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Dividing A Square Into Similar Rectangles
Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics. Three rectangles There is only one way (up to rotation and reflection) to divide a square into two similar rectangles. However, there are three distinct ways of partitioning a square into three similar rectangles: #The trivial solution given by three congruent rectangles with aspect ratio 3:1. #The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2. #The solution in which the three rectangles are all of different sizes and where they have aspect ratio ''ρ''2, where ''ρ'' is the plastic ratio. The fact that a rectangle of aspect ratio ''ρ''2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ''ρ''2 related to the Routh–Hurwitz theorem: all of its conjugates have po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Square Partitions
Plastics are a wide range of synthetic polymers, synthetic or Semisynthesis, semisynthetic materials composed primarily of Polymer, polymers. Their defining characteristic, Plasticity (physics), plasticity, allows them to be Injection moulding, molded, Extrusion, extruded, or Compression molding, pressed into a diverse range of solid forms. This adaptability, combined with a wide range of other properties such as low weight, durability, flexibility, chemical resistance, low toxicity, and low-cost production, has led to their widespread use around the world. While most plastics are produced from natural gas and petroleum, a growing minority are produced from renewable resources like polylactic acid. Between 1950 and 2017, 9.2 billion metric tons of plastic are estimated to have been made, with more than half of this amount being produced since 2004. In 2023 alone, preliminary figures indicate that over 400 million metric tons of plastic were produced worldwide. If global trends ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, " is unique up to " means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation that relates two lists if one can be obtained by reordering (permutation, permuting) the other. As another example, the statement "the solution to an indefinite integral is , up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degree (angle), degrees, or Pi, /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called square (algebra), squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similar (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aspect Ratio
The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangle is oriented as a "landscape format, landscape". The aspect ratio is most often expressed as two integer numbers separated by a colon (x:y), less commonly as a simple or decimal Fraction (mathematics), fraction. The values x and y do not represent actual widths and heights but, rather, the proportion between width and height. As an example, 8:5, 16:10, 1.6:1, and 1.6 are all ways of representing the same aspect ratio. In objects of more than two dimensions, such as hyperrectangles, the aspect ratio can still be defined as the ratio of the longest side to the shortest side. Applications and uses The term is most commonly used with reference to: * Graphic / image ** Aspect ratio (image), Image aspect ratio ** Display aspect ratio ** Pape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Ratio
In mathematics, the plastic ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins as . The adjective ''plastic'' does not refer to Plastic, the artificial material, but to the formative and sculptural qualities of this ratio, as in ''plastic arts''. Definition Three quantities are in the plastic ratio if \frac =\frac =\frac The ratio is commonly denoted Substituting b=\rho c \, and a=\rho b =\rho^2 c \, in the middle fraction, \rho =\frac. It follows that the plastic ratio is the unique real solution of the cubic equation \rho^3 -\rho -1 =0. Solving with Cubic equation#Cardano's formula, Cardano's formula, \begin w_ &=\frac12 \left( 1 \pm \frac13 \sqrt \right) \\ \rho &=\sqrt[3] +\sqrt[3] \end or, using the Cubic equation#Trigonometric and hyperbolic solutions, hyperbolic cosine, :\rho =\frac \cosh \left( \frac \operatorname \left( \frac \right) \right). is the superstabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable, linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh–Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz. Notations Let be a polynomial (with complex coefficients) of degree with no roots on the imaginary axis (i.e. the line where is the imaginary unit and is a real number). Let us define real polynomials and by , respectively the real a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * '' Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sections. History ''Current Contents ...'' Notable articles Two articles published in ''Discrete & Computational Geometry'', one by Gil Kalai in 1992 with a proof of a subexponential upper bound on the diameter of a polytope and another by Samuel Ferguson in 2006 on the Kepler conjecture on optimal three-dimensional sphere packing, earned their authors the Fulk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John C
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died ), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (died ), one of the twelve apostles of Jesus Christ * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope John ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squaring The Square
Squaring the square is the problem of tessellation, tiling an integral square using only other integral squares. (An integral square is a square (geometry), square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares. Perfect squared squares A "perfect" squared square is a square such that each of the smaller squares has a different size. Perfect squared squares were studied by R. Leonard Brooks, R. L. Brooks, Cedric Smith (statistician), C. A. B. Smith, Arthur Harold Stone, A. H. Stone and W. T. Tutte (wr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
