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Heronian Tetrahedron
A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers. Examples An example known to Leonhard Euler is a Heronian birectangular tetrahedron, a tetrahedron with a path of three edges parallel to the three coordinate axes and with all faces being right triangles. The lengths of the edges on the path of axis-parallel edges are 153, 104, and 672, and the other three edge lengths are 185, 680, and 697, forming four right triangle faces described by the Pythagorean triples (153,104,185), (104,672,680), (153,680,697), and (185,672,697). Eight examples of Heronian tetrahedra were discovered in 1877 by Reinhold Hoppe. 117 is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
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 and the only one that has fewer than 5 faces. 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 anot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disphenoid
In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are isotetrahedron,. sphenoid,. bisphenoid, isosceles tetrahedron,. equifacial tetrahedron, almost regular tetrahedron, and tetramonohedron. All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths. Special cases and generalizations If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with Td tetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science News
''Science News (SN)'' is an American bi-weekly magazine devoted to articles about new scientific and technical developments, typically gleaned from recent scientific and technical journals. History ''Science News'' has been published since 1922 by Society for Science & the Public, a non-profit organization founded by E. W. Scripps in 1920. American chemist Edwin Slosson served as the publication's first editor. From 1922 to 1966, it was called ''Science News Letter''. The title was changed to ''Science News'' with the March 12, 1966 issue (vol. 89, no. 11). Tom Siegfried was the editor from 2007 to 2012. In 2012, Siegfried stepped down, and Eva Emerson became the Editor in Chief of the magazine. In 2017, Eva Emerson stepped down to become the editor of a new digital magazine, Annual Reviews. On February 1, 2018 Nancy Shute became the Editor in Chief of the magazine. In April 2008, the magazine changed from a weekly format to the current biweekly format, and the website w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching. Its publisher is the Mathematical Association. William John Greenstreet was its editor for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors * Edward Mann Langley: 1894-1896 * Francis Sowerby Macaulay: 1896-1897 * William John Greenstreet: 1897-1930 * Alan Broadbent: 1930-1955 * Reuben Goodstein: 1956-1962 * Edwin A. Maxwell Edwin Arthur Maxwell (12 January 1907 – 27 August 1987) was a Scottish mathematician, who worked at Cambridge University for most of his career. Although his contributions to original research were li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheels, Life And Other Mathematical Amusements
''Wheels, Life and Other Mathematical Amusements'' is a book by Martin Gardner published in 1983. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Contents ''Wheels, Life and Other Mathematical Amusements'' is a book of 22 mathematical games columns that were revised and extended after being previously published in ''Scientific American''. It is Gardner's 10th collection of columns, and includes material on Conway's Game of Life, supertasks, intransitive dice, braided polyhedra, combinatorial game theory, the Collatz conjecture, mathematical card tricks, and Diophantine equations such as Fermat's Last Theorem. Reception Dave Langford reviewed ''Wheels, Life and Other Mathematical Amusements'' for ''White Dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Number Theory
The ''Journal of Number Theory'' (''JNT'') is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State University). It is currently published monthly by Elsevier and the editor-in-chief is Dorian Goldfeld ( Columbia University). According to the '' Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.72. References External links * Number theory Mathematics journals Publications established in 1969 Elsevier academic journals Monthly journals English-language journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trirectangular Tetrahedron
In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one 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. Only the bifurcating graph of the B_3 Affine Coxeter group has a Trirectangular tetrahedron fundamental domain. Metric formulas If the legs have lengths ''a, b, c'', then the trirectangular tetrahedron has the volume :V=\frac. The altitude ''h'' satisfies :\frac=\frac+\frac+\frac. The area T_0 of the base is given by :T_0=\frac. De Gua's theorem If the area of the base is T_0 and the areas of the three other (right-angled) faces are T_1, T_2 and T_3, then :T_0^2=T_1^2+T_2^2+T_3^2. This is a generalization of the Pythagorean theorem to a tetrahedron. Integer sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Brick
In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick is one whose space diagonal is also an integer but such a brick has not yet been found. Definition The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations: :\begin a^2 + b^2 = d^2\\ a^2 + c^2 = e^2\\ b^2 + c^2 = f^2\end where are the edges and are the diagonals. Properties * If is a solution, then is also a solution for any . Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths , the triple constitutes an Euler brick as well.Wacław Sierpiński, ''Pythagorean Triangles'', Dover Publications, 2003 (orig. ed. 1962). * Exactly one edge and two face diagonals of a ''primitive'' E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Powers
In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. *Faulhaber's formula expresses 1^k + 2^k + 3^k + \cdots + n^k as a polynomial in ''n'', or alternatively in term of a Bernoulli polynomial. *Fermat's right triangle theorem states that there is no solution in positive integers for a^2=b^4+c^4 and a^4=b^4+c^2. *Fermat's Last Theorem states that x^k+y^k=z^k is impossible in positive integers with ''k''>2. *The equation of a superellipse is , x/a, ^k+, y/b, ^k=1. The squircle is the case k=4, a=b. *Euler's sum of powers conjecture (disproved) concerns situations in which the sum of ''n'' integers, each a ''k''th power of an integer, equals another ''k''th p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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