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Moser's Circle Problem
file:circle_division_by_chords.svg, The number of and for first 6 terms of Moser's circle problem In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method. The greatest possible number of regions, r_G = + + 1, giving the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163 (number), 163, 256, ... (). Though the first five terms match the geometric progression , it deviates at , showing the risk of generalising from only a few observations. Lemma If there are ''n'' points on the circle and one more point is added, ''n'' lines can be drawn from the new point to the previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard K
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", " Dick", " Dickon", " Dickie", " Rich", " Rick", "Rico (name), Rico", " Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Anders ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pizza Theorem
In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a Disk (mathematics), disk in a certain way. The theorem is so called because it mimics a traditional pizza slicing technique. It shows that if two people share a pizza sliced into 8 pieces (or any multiple of 4 greater than 8), and take alternating slices, then they will each get an equal amount of pizza, irrespective of the central cutting point. Statement Let ''p'' be an interior point of the disk, and let ''n'' be a multiple of 4 that is greater than or equal to 8. Form ''n'' circular sector, sectors of the disk with equal angles by choosing an arbitrary line (geometry), line through ''p'', rotating the line times by an angle of radians, and slicing the disk on each of the resulting lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states : History The pizza theorem was originally proposed as a challenge proble ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazy Caterer's Sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a Disk (mathematics), disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point inside the circle, but up to seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines; for generalizations to higher dimensions, ''see'' arrangement of hyperplanes. The analogue of this integer sequence, sequence in three dimensions is the cake numbers. Formula and sequence The maximum number ''p'' of pieces that can be created with a given number of cuts (where ) is given by the formula :p = \frac. Using binomial coefficients, the formula can be expressed as :p = 1 + \dbinom = \dbinom+\dbinom+\dbinom. Simply put, each number equals a triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake Number
In mathematics, the cake number, denoted by ''Cn'', is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly ''n'' planes. The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of ''Cn'' for are given by . General formula If ''n''! denotes the factorial, and we denote the binomial coefficients by : = \frac, and we assume that ''n'' planes are available to partition the cube, then the ''n''-th cake number is: : C_n = + + + = \tfrac\!\left(n^3 + 5n + 6\right) = \tfrac(n+1)\left(n(n-1) + 6\right). Properties The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence. The difference between successive cake numbers also gives the lazy caterer's sequence. The fourth column of Bernoulli's triangle (''k'' = 3) gives the cake numbers for ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Division By Chords 6 Points
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus: a ring-shaped object, the region bounded by two concentric circles. * Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Arc
A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than radians (180 degrees); and the other arc, the major arc, subtends an angle greater than radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a '' chord'' of a circle. If the length of an arc is exactly half of the circle, it is known as a '' semicircular arc''. Length The length (more precisely, arc length) of an arc of a circle with radius ''r'' and subtending an angle ''θ'' (measured in radians) with the circle center — i.e., the central angle — is : L = \theta r. This is because :\frac=\frac. Substituting in the circumference :\frac=\frac, and, with ''α'' being the same angle measured in degrees, since ''θ'' = , the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
