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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] |
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] |
Arbelos Proof2
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] |
Arbelos Sculpture Netherlands 1
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] |
Arbelos Diagram With Points Marked
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] |
Parbelos
The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases. The two smaller parabola segments are placed next to each other with their bases on a common line and the largest parabola is placed over the two smaller ones such that its width is the sum of the widths of the smaller ones (see graphic). The parbelos has a number of properties which are somewhat similar or even identical to the some of the properties of the Arbelos. For instance the following two properties are identical to those of the arbelos:Michał Różański, Alicja Samulewicz, Marcin Szweda, Roman Wituła: "Variations on the arbelos". In: ''Journal of Applied Mathematics and Computational Mechanics'', Volume 16, Issue 2, 2017, pp. 123-133online copy *The arc length of the outer parabola is equal to the sum of the arc lengths of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes' Circles
In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of , , and , perpendicular to line , then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment. These circles first appeared in the ''Book of Lemmas'', which showed (Proposition V) that the two circles are congruent. Thābit ibn Qurra, who translated this book into Arabic, attributed it to Greek mathematician Archimedes. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called Archimedes's circles. However, this attribution has been questioned by later scholarship. Construction Specifically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Chain
In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction The arbelos is defined by two circles, ''C''U and ''C''V, which are tangent at the point A and where ''C''U is enclosed by ''C''V. Let the radii of these two circles be denoted as ''r''U and ''r''V, respectively, and let their respective centers be the points U and V. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to ''C''U (the inner circle) and internally tangent to ''C''V (the outer circle). Let the radius, diameter and center point of the ''n''th circle of the Pappus chain be denoted as ''r''''n'', ''d''''n'' and P''n'', respectively. Properties Centers of the circles Ellipse All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the ''n''th circle of the Pappus chain to the two centers U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions (lemmas) on circles. History Translations The ''Book of Lemmas'' was first introduced in Arabic by Thābit ibn Qurra; he attributed the work to Archimedes. In 1661, the Arabic manuscript was translated into Latin by Abraham Ecchellensis and edited by Giovanni A. Borelli. The Latin version was published under the name ''Liber Assumptorum''. T. L. Heath translated Heiburg's Latin work into English in his ''The Works of Archimedes''. A more recently discovered manuscript copy of Thābit ibn Qurra's Arabic translation was translated into English by Emre Coşkun in 2018. Authorship The original authorship of the ''Book of Lemmas'' has been in question because in proposition four, the book refers to Archimedes in third person; however, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle. Uses A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straight Line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two Point (geometry), points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as Non-Euclidean geometry, non-Euclidean, Projective geometry, projective and affine geometry). In modern mathematic ... [...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] |
