Letteratura scientifica selezionata sul tema "Heron tetrahedra"

We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the base is a Heronian triangle. Examples of rectangular tetrahedra are given, in which all sides are integer numbers, but the area of the base is irrational. The example of the rectangular tetrahedron is also given, which has lengths of one side irrational and the other integer, but the area of the base is integer.

Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have.
Masters Thesis

Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have.
Masters Thesis

The catalytic reduction of the alkene 1 gave the cis-fused product (not illustrated), by kinetic H₂ addition to the less congested face of the alkene. Ryan A. Shenvi of Scripps La Jolla found (J. Am. Chem. Soc. 2014, 136, 1300) conditions for stepwise HAT, con­verting 1 to the thermodynamically-favored trans-fused ketone 2. Seth B. Herzon of Yale University devised (J. Am. Chem. Soc. 2014, 136, 6884) a protocol for the reduc­tion, mediated by 4, of the double bond of a haloalkene 3 to give the saturated halide 5. The Shenvi conditions also reduced a haloalkene to the saturated halide. Daniel J. Weix of the University of Rochester and Patrick L. Holland, also of Yale University, established (J. Am. Chem. Soc. 2014, 136, 945) conditions for the kinetic isomerization of a terminal alkene 6 to the Z internal alkene 7. Christoforos G. Kokotos of the University of Athens showed (J. Org. Chem. 2014, 79, 4270) that the ketone 9, used catalytically, markedly accelerated the Payne epoxidation of 8 to 10. Note that Helena M. C. Ferraz of the Universidade of São Paulo reported (Tetrahedron Lett. 2000, 41, 5021) several years ago that alkene epoxidation was also easily carried out with DMDO generated in situ from acetone and oxone. Theodore A. Betley of Harvard University prepared (Chem. Sci. 2014, 5, 1526) the allylic amine 12 by reacting the alkene 11 with 1-azidoadamantane in the presence of an iron catalyst. Rodney A. Fernandes of the Indian Institute of Technology Bombay developed (J. Org. Chem. 2014, 79, 5787) efficient conditions for the Wacker oxida­tion of a terminal alkene 6 to the methyl ketone 13. Yong-Qiang Wang of Northwest University oxidized (Org. Lett. 2014, 16, 1610) the alkene 6 to the enone 14. Peili Teo of the National University of Singapore devised (Chem. Commun. 2014, 50, 2608) conditions for the Markovnikov hydration of the alkene 6 to the alcohol 15. Internal alkenes were inert under these conditions, but Yoshikazo Kitano of the Tokyo University of Agriculture and Technology effected (Synthesis 2014, 46, 1455) the Markovnikov amination (not illustrated) of more highly substituted alkenes.