Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 2.779
1.
120-Cell -- from MathWorld - Weisstein, Eric W.
The 120-cell is a finite regular four-dimensional polytope with Schläfli symbol \{5, 3, 3\}. It is also known as the hyperdodecahedron or hecatonicosachoron, and is composed of 120 dodecahedra, with 3 to an edge. The 120-cell has 600 vertices (Coxeter 1969), and consists of 120 dodecahedra and 720 pentagons (Coxeter 1973, p. 264). In the plate following p. 176, Coxeter (1973) illustrates the polytope. The dual of the 120-cell is the 600-cell. The 120-cell has graph spectrum ...
2.
16-Cell -- from MathWorld - Weisstein, Eric W.
The 16-cell \beta_4 is the finite regular four-dimensional cross polytope with Schläfli symbol \{3, 3, 4\}. It is also known as the hyperoctahedron (Buekenhout and Parker 1998) or hexadecachoron, and its composed of 16 tetrahedra, with 4 to an edge. The 16-cell is a four-dimensional dipyramid based on the three-dimensional square dipyramid with its two apices in opposite directions along the fourth dimension (Coxeter 1973, p. 121). The 16-cell is the dual of the tesseract. Its vertices are...
4.
18-Point Problem -- from MathWorld - Weisstein, Eric W.
Place a point somewhere on a line segment. Now place a second point and number it 2 so that each of the points is in a different half of the line segment. Continue, placing every Nth point so that all N points are on different (1/N)th of the line segment. Formally, for a given N, does there exist a sequence of real numbers x_1, x_2, ..., x_N such that for every n\in\{1,\dots,N\} and every k\in\{1,\dots,n\}, the inequality {k-1\over n}\leq x_i<{k\over n} holds for some ...
5.
24-Cell -- from MathWorld - Weisstein, Eric W.
The 24-cell is a finite regular four-dimensional polytope with Schläfli symbol \{3, 4, 3\}. It is also known as the hyperdiamond or icositetrachoron, and is composed of 24 octahedra, with 3 to an edge. Coxeter (1969) gives a list of the polyhedron vertex positions. The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog. The even coefficients of the D_4 lattice are 1, 24, 24, 96, ... (Sloane's A004011), and the 24 shortest vectors...
6.
257-gon -- from MathWorld - Weisstein, Eric W.
257 is a Fermat prime, and the 257-gon is therefore a constructible polygon using compass and straightedge, as proved by Gauss. An illustration of the 257-gon is not included here, since its 257 segments so closely resemble a circle. Richelot and Schwendenwein found constructions for the 257-gon in 1832 (Coxeter 1969). De Temple (1991) gives a construction using 150 circles (24 of which are Carlyle circles) which has geometrography symbol 94S_1+47S_2+275C_1+0C_2+150C_3 and simplicity 566. See...
7.
30-60-90 Triangle -- from MathWorld - Weisstein, Eric W.
A right triangle having angles of 30°, 60°, and 90°. For a 30-60-90 triangle with hypotenuse of length a, the legs have lengths b = a\sin(60^\circ) = {{1\over 2}}a\sqrt{3} c = a\sin(30^\circ) = {{1\over 2}}a, and the area is A = {{1\over 2}}bc ={{1\over 8}}\sqrt{3}\,a^2. The inradius r and circumradius R are r = {{1\over 4}}(\sqrt{3}-1)a R = {{1\over 2}}a. 30-60-90 triangles are used in drafting, as illustrated above. This allows lines of 0°, 30°, 60°, and 90° to...
8.
6-Sphere Coordinates -- from MathWorld - Weisstein, Eric W.
The coordinate system obtained by inversion of Cartesian coordinates, with u,v,w\in(-\infty,\infty). The transformation equations are x = {u\over u^2+v^2+w^2} y = {v\over u^2+v^2+w^2} z = {w\over u^2+v^2+w^2}. The equations of the surfaces of constant coordinates are given by \left({x-{1\over 2u}}\right)^2+y^2+z^2={1\over 4u^2}, which gives spheres tangent to the yz-plane at the origin for u constant, x^2+\left({y-{1\over 2v}}\right)^2+z^2={1\over 4v^2}, which gives spheres tangent to...
9.
65537-gon -- from MathWorld - Weisstein, Eric W.
65537 is the largest known Fermat prime, and the 65537-gon is therefore a constructible polygon using compass and straightedge, as proved by Gauss. The 65537-gon has so many sides that it is, for all intents and purposes, indistinguishable from a circle using any reasonable printing or display methods. Hermes spent 10 years on the construction of the 65537-gon at Königsberg around (1900). After the Second World War, his manuscripts were moved to the Mathematical Institute in Göttingen,...
10.
AAA Theorem -- from MathWorld - Weisstein, Eric W.
Specifying three angles A, B, and C does not uniquely define a triangle, but any two triangles with the same angles are similar. Specifying two angles of a triangle automatically gives the third since the sum of angles in a triangle sums to 180° (\pi radians), i.e., C=\pi-A-B. See also: AAS Theorem, ASA Theorem, ASS Theorem, SAS Theorem, SSS Theorem, Triangle
11.
AAS Theorem -- from MathWorld - Weisstein, Eric W.
Specifying two angles A and B and a side a uniquely determines a triangle with area K = {a^2\sin B\sin C\over 2\sin A} = {a^2\sin B\sin(\pi-A-B)\over 2\sin A}. The third angle is given by C=\pi-A-B, since the sum of angles of a triangle is 180° (\pi radians). Solving the law of sines {a\over\sin A}={b\over\sin B} for b gives b=a{\sin B\over\sin A}. Finally, c = b\cos A+a\cos B=a(\sin B\cot A+\cos B) = a\sin B(\cot A+\cot B). See also: AAA Theorem, ASA Theorem, ASS Theorem, SAS...
12.
ASA Theorem -- from MathWorld - Weisstein, Eric W.
Specifying two adjacent angles A and B and the side between them c uniquely (up to geometric congruence) determines a triangle with area K={c^2\over 2(\cot A+\cot B)}. The angle C is given in terms of A and B by C=\pi-A-B, and the sides a and b can be determined by using the law of sines {a\over\sin A}={b\over\sin B}={c\over\sin C} to obtain a = {\sin A\over\sin(\pi-A-B)}c b = {\sin B\over\sin(\pi-A-B)}c. See also: AAA Theorem, AAS Theorem, ASS Theorem, SAS Theorem, SSS Theorem, Triangle...
13.
ASS Theorem -- from MathWorld - Weisstein, Eric W.
Specifying two adjacent side lengths a and c of a triangle (with a < c) and one acute angle A opposite a does not, in general, uniquely determine a triangle. If \sin A < a/c, there are two possible triangles satisfying the given conditions. If \sin A = a/c, there is one possible triangle. If \sin A > a/c, there are no possible triangles. Remember: don't try to prove congruence with the ASS theorem or you will make an ASS out of yourself. See also: AAA Theorem, AAS Theorem, SAS...
14.
Abscissa -- from MathWorld - Weisstein, Eric W.
The x- (horizontal) coordinate of a point in a two dimensional coordinate system. Physicists and astronomers sometimes use the term to refer to the axis itself instead of the distance along it. See also: Axis, Ordinate, Real Line, x-Axis, y-Axis, z-Axis
15.
Absolute Geometry -- from MathWorld - Weisstein, Eric W.
Geometry which depends only on the first four of Euclid's postulates and not on the parallel postulate. Euclid himself used only the first four postulates for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. See also: Affine Geometry, Elements, Euclid's Postulates, Geometry, Ordered Geometry, Parallel Postulate
16.
Acoptic Polyhedron -- from MathWorld - Weisstein, Eric W.
A term invented by B. Grünbaum in an attempt to promote concrete and precise polyhedron terminology. The word "coptic" derives from the Greek for "to cut," and acoptic polyhedra are defined as polyhedra for which the faces do not intersect (cut) themselves, making them 2-manifolds. See also: Honeycomb, Nolid, Polyhedron, Sponge
18.
Acute Triangle -- from MathWorld - Weisstein, Eric W.
A triangle in which all three angles are acute angles. A triangle which is neither acute nor a right triangle (i.e., it has an obtuse angle) is called an obtuse triangle. From the law of cosines, for a triangle with side lengths a, b, and c, \cos C={a^2+b^2-c^2\over 2ab}, with C the angle opposite side C. For an angle to be acute, \cos C>0. Therefore, an acute triangle satisfies a^2+b^2>c^2, b^2+c^2>a^2, and c^2+a^2>b^2. The smallest number of acute triangles into which an...
19.
Adams' Circle -- from MathWorld - Weisstein, Eric W.
Given a triangle \Delta ABC, construct the contact triangle \Delta T_AT_BT_C. Now extend lines parallel to the sides of the contact triangle from the Gergonne point. These intersect the triangle \Delta ABC in the six points P, Q, R, S, T, and U. C. Adams proved in 1843 that these points are concyclic in a circle now known as the Adams' circle. Adams' circle is a central circle with circle function l={a(a-b-c)^3(ab+ac-2bc-b^2-c^2)\over bc(a^2+b^2+c^2-2ab-2bc-2ca)}, which does not correspond...
20.
Affine -- from MathWorld - Weisstein, Eric W.
The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the n-dimensional affine space \mathbb{R}^n is determined by any basis of n vectors, which are not necessarily orthonormal. Therefore, the resulting axes are not necessarily mutually perpendicular nor have the same unit measure. In this sense, affine is a generalization of Cartesian or Euclidean. An example of an affine property is the average area of a random triangle...
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