1,722,420 research outputs found
Pappus configurations in finite Hall affine planes
No full textLazebnik, FelixIn the classical projective planes, both Desargues's theorem and Pappus's theorem hold. According to a result of Ostrom, the Desargues configuration can also be found in every finite projective plane on at least twenty-one points, classical or not. In fact, Ostrom's argument shows that the number of Desargues configurations in every finite plane is actually quite large. The result is also true in the finite projective plane on thirteen points. The existence of Pappus configurations in every non-classical finite affine or projective plane is unknown. We study whether the Pappus configuration is present in such planes. ☐ In particular, we endeavor to prove that in finite Hall affine planes, the following strong version for the existence of Pappus configurations holds: For every pair of lines ℓ1, ℓ2 and every triple of points on ℓ1 and every choice of a single point on ℓ2, a pair of points on ℓ2 can be found to complete a Pappus configuration. This statement is not proven in every case. When it is not, weaker versions for existence are shown. Hall planes are not Pappian, yet this work implies that the number of Pappus configurations in Hall planes is actually quite large.University of Delaware, Department of Mathematical SciencesPh.D
Pappus of Alexandria, Book VIII of the Mathematical Collection
Full text linkJohn B. Little is the translator.
This is a Greek-to-English translation of one section or chapter of a larger ancient Greek work called the Mathematical Collectionby Pappus of Alexandria (ca. 290 - 350 CE). Specifically, this is “Book VIII”\u27 of that work. To date and to the translator’s knowledge, no complete English translation of Book VIII has been published.
Pappus was very influential as a bridge between the knowledge that had been preserved from ancient mathematics and European mathematicians in the Renaissance. This specific part of Pappus\u27 work deals with applications of geometry to questions in mechanics.
The format includes an introductory essay, followed by the translated text, with numerous footnotes giving context, annotations, etc. The translator used of a number of figures from a scanned version of the classic scholarly edition of the Greek text of Pappus (and a Latin translation) edited by Friedrich Hultsch and published in the 1870\u27s. These come from gallica.bnf.fr, the digital library of the Bibliothèque Nationale de France and its partners.https://crossworks.holycross.edu/hc_books/1061/thumbnail.jp
Pappus of Alexandria, Book III of the Mathematical Collection
Full text linkJohn B. Little is the translator.
This is a translation of Book III of the Mathematical Collection by Pappus of Alexandria (ca. 290 - 350 CE) from the original Greek to English, following the edition of Friedrich Hultsch. While other books of the Mathematical Collection have been translated into English and short quotations from Book III have appeared in a number of places (see the Introduction), to my knowledge, no complete English translation of Book III has been published. Pappus was very influential as a sort of conduit between knowledge preserved from ancient Greek mathematics and European mathematicians in the Renaissance. This is evident here in the way Pappus discusses several solutions for the problem of the duplication of the cube in the first section of this book. The rest of Book III deals with a number of other problems in plane and solid geometry. The opening section is addressed to a certain Pandrosion, who was apparently a fellow teacher of mathematics in Alexandria, and the earliest female mathematician of whom any record has survived.https://crossworks.holycross.edu/hc_books/1062/thumbnail.jp
Pappus of Alexandria and the mathematics of late antiquity
No full textBook synopsis: This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. This is one of very few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus' text as part of a wider context and relates it to other contemporary cultural practices and opens new avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity
Book V of the Mathematical Collection of Pappus of Alexandria, translated by John B. Little
Full text linkJohn B. Little is the translator.
Book V of the Mathematical Collection is addressed to a certain Megethion, about whom we know nothing else. From the context he may have been a student or patron of Pappus in Alexandria. In a heading at the start, Pappus says that the general theme will be comparisons between different geometric figures. The overall structure brings interesting relations and connections to the fore. The book opens with a very well-known and charming discussion of how the importance of such comparisons can be seen by considering the structures built by non-human creatures such as bees. This might seem surprising since the bees do not seem to have the power to reason about geometry in the ways that humans do. But Pappus starts by identifying tilings of the plane as plans for the cross-section of a structure like a honeycomb. He points out that only equilateral triangles, squares, and regular hexagons can form regular tilings of a plane and those are the only conceivable possibilities for the tidy and systematic bees. Among these options, the bees have settled on the hexagon for their honeycombs because a hexagon encloses a greater area than a triangle or a square of the same perimeter. Since the perimeter of the cell gives a measure of the quantity of material needed to construct the cell, while the area is related to the storage capacity, the hexagonal shape is the most economical one for storing honey. The bees possess a certain kind of ``geometric foresight\u27\u27 aimed at providing for the material needs of their lives. On the other hand, humans can reason and are convinced by logical demonstrations and this leads to a greater form of wisdom. Hence Pappus proceeds to give proofs that the rectilinear plane figure of maximal area with a given number of sides and a given perimeter must be equilateral and equiangular. In addition, among regular polygons with the same perimeter, polygons with more sides always contain more area. Moreover, the circle with the same perimeter or circumference is larger in area than all the regular polygons. These theorems were apparently first established by Zenodorus (ca. 200 - 140 BCE (?)) although Pappus does not mention him by name. After considering a somewhat similar result for regions bounded by arcs of circles, Pappus shifts the discussion to the realm of three-dimensional figures and results he explicitly attributes to Archimedes (ca. 287 - 212 BCE). The main theme is comparisons between figures such as polyhedra, cones, and cylinders on the one hand, and the sphere on the other hand. Pappus introduces the five regular (Platonic) solids, then provides a quite detailed discussion of the thirteen ``semi-regular\u27\u27 or Archimedean solids. No surviving work of Archimedes describes these solid figures, though, so Pappus\u27s account is interesting from the historical point of view. Pappus then shows that the sphere is greater than any regular polyhedron of the same surface area. The final sections of Book V contain an exposition of famous results from Book I of Archimedes\u27 On the sphere and cylinder, and a proof that the regular polyhedra have a property parallel to what was seen earlier for plane polygons. Namely, if the surface areas are equal, the polyhedron with more faces encloses a greater volume.https://crossworks.holycross.edu/hc_books/1064/thumbnail.jp
Book VI of the Mathematical Collection of Pappus of Alexandria, translated by John B. Little
Full text linkBook VI of the Mathematical Collection is a collection of comments or notes about various points treated in parts of a collection of other texts sometimes known as the “Little Astronomy.” Pappus uses these works as sources and frequently quotes from them in this book. In the teaching of mathematical astronomy in late antiquity, and this would include the time of Pappus, the “Little Astronomy” is often understood to have been a follow-up to Euclid’s Elements and a preliminary to the study of the Almagest of Claudius Ptolemy (ca. 100–165 CE). The “Little Astronomy” included works by a group of Classical and Hellenistic authors including Autolycus of Pitane (ca. 360–290 BCE), Euclid of Alexandria (ca. 300 BCE), Aristarchus of Samos (ca. 310–230 BCE), Hypsicles (ca. 190–120 BCE), and Theodosius of Bithynia (ca. 160–100 BCE). Some of these were clearly what we would call elementary textbooks and Pappus’s Book VI seems to have been written as a sort of “guide for the perplexed” addressing subtle points that Pappus thought had not been treated sufficiently clearly or completely in those sources. In the introductory paragraph at the start, in fact, he says somewhat polemically, “many of those teaching astronomy, when they understand statements in a more careless way, include some things as necessary, while omitting others as unnecessary.”https://crossworks.holycross.edu/hc_books/1065/thumbnail.jp
Mathematicae collectiones
No full textAlexandrinus Pappus ; Federico Commandino conversae et commentarijs illustratae ...Aus dem Griech. übers
Rerum Germanicarum ab anno M.DC.XVII. ad annum M.DC.XLVIII. gestarum epitome
No full text[Leonhard Pappus oder Thomas Carve oder Johan Adler Salvius]Verfasser gemäss VD17 Leonhard Pappus oder Thomas Carve oder Johan Adler Salviu
Papiers de J. Hermann EISENMANN contenant les matériaux d'une édition, restée manuscrite, de Pappus d'Alexandrie, collectionum mathematicarum libri octo. XIXe siècle. Supplément grec 999
No full textPAPPUS D'ALEXANDRIE. collectionum mathematicarum libri octoNumérisation effectuée à partir d'un document de substitution.1(ff. 1-2v) « Pappi Alexandrini lemmata in tertium librum conicorum Apollonii Pergaei » (pp. 153-216 de l'édition d'Apollonios de Perga par Edm. Halley, Oxford 1710, avec notes manuscrites au crayon dans les marges). 2(ff. 3-49v) « Apollonii Pergaei conicorum liber tertius cum commentariis Eutocii Ascalonitae » (pp. 217-250 de la même édition, avec annotations manuscrites au crayon dans les marges). 3(ff. 50-205) extraits et notes divers
Pappus Chain
No full textNoneThe Pappus chain extends across at least two millennia of mathematics. Its origins trace back to the ancient Greek mathematician Archimedes and his studies of circles inscribed within the figure of an arbelos (or shoemaker's knife). The inversive geometry trick for efficiently computing the positions of pairwise tangent inscribed circles, or a Pappus chain, is apparently a modern invention. The Apollonian gasket or curvilinear Sierpinski sieve is constructed by the same iterative process of inscribing a circle in triplets of tangent circles. Thus the Pappus chain construction anticipates the class-2 nested behavior of many elementary cellular automataComponente Curricular::Ensino Fundamental::Séries Finais::Matemátic
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