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Langley’s Adventitious Angles
Langley’s Adventitious Angles is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by Edward Mann Langley in ''The Mathematical Gazette'' in 1922.. The problem In its original form the problem was as follows: ::ABC is an isosceles triangle with \angle=\angle=80^\circ. ::CF at 30^\circ to AC cuts AB in F. ::BE at 20^\circ to AB cuts AC in E. ::Prove \angle=30^\circ. . Solution The problem of calculating angle \angle is a standard application of Hansen's problem, Hansen's resection. Such calculations can establish that \angle is within any desired precision of 30^\circ, but being of only finite precision, always leave doubt about the exact value. A direct proof using classical geometry was developed by James Mercer (mathematician), James Mercer in 1923. This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adventitious Angles
Important structures in plant development are buds, shoots, roots, leaf, leaves, and flowers; plants produce these tissues and structures throughout their life from meristems located at the tips of organs, or between mature tissues. Thus, a living plant always has embryonic tissues. By contrast, an animal embryo will very early produce all of the body parts that it will ever have in its life. When the animal is born (or hatches from its egg), it has all its body parts and from that point will only grow larger and more mature. However, both plants and animals pass through a phylotypic stage that evolved independently and that causes a developmental constraint limiting morphological diversification. According to plant physiology, plant physiologist A. Carl Leopold, the properties of organization seen in a plant are emergence, emergent properties which are more than the sum of the individual parts. "The assembly of these tissues and functions into an integrated multicellular organism y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward Mann Langley
Edward Mann Langley (22 January 1851 – 9 June 1933) was a British mathematician, author of mathematical textbooks and founder of the Mathematical Gazette. He created the mathematical problem known as Langley’s Adventitious Angles. Biography Langley was born in Buckden, Cambridgeshire, Buckden on 22 January 1851. He was educated at Bedford Modern School, the University of London and Trinity College, Cambridge''Cambridge University Alumni, 1261-1900'' where he was eleventh Senior Wrangler (University of Cambridge), Wrangler (1878). After University of Cambridge, Cambridge, Langley taught mathematics at Bedford Modern School (1878-1918) where he wrote numerous mathematical text books and his pupils included the famous future mathematician Eric Temple Bell.''The Mathematical Gazette'', October 1933 Langley became Secretary of the Mathematical Association (1885-1893), founded the Mathematical Gazette (1894) and became its editor (1894–95). In addition to mathematics, EM Lang ... [...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: 1962-1971 * Douglas Quadling Douglas Arthur Quadling (1926–2015) was an English mathematician, school master and educationalist who was one of the four drivers behind the School Mathematics Project (SMP) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hansen's Problem
Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points ''A'' and ''B'', and two unknown points ''P''1 and ''P''2. From ''P''1 and ''P''2 an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of ''P''1 and ''P''2. See figure; the angles measured are (''α''1, ''β''1, ''α''2, ''β''2). Since it involves observations of angles made at unknown points, the problem is an example of Resection (orientation), resection (as opposed to intersection). Solution method overview Define the following angles: ''γ'' = ''P''1''AP''2, ''δ'' = ''P''1''BP''2, ''φ'' = ''P''2''AB'', ''ψ'' = ''P''1''BA''. As a first step we will solve for ''φ'' and ''ψ''. The sum of these two unknown ang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Mercer (mathematician)
James Mercer FRS (15 January 1883 – 21 February 1932) was a mathematician, born in Bootle, close to Liverpool, England. He was educated at University of Manchester, and then University of Cambridge. He became a Fellow, saw active service at the Battle of Jutland in World War I and, after decades of ill health, died in London. He proved Mercer's theorem, which states that positive-definite kernels can be expressed as a dot product in a high-dimensional space. This theorem is the basis of the kernel trick In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example ... ( applied by Aizerman), which allows linear algorithms to be easily converted into non-linear algorithms. References 1883 births 1932 deaths 19th-century British mathematicians 20th-century British mathematicians Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adventitious Quadrangles2
Important structures in plant development are buds, shoots, roots, leaves, and flowers; plants produce these tissues and structures throughout their life from meristems located at the tips of organs, or between mature tissues. Thus, a living plant always has embryonic tissues. By contrast, an animal embryo will very early produce all of the body parts that it will ever have in its life. When the animal is born (or hatches from its egg), it has all its body parts and from that point will only grow larger and more mature. However, both plants and animals pass through a phylotypic stage that evolved independently and that causes a developmental constraint limiting morphological diversification. According to plant physiologist A. Carl Leopold, the properties of organization seen in a plant are emergent properties which are more than the sum of the individual parts. "The assembly of these tissues and functions into an integrated multicellular organism yields not only the characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerrit Bol
Gerrit Bol (May 29, 1906 in Amsterdam – February 21, 1989 in Freiburg) was a Dutch mathematician who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points. Life Bol earned his PhD in 1928 at Leiden University under Willem van der Woude. In the 1930s, he worked at the University of Hamburg on the geometry of webs under Wilhelm Blaschke and later projective differential geometry. In 1931 he earned a habilitation. In 1933 Bol signed the '' Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State''. In 1942–1945 during World War II, Bol fought on the Dutch side, and was taken prisoner. On the authority of Blaschke, he was released. After the war, Bol became professor at the Albert-Ludwigs-University of Freiburg The University of Freiburg (colloquially german: Uni Freiburg), officially the Albert Ludwig University of Freiburg (german: Albert-Ludwigs-Universität Freiburg), is a public ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triacontagon
In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. Regular triacontagon The '' regular triacontagon'' is a constructible polygon, by an edge- bisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t. A truncated triacontagon, t, is a hexacontagon, . One interior angle in a regular triacontagon is 168 degrees, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the equilateral triangle (60°) and the regular pentagon (108°). The area of a regular triacontagon is (with ) :A = \frac t^2 \cot \frac = \frac t^2 \left(\sqrt + 3\sqrt + \sqrt\sqrt\right) The inradius of a regular triacontagon is :r = \frac t \cot \frac = \frac t \left(\sqrt + 3\sqrt + \sqrt\sqrt\right) The circumradius of a regular triac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
