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Schoch Circles
In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch. History In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to ''Scientific Americans "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to Professor Clayton Dodge of the University of Maine in 1996. The two were planning to write an article about the Arbelos, in which the Schoch circles would be included; however, Bankoff died the year after. In 1998, Peter Y. Woo of Biola University Biola University () is a private, nondenominational, evangelical Christian university in La Mirada, California. It was founded in 1908 as the Bible Institute of Los Angeles. It has over 150 programs of study in nine schools offering bachelor's ... published Schoch's findings on his website. By generalizing two of Schoch's circles, Woo discovered an infinite family of Archimedean circles name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Circle
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and ''r'' denotes the radiius of any of the inner half circles, then the radius ''ρ'' of such an Archimedean circle is given by :\rho=\fracr\left(1-r\right), There are over fifty different known ways to construct Archimedean circles. Origin An Archimedean circle was first constructed by Archimedes in his ''Book of Lemmas''. In his book, he constructed what is now known as Archimedes' twin circles. Radius If a and b are the radii of the small semicircles of the arbelos, the radius of an Archimedean circle is equal to :R = \frac This radius is thus \frac 1R = \frac 1a + \frac 1b. The Archimedean circle with center C (as in the figure at right) is tangent to the tangents from the centers of the small semicircles to the other small semicir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which in turn is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large format newspaper was released August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, and the universal joint which now can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was also a leading authority on Lewis Carroll. ''The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematicsand by extension, mathematics in generalthroughout the latter half of the 20th century, principally through his "Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Bankoff
Leon Bankoff (December 13, 1908 – February 16, 1997), born in New York City, New York, was an American dentist. As an amateur mathematician he constructed the Bankoff circle. He was also an Esperantist. Life After a visit to the City College of New York, Bankoff studied dentistry at New York University. Later, he moved to Los Angeles, California, where he taught at the University of Southern California; while there, he completed his studies. He practiced over 60 years as a dentist in Beverly Hills. Many of his patients were celebrities. Along with Bankoff's interest in dentistry were the piano and the guitar. He was fluent in Esperanto, created artistic sculptures, and was interested in the progressive development of computer technology. Above all, he was a specialist in the mathematical world and highly respected as an expert in the field of flat geometry. Since the 1940s, he lectured and published many articles as a co-author. Bankoff collaborated with Paul Erdős in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Maine
The University of Maine (UMaine or UMO) is a Public university, public Land-grant university, land-grant research university in Orono, Maine. It was established in 1865 as the land-grant college of Maine and is the Flagship universities, flagship university of the University of Maine System. It is Carnegie Classification of Institutions of Higher Education, classified by Carnegie as among "R1: Doctoral Universities – Very high research activity". With an enrollment of approximately 11,500 students, UMaine is the state's largest college or university. The University of Maine's athletic teams, nicknamed the Maine Black Bears, Black Bears, are Maine's only NCAA Division I, Division I athletics program. Maine Black Bears men's ice hockey, Maine's men's ice hockey team has won two national championships. History The University of Maine was founded in 1862 as a function of the Morrill Land-Grant Colleges Act, Morrill Act, signed by President Abraham Lincoln. Established in 1865 as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biola University
Biola University () is a private, nondenominational, evangelical Christian university in La Mirada, California. It was founded in 1908 as the Bible Institute of Los Angeles. It has over 150 programs of study in nine schools offering bachelor's, master's, and doctoral degrees. The university hosts the annual Missions Conference, the largest annual missions conference and the second largest missions conference in the world. It has also played a significant role in the development of intelligent design. History Biola University was founded in 1908 as the Bible Institute of Los Angeles by Lyman Stewart, President of the Union Oil Company of California (subsequently known as Unocal and later purchased by the Chevron Corporation); Thomas C. Horton, a Presbyterian minister and Christian author; and Augustus B. Prichard, also a Presbyterian minister.William Jeynes and David W. Robinson (2012), ''International Handbook of Protestant Education'', Springer, p. 127./ref> In 1912, the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woo Circles
In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point ''C''. Let ''m'' denote any nonnegative real number. Draw two circles, with radii ''m'' times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point ''C'' and with radius ''m'' times the radius of the corresponding small arbelos arc. Any circle centered on the Schoch line and externally tangent to the circles is a Woo circle. See also *Schoch circles In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch. History In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to ''Scientific Americans "Mathematical Games" editor Ma ... References Arbelos Circles {{Elementary-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoch Line
In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles. Construction An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) ''A'', ''B'', and ''C''. Let ''K''1 and ''K''2 be two more arcs, centered at ''A'' and ''C'', respectively, with radii ''AB'' and ''CB'', so that these two arcs are tangent at ''B''; let ''K''3 be the largest of the three arcs of the arbelos. A circle, with the center ''A''1, is then created tangent to the arcs ''K''1, ''K''2, and ''K''3. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle; it is one of the Schoch circles. The Schoch line is perpendicular In elementary geometry, two geometric objects are perpendicular if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that contains their diameters. The earliest known reference to this figure is in Archimedes's ''Book of Lemmas'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. Properties Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is convex, with diameter Area The area of the arbelos is equal to the area of a circle with diameter . Proof: For the proof, reflect the arbelos over the line through the points and , and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters , ) are subtracted from the area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
