Inappropriate tone date February 2008 A porism is a mathematical proposition or corollary . In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem . History Beginnings The treatise which has given rise to this subject ... to constructing what is proposed, and finally a porism as directed to finding what is proposed ... was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic ..., as a result unsought, as it were, but seen to follow from a theorem. On the porism ... , ed. Friedlein, p. 301 . Pappus on Euclid s porism Pappus gives a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case. This porism, expressed in modern ... a straight line. Pappus gives also a complete enunciation of one porism of the first book of Euclid ... with definitions of theorem, problem, datum, porism and locus. Respecting the porism Simson says ... proponantur. blockquote A locus says Simson is a species of porism. Then follows a Latin ... between theorems and problems, and were called porisms. Playfair accordingly defined a porism ... problem indeterminate or capable of innumerable solutions. Though this definition of a porism appears ... of the nature of a porism more closely conforming to the definitions in Pappus. This was followed in the same ... of porism to one section. According to Frans van Schooten , if the various relations between straight ... seven Porisms. Again, Chasles seems to have been wrong in making the ten cases of the four line Porism begin the book, instead of the intercept Porism fully enunciated by Pappus, to which the lemma to the first Porism relates intelligibly, being a particular case of it. An interesting hypothesis as to the Porisms ... im Altertum , 1886, ch. viii. . Observing, e.g., that the intercept Porism is still true if the two ... porism as defined in Pappus and Proclus . See also Poncelet s porism Steiner s porism References ... more details
Image PonceletPorism.gif thumb right Illustration of Poncelet s porism for n 3, a triangle that is inscribed in one circle and circumscribes another. In geometry , Poncelet s porism sometimes referred to as Poncelet s closure theorem , named after French engineer and mathematician Jean Victor Poncelet , states the following Let C and D be two plane conic s. If it is possible to find, for a given n 2, one n sided polygon that is simultaneously inscribed in C and circumscribed around D , then it is possible to find infinitely many of them. Poncelet s porism can be proved via elliptic curve s geometrically this depends on the representation of an elliptic curve as the Double cover topology double cover of C with four ramification point s. Note that C is isomorphic to the complex projective line . The relevant ramification is over the four points of C where the conics intersect. There are four such points by B zout s theorem . One can also describe the elliptic curve as a double cover of D in this case, the ramification is over the contact points of the four bitangents. Proof Let p be a point of P sup 2 sup and a line of the dual projective plane. The key tool ... point does, as well. See also Steiner s porism Tangent lines to circles References Bos, H. J ..., 289 364. External links http sbseminar.wordpress.com 2007 07 16 poncelets porism David Speyer on Poncelet s Porism D. Fuchs, S. Tabachnikov, Mathematical Omnibus Thirty Lectures on Classic Mathematics ... PonceletsPorismEllipseParabolaOrder3.html Java applet by Michael Borcherds showing Poncelet s Porism ... Poncelet s Porism for 2 general ellipses order 3 made using http www.geogebra.org webstart GeoGebra ... showing Poncelet s Porism for 2 general ellipses order 5 made using http www.geogebra.org webstart ... Borcherds showing Poncelet s Porism for 2 general ellipses order 6 made using http www.geogebra.org ... PonceletsPorism.html Article on Poncelet s Porism at Mathworld. Category Conic sections Category ... more details
Image Pivot theorem.svg thumb right 300px Image Pivot theorem d.png thumb right 300px The three dimensional case the four spheres intercepts other spheres on black circles. In geometry , the pivot theorem states that, given any three points P, Q, and R on each respective side of a triangle ABC, the three circles through the points AQR, BPR and CPQ share a common point M. Conversely, this is equivalent to a porism given any three circles sharing a common point M, there are an infinite number of triangles such that one point lies on each circle and the sides of the triangle pass through the intersection points of the circles. There is also a three dimensional analog, in which the four spheres passing through a point of a tetrahedron and points on the edges of the tetrahedron intersect in a common point. See also Miquel s theorem Bibliography cite book author Wells D year 1991 title The Penguin Dictionary of Curious and Interesting Geometry publisher Penguin Books location New York isbn 0 14 011813 6 pages pp. 227&ndash 228 External links MathWorld title Pivot theorem urlname PivotTheorem Category Circles Category Theorems in plane geometry ... more details
About the Scottish mathematician other people with a similar name Robert Simpson disambiguation File Robert Simson.jpg thumb Robert Simson File Robert Simson memorial.jpg thumb Memorial to Robert Simson in West Kilbride cemetery. The memorial plate reads To Dr. Robert Simson of the University of Glasgow, the Restorer of Grecian Geometry and by his works, the great promoter of its study in the Schools. A Native of this Parish. Robert Simson 14 October 1687 1 October 1768 was a Scotland Scottish mathematics mathematician and Professor of Mathematics, Glasgow professor of mathematics at the University of Glasgow . The pedal line of a triangle is sometimes called the Simson line after him. ref name uni http www.universitystory.gla.ac.uk biography ?id WH0065&type P Robert Simson . University of Glasgow multi tab page ref Life The eldest son of John Simson of Kirktonhall, West Kilbride in Ayrshire , Robert Simson was intended for the Church, but the bent of his mind was towards mathematics. He was educated at the University of Glasgow and graduated M.A. When the prospect opened of his succeeding to the Professor of Mathematics, Glasgow mathematical chair at the University of Glasgow , Simson proceeded to London for further study. After a year in London, he returned to Glasgow and, in 1711, was appointed by the university to the professorship of mathematics, an office which he retained until 1761. Works Simson s contributions to mathematical knowledge took the form of critical editions and commentaries on the works of the ancient geometer s. The first of his published writings is a paper in the Philosophical Transactions 1723, vol. xl. p. 330 on Euclid s Porism s . Then followed Sectionum conicarum libri V. Edinburgh, 1735 , a second edition of which, with additions, appeared in 1750. The first three books of this treatise were translated into English and, several times, printed as The Elements of the Conic section Conic Sections . In 1749, was published Apollonii Per ... more details
wikify date November 2010 Henry Martyn Taylor , Fellow of the Royal Society F.R.S. , Fellow of the Royal Astronomical Society F.R.A.S. 6 June 1842, Bristol &ndash 16 October 1927, Cambridge , was an England English mathematician and barrister . ref cite journal last1 F. first1 A. R. title Mr. H. M. Taylor, F.R.S. journal Nature date 1927 11 05 volume 120 issue 3027 pages 664&ndash 665 doi 10.1038 120664a0 ref ref cite journal title sm Henry Martyn Taylor journal Monthly Notices of the Royal Astronomical Society year 1929 volume 89 pages 324&ndash 325 bibcode 1929MNRAS..89..324. ref ref cite journal author1 Addison, Henry Robert author2 Oakes, Charles Henry author3 Lawson, William John author4 Sladen, Douglas Brooke Wheelton title TAYLOR, Henry Martyn journal Who s Who, year 1907 volume 59 pages p. 1723 url http books.google.com books?id yEcuAAAAYAAJ&pg PA1723 ref Henry Martyn Taylor was the second son of the Rev. James Taylor and Eliza Johnson. He was educated in Wakefield and at Trinity College, Cambridge , where he graduated B.A. as 3rd Wrangler University of Cambridge Wrangler in 1865. ref Venn id TLR861HM name Taylor, Henry Martyn ref He was elected a Fellow of the Royal Society in Jun 1898. His candidacy citation read that he was Barrister at Law. Fellow of Trinity College, Cambridge. Ex Tutor of Trinity College, Cambridge. Third Wrangler and Second Smith s Prizeman in 1865. Author of papers in the Mathematical Messenger, as follows Vol iii, p 189, Geometrical Explanation of the Equations for the Longitude of the Node and the Inclination of the Orbit vol v, p 1, 1876, On the Generation of Developable Surface through Two given Curves vol vii, p 22, 1877, On Certain Series in Trigonometry vol vii, p 145, 1877, On the Porism of the Ring of Circles touching Two Circles vol xi, p 177, On a Six point Circle connected with a Triangle vol xiii, p 145, On a Cubic Surface vol xvi, p 39, On a Geometrical Interpretation of the Algebraic Expression which, equated to Zero, r ... more details
Pole and polar Talk Pole and polar Polygon circle graph Polygon circle graph Poncelet s porism Talk Poncelet s porism Power center geometry Talk Power center geometry Power of a point Talk Power of a point ... more details
who used Ptolemy s theorem extensively in his trigonometrical work refers to this result as a Porism ... for the pentagon chord. The remaining pair of chords were calculated using the Porism now referred ... by application of the Porism Pythagoras Thm . Then math overline CD frac overline EC cdot overline ... by application of the Porism . http articles.adsabs.harvard.edu cgi bin nph iarticle query?bibcode ... to Pythagoras Theorem by name but uses the term Porism a word which in this particular context would appear to denote an observation on or obvious consequence of another existing theorem. The Porism ... more details
Technical date June 2011 In mathematics , the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane . The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper ref name SCH1 cite journal title The Pentagram Map url http www.expmath.org expmath volumes 1 1.html author Schwartz, Richard Evan journal Experimental Mathematics journal Journal of Experimental Math year 1992 volume 1 pages 90 95 ref though it seems that the special case, in which the map is defined for pentagons only, goes back at least to a 1945 paper of Theodore Motzkin . ref name MOT cite journal doi 10.1090 S0002 9904 1945 08488 2 title The pentagon in the projective plane, with a comment on Napier s rule journal Bulletin of the American Mathematical Society volume 51 issue 12 year 1945 pages 985 989 author Th. Motzkin authorlink Theodore Motzkin ref The pentagram map is similar in spirit to the constructions underlying Desargues Theorem and Poncelet s porism . It echoes the rationale and construction underlying a conjecture of Branko Gr nbaum concerning the diagonals of a polygon. ref name ZAK cite journal title On the products of cross ratios on diagonals of polygons author Zaks, Joseph url http www.springerlink.com content p592345k82444x61 journal Geometriae Dedicata volume 60 number 2 pages 145 151 doi 10.1007 BF00160619 accessdate 2010 02 12 ref Definition of the map Basic construction Suppose that the vertex geometry vertices of the polygon P are given by math P 1,P 3,P 5, ldots math The image of P under the pentagram map is the polygon Q with vertices math Q 2,Q 4,Q 6, ldots math as shown in the figure. Here math Q 4 math is the intersection of the diagonals math P 1P 5 math and math P 3P 7 math , and so on. File penga3.svg border right 300px test On a basic level, ... more details
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