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Neusis
In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematics, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Construction
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 idealized ruler and a 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 means: see Markable rulers below.) More formally, the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icositrigon
In geometry, an icositrigon (or icosikaitrigon) or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not Neusis construction, neusis constructible. Regular icositrigon A ''regular polygon, regular icositrigon'' is represented by Schläfli symbol . A regular icositrigon has internal angles of \frac degrees, with an area of A = \fraca^2 \cot \frac = 23r^2 \tan \frac \simeq 41.8344\,a^2, where a is side length and r is the inradius, or apothem. The regular icositrigon is not Constructible polygon, constructible with a Straightedge and compass construction, compass and straightedge or angle trisection, on account of the 23 (number), number 23 being neither a Fermat prime, Fermat nor Pierpont prime#Polygon construction, Pierpont prime. In addition, the regular icositrigon is the Neusis construction#Use of the neusis, smallest regular polygon that is not constructible even with neusis. Concerning the nonconstructability of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridecagon
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. Regular tridecagon A ''regular polygon, regular tridecagon'' is represented by Schläfli symbol . The measure of each internal angle of a Regular polygon, regular tridecagon is approximately 152.308 degree (angle), degrees, and the area with side length ''a'' is given by :A = \fraca^2 \cot \frac \simeq 13.1858\,a^2. Construction As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructible polygon, constructed using a compass and straightedge. However, it is constructible using neusis construction, neusis, or an angle trisector. The following is an animation from a ''neusis construction'' of a regular tridecagon with radius of circumcircle \overline = 12, according to Andrew M. Gleason, based on the angle trisection by means of the Tomahawk (geometry), Tomahawk (light blue). An approximate construction of a regular tridecagon using straightedge and compass (d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonagon
In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix Hybrid word, hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogone'' and in English from the 17th century. The name ''enneagon'' comes from Greek language, Greek ''enneagonon'' (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common. Regular nonagon A ''regular polygon, regular nonagon'' is represented by Schläfli symbol and has internal angles of 140°. The area of a regular nonagon of side length ''a'' is given by :A = \fraca^2\cot\frac=(9/2)ar = 9r^2\tan(\pi/9) :::= (9/2)R^2\sin(2\pi/9)\simeq6.18182\,a^2, where the radius ''r'' of the inscribed circle of the regular nonagon is :r=(a/2)\cot(\pi/9) and where ''R'' is the radius of its circumscribed circle: :R = \sqrt=r\sec(\pi/9)=(a/2)\csc(\pi/9). Construction Although a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ''Wikt:hepta-, hepta-'', a Greek language, Greek-derived numerical prefix (both are cognate), together with the suffix ''-gon'' for , meaning angle. Regular heptagon A regular polygon, regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degree (angle), degrees). Its Schläfli symbol is . Area The area (''A'') of a regular heptagon of side length ''a'' is given by: :A = \fraca^2 \cot \frac \simeq 3.634 a^2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with Vertex (geometry), vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of \pi/7 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetradecagon
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. Regular tetradecagon A ''regular polygon, regular tetradecagon'' has Schläfli symbol and can be constructed as a quasiregular Truncation (geometry), truncated heptagon, t, which alternates two types of edges. The area of a Regular polygon, regular tetradecagon of side length ''a'' is given by :A = \fraca^2\cot\frac \approx 15.3345a^2 Construction As 14 = 2 × 7, a regular tetradecagon cannot be constructible polygon, constructed using a compass and straightedge. However, it is constructible using neusis construction, neusis with use of the angle trisector, or with a marked ruler, as shown in the following two examples. Symmetry The ''regular tetradecagon'' has dihedral symmetry, Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: Dih7, Dih2, and Dih1, and 4 cyclic group symmetries: Z14, Z7, Z2, and Z1. These 8 symmetries can be seen in 10 distinct symmetries on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trisection Of Any 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] |
Doubling The Cube
Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other methods. According to Eutocius, Archytas was the first to solve the problem of doubling the cube (the so-called Delian problem) with an ingenious geometric construction. The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, , the cube root of two. This is because a cube of side length 1 has a volume of , and a cube of twice tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Mathematics
Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Ancient Greek, Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in Mathematical proof, proofs is an important difference between Greek mathematics and those of preceding civilizations. The early history of Greek mathematics is obscure, and traditional narratives of Theorem, mathematical theorems found before the fifth century BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subje ... [...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] |
