
File Coxeter Dynkin 3 space groups.png 480px thumb For 3 space, there are 3 simple Goursat tetrahedra, represented by 4,3,4 , 4,3 sup 1,1 sup , and a square graph. In geometry , a Goursat tetrahedron is a tetrahedron tetrahedral fundamental domain of a Wythoff construction . Each tetrahedral face represents a reflection hyperplane on 3 dimensional surfaces the 3 sphere , the Euclidean 3 space, and hyperbolic 3 space. Coxeter named after Edouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangle s for Wythoff constructions on the sphere. Graphical representation A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face mirror of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being dihedral angle . File General Goursat tetrahedron.png 100px A 4 node Coxeter Dynkin diagram represents this tetrahedral graphs with order 2 edges hidden. If some edges are order 2, the Coxeter group can be used for a simpler notation. Existence requires each of the 3 node subgraphs of this graph, p q r , p u s , q t u , and r s t , must correspond to a Schwarz triangle . Solutions There are many hundreds of solutions with finite density polytope densities . 3 sphere finite solutions The solutions for the 3 sphere with density 1 solutions are Duoprism s p x q , CDD node p node 2 node q node Hyperprism s 3,3 x , CDD node 3 node 3 node 2 node 3,4 x , CDD node 3 node 4 node 2 node 3,5 x , CDD node 3 node 5 node 2 node Linear graphs Uniform polychoron Uniform polychora 3,3,3 , CDD node 3 node 3 node 3 node 3,3,4 , CDD node 3 node 3 node 4 node 3,4,3 , CDD node 3 node 4 node 3 node 3,3,5 , CDD node 3 node 3 node 5 node Tri dental graphs 3 sup 1,1,1 sup , CDD nodes split2 node 3 node There are hundreds of rational solutions for the 3 sphere , including these 6 l ... more details
Tetrahedron Provincial Park is a provincial park in British Columbia , Canada , located northeast of Sechelt, British Columbia Sechelt in the Sunshine Coast, British Columbia Sunshine Coast area of the province. The park was created in 1995. ref http www.wildernesscommittee.org campaigns historic sunshine Wilderness Committee Sunshine Coast ref coord 49 36 N 123 35 W display title name Tetrahedron Provincial Park External links BCGNIS 41350 Tetrahedron Park http www.env.gov.bc.ca bcparks explore parkpgs tetra.html British Columbia Ministry of the Environment Tetrahedron Provincial Park References reflist Pacific Ranges Category Provincial Parks of British Columbia Category Sunshine Coast Regional District Category Protected areas established in 1995 BritishColumbia park stub ... more details
Infobox Polyhedron with net Image File augmented truncated tetrahedron.png Polyhedron Type Johnson solid Johnson br augmented tridiminished icosahedron J sub 64 sub J sub 65 sub augmented truncated cube J sub 66 sub Face List 2 2x3 triangle s br 3 Square geometry square s br 3 hexagon s Edge Count 27 Vertex Count 15 Symmetry Group Point groups in three dimensions C sub 3v sub Vertex List 2x3 3.6 sup 2 sup br 3 3.4.3.4 br 6 3.4.3.6 Dual Property List Convex set convex Net Image File Johnson solid 65 net.png In geometry , the augmented truncated tetrahedron is one of the Johnson solid s J sub 65 sub . It is created by attaching a triangular cupola to one hexagon al face of an truncated tetrahedron . External links MathWorld urlname JohnsonSolid title Johnson Solid MathWorld urlname AugmentedTruncatedTetrahedron title Augmented truncated tetrahedron Polyhedron stub Category Johnson solids es Tetraedro truncado aumentado eo Pligrandigita senpintigita kvaredro fr T tra dre tronqu augment nl Verhoogde afgeknotte tetra der th ... more details
wikt sphenoid Sphenoid may refer to In anatomy, the sphenoid bone In geometry, a tetrahedron with mirror symmetry dab Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
form shown above. Continuing the flip move completely inverts the tetrahedron into its dual tetrahedron. The elements that were face elements now comprise the apices of the new tetrahedron ... orange violet magenta ROVM tetrahedron and a yellow blue light green dark green YBLD tetrahedron. Solutions ... to solve for the ROVM tetrahedron while leaving the YBLD tetrahedron shuffled and vice ... more details
In geometry, the Murakami Yano formula , introduced by harvtxt Murakami Yano 2005 , is a formula for the volume of a hyperbolic or spherical tetrahedron given in terms of its dihedral angles. References Citation last1 Murakami first1 Jun last2 Yano first2 Masakazu title On the volume of a hyperbolic and spherical tetrahedron url http intlpress.com CAG CAG v13.php v13n2 id MR 2154824 year 2005 journal Communications in Analysis and Geometry issn 1019 8385 volume 13 issue 2 pages 379 400 expand further date February 2012 Category Geometry ... more details
Styles The ACS style is a set of standards for writing documents relating to chemistry, including a standard method of citation in academic journal academic publication s, developed by the American Chemical Society ACS . The printed versions of the ACS style manual are entitled ACS Style Guide Effective Communication of Scientific Information , 3rd ed. 2006 , edited by Anne M. Coghill and Lorrin R. Garson, and ACS Style Guide A Manual for Authors and Editors 1997 . Citation format Abbreviations Titles of journals are abbreviated e.g. J. Am. Chem. Soc. Journal of the American Chemical Society J. Phys. Chem. Journal of Physical Chemistry J. Phys. Chem. A Journal of Physical Chemistry A, B, or C J. Org. Chem. Journal of Organic Chemistry Org Lett Organic Letters Phys. Rev. Lett. Physical Review Letters Tetrahedron tetrahedron journal Tetrahedron Tetrahedron Lett Tetrahedron Letters Acc. Chem. Res. Accounts of Chemical Research Article published in a journal Last Name, First Initial. Last Name, First Initial. Journal . Year , Volume , Pages. Example of a journal citation quotation Deno, N. C. Richey, H. G. Liu, J. S. Lincoln, D. N. Turner, J. O. J. Am. Chem. Soc. 1965 , 87 , 4533 4538. Further reading Anne M. Coghill Editor , Lorrin R. Garson Editor . The ACS Style Guide Effective Communication of Scientific Information. 3rd ed. American Chemical Society, 2006 . External links http chemistry.library.wisc.edu instruction acstyle.htm ACS Style Guidelines Category American Chemical Society Category Bibliography Category Style guides for American English Category Academic style guides Category Style guides for technical and scientific writing ... more details
An Oxo Diels Alder reaction is an organic reaction and a variation of the Diels Alder reaction in which a suitable diene reacts with an aldehyde to form a dihydropyran ring. This reaction is of some importance to synthetic organic chemistry. Image Oxo Diels Alder general.svg Oxo Diels Alder reaction general The oxo DA reaction was first reported in 1949 ref A Diels Alder Type Reaction with Formaldehyde Thomas L. Gresham, Thomas R. Steadman J. Am. Chem. Soc. , 1949 , 71 2 , pp 737 738 DOI 10.1021 ja01170a101 ref using a methylpentadiene and formaldehyde as reactants. Image Oxo Diels Alder reaction Gresham 1949.svg Oxo Diels Alder reaction Gresham 1949 Asymmetric synthesis Asymmetric oxo DA reactions including catalytic reactions are well known. ref Tetrahedron Report number 869 Asymmetric hetero Diels Alder reactions of carbonyl compounds Helene Pellissier tetrahedron journal Tetrahedron 65 2009 2839 2877 doi 10.1016 j.tet.2009.01.068 ref Many strategies rely on coordinating a chiral Lewis acid to the carbonyl group. See also Aza Diels Alder reaction References Reflist Category Cycloadditions nl Oxo Diels Alder reactie ... more details
The Prada Transformer is an unusual building which has one of four different apparent shapes, depending on the function for which the building is needed at the moment. The building is roughly in the shape of a tetrahedron . Crane machine Cranes rotate the building so that different surfaces of the tetrahedron face downward, thereby changing the building s form and function. The different faces of the tetrahedron are actually shapes other than triangles. The building s base is a hexagon when used for a fashion exhibition, a rectangle when used as a movie theater , a cross when used for an art exhibition and a circle when used for a special event. The building was funded by Prada , and designed by Rem Koolhaas architecture firm Office for Metropolitan Architecture . The building is located in Seoul , South Korea , next to the Gyeonghui Palace . The building was first used for the fashion exhibition Waist Down Skirts by Miuccia Prada , which began April 25, 2009. Its form and function was first changed on June 26, 2009, into a movie theater. External links http prada transformer.com Prada Transformer official web site coord missing South Korea Category Buildings and structures completed in 2009 Category Rem Koolhaas buildings Category Buildings and structures in Seoul SouthKorea struct stub ... more details
8. p. 15 ref It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names are isosceles tetrahedron and equifacial tetrahedron . All ... triangle s, one obtains a regular tetrahedron , which is not normally considered a disphenoid. Characterizations A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right angled. ref name Andreescu We also have that a tetrahedron is a disphenoid if and only if the Center ... perpendiculars of AB and CD AC and BD and AD and BC respectively in a tetrahedron ABCD , then the tetrahedron ... journal Mathematical Gazette pages 269 271 title Some properties of the isosceles tetrahedron volume ... 16T 2R 2 l 2m 2n 2 9V 2. math The square of the lengths of the Tetrahedron Properties of a generalized tetrahedron bimedians are ref name Leech math tfrac 1 2 l 2 m 2 n 2 , quad tfrac 1 2 l 2 m 2 n 2 , quad tfrac 1 2 l 2 m 2 n 2 . math Other properties If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid. ref name Brown If the four faces of a tetrahedron ... scalenohedron. See also Orthocentric tetrahedron Snub disphenoid A Johnson solid with 12 equilateral triangle faces and D sub 2d sub symmetry. Trirectangular tetrahedron References references ... more details
Encyclopedia
top
