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Tomahawk (geometry)
The tomahawk is a tool in geometry for angle trisection, the problem of splitting an angle into three equal parts. The boundaries of its shape include a semicircle and two line segments, arranged in a way that resembles a tomahawk, a Native American axe. The same tool has also been called the shoemaker's knife,. but that name is more commonly used in geometry to refer to a different shape, the arbelos (a curvilinear triangle bounded by three mutually tangent semicircles). Description The basic shape of a tomahawk consists of a semicircle (the "blade" of the tomahawk), with a line segment the length of the radius extending along the same line as the diameter of the semicircle (the tip of which is the "spike" of the tomahawk), and with another line segment of arbitrary length (the "handle" of the tomahawk) perpendicular to the diameter. In order to make it into a physical tool, its handle and spike may be thickened, as long as the line segment along the handle continues to be part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomahawk Filled
A tomahawk is a type of single-handed axe used by the many Native Americans in the United States, Indian peoples and nations of North America, traditionally resembles a hatchet with a straight shaft. Etymology The name comes from Powhatan language, Powhatan , derived from the Proto-Algonquian root 'to cut off by tool'. Algonquian languages, Algonquian cognates include Lenape language, Lenape , Malecite-Passamaquoddy language, Malecite-Passamaquoddy , and Abenaki language, Abenaki , all of which mean 'axe'. The term came into the English language in the 17th century as an Anglicisation#Anglicisation of loanwords, adaptation of the Powhatan (Virginian Eastern Algonquian languages, Algonquian) word. History The Algonquian people created the tomahawk. Before Europeans came to the continent, Native Americans would use stones, sharpened by a process of knapping and pecking, attached to wooden handles, secured with strips of Rawhide (material), rawhide. The tomahawk quickly spread ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisection
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'', a line that passes through the midpoint of a given segment, and the ''angle bisector'', a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the ''bisector''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly. *The perpendicular bisector of a line segment AB also has the property that each of its points X is equidistant from segment AB's endpoints: (D)\quad , XA, = , XB, . The proof follows from , MA, =, MB, and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Lucien Bergery
Claude Lucien Bergery (1787–1863) was a French economist and management theorist. He was a founder of scientific management. Life The son of an innkeeper, Bergery was born in Orléans. He was a student at the École Polytechnique which he entered in 1806, He became an artillery captain, serving in Spain, and was decorated by Napoleon I during the Hundred Days. Demobilised, he taught applied science at the École royale de l'artillerie in Metz from 1817, then transferred to teacher training in the same city. Bergery believed the study of applied geometry was improving. With Jean-Victor Poncelet, he created free courses for workers and artisans, in response to a call from Charles Dupin. Courses were given in 1826 by Bergery, Poncelet, Libre-Irmond Bardin and Jean-Louis Woisard, all past ''polytechniciens''. In the aftermath of the July Revolution of 1830 Bergery had a chance to move to Paris. He had supporters who wished to keep out Adolphe Blanqui, who became successor to Jean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neusis Construction
In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line element of given length () in between two given lines ( and ), in such a way that the line element, or its extension, passes through a given point . That is, one end of the line element has to lie on , the other end on , while the line element is "inclined" towards . Point is called the pole of the neusis, line the directrix, or guiding line, and line the catch line. Length is called the ''diastema'' (). A neusis construction might be performed by means of a marked ruler that is rotatable around the point (this may be done by putting a pin into the point and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel proved that the problems of # doubling the cube, and # trisecting the angle are impossible to solve if one uses only a compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: # a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e. that the sufficient conditions given by Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ... are also necessary) The solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compass And Straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealization (science philosophy), idealized ruler and a Compass (drawing tool), compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypotenuse
In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided into a pair of right triangles by cutting it along either diagonal; the diagonals are the hypotenuses of these triangles. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as a^2 + b^2 = c^2, where ''a'' is the length of one leg, ''b'' is the length of another leg, and ''c'' is the length of the hypotenuse. For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4. Since 25 is the square of the hypotenuse, the length of the hypotenuse is the square r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruent Triangles
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle is called the '' hypotenuse'' (side c in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: '' cathetus''). Side a may be identified as the side ''adjacent'' to angle B and ''opposite'' (or ''opposed to'') angle A, while side b is the side adjacent to angle A and opposite angle B. Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trisect An Angle
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomahawk2
A tomahawk is a type of single-handed axe used by the many Indian peoples and nations of North America, traditionally resembles a hatchet with a straight shaft. Etymology The name comes from Powhatan , derived from the Proto-Algonquian root 'to cut off by tool'. Algonquian cognates include Lenape , Malecite-Passamaquoddy , and Abenaki , all of which mean 'axe'. The term came into the English language in the 17th century as an adaptation of the Powhatan (Virginian Algonquian) word. History The Algonquian people created the tomahawk. Before Europeans came to the continent, Native Americans would use stones, sharpened by a process of knapping and pecking, attached to wooden handles, secured with strips of rawhide. The tomahawk quickly spread from the Algonquian culture to the tribes of the South and the Great Plains. Native Americans created a ''tomahawk’s poll'', the side opposite the blade, which consisted of a hammer, spike or pipe. These became known as pipe tomah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
