Straightedge and compass construction
Straightedge and compass construction known as ruler-and-compass construction or classical construction, is the construction of lengths and other geometric figures using only an idealized ruler and compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, no markings on it; the compass is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates, it turns out to be the case that every point constructible using straightedge and compass may be constructed using compass alone. The ancient Greek mathematicians first conceived straightedge and compass constructions, a number of ancient problems in plane geometry impose this restriction; the ancient Greeks developed many constructions. Gauss showed that most are not; some of the most famous straightedge and compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields.
In spite of existing proofs of impossibility, some persist in trying to solve these problems. Many of these problems are solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using straightedge and compass alone. In terms of algebra, a length is constructible if and only if it represents a constructible number, an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots; the "straightedge" and "compass" of straightedge and compass constructions are idealizations of rulers and compasses in the real world: The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment; the compass can be opened arbitrarily wide.
Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it is not drawing a circle. Actual compasses do not collapse and modern geometric constructions use this feature. A'collapsing compass' would appear to be a less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements, no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a checkered history; each construction must be exact. "Eyeballing" it and getting close does not count as a solution. Each construction must terminate; that is, it must have a finite number of steps, not be the limit of closer approximations. Stated this way and compass constructions appear to be a parlour game, rather than a serious practical problem; the ancient Greek mathematicians first attempted straightedge and compass constructions, they discovered how to construct sums, products and square roots of given lengths.
They could construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, a regular polygon with 3, 4, or 5 sides. But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle, or a regular polygon with other numbers of sides. Nor could they construct the side of a cube whose volume would be twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas, but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square the circle, Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed.
In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He showed that Gauss's sufficient constructibility condition for regular polygons is necessary. In 1882 Lindemann showed that π is a transcendental number, thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge and compass constructions consist of repeated application of five basic constructions using the points and circles that have been constructed; these are: Creating the line through two existing points Creating the circle through one point with centre another point Creating the point, th
A Pierpont prime is a prime number of the form 2 u 3 v + 1 for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth, they are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections. A Pierpont prime with v = 0 is of the form 2 u + 1, is therefore a Fermat prime. If v is positive u must be positive, therefore the non-Fermat Piermont primes all have the form 6k + 1, when k is a positive integer; the first few Pierpont primes are: 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329... Empirically, the Pierpont primes do not seem to be rare or sparsely distributed. There are 36 Pierpont primes less than 106, 59 less than 109, 151 less than 1020, 789 less than 10100. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime.
Thus, it is expected that among n-digit numbers of the correct form 2 u 3 v + 1, the fraction of these that are prime should be proportional to 1/n, a similar proportion as the proportion of prime numbers among all n-digit numbers. As there are Θ numbers of the correct form in this range, there should be Θ Pierpont primes. Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, more that there should be 9n Pierpont primes up to 10n. According to Gleason's conjecture there are Θ Pierpont primes smaller than N, as opposed to the smaller conjectural number O of Mersenne primes in that range; when 2 u > 3 v, the primality of 2 u 3 v + 1 can be tested by Proth's theorem. On the other hand, when 2 u < 3 v alternative primality tests for p = 2 u 3 v + 1 are possible based on the factorization of p − 1 as a small number multiplied by a large power of three. As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors.
The following table gives values of m, k, n such that k ⋅ 2 n + 1 divides 2 2 m + 1. The left-hand side is a Pierpont prime when k is a power of 3; as of 2018, the largest known Pierpont prime is 3 × 210829346 + 1, whose primality was discovered by Sai Yik Tang and PrimeGrid in 2014. In the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible, it has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of N sides to be formed, as long as N ≥ 3 and of the form 2m3nρ, where ρ is a product of distinct Pierpont primes; this is the same class of regular polygons as those that can be constructed with a compass and angle-trisector. Regular polygons which can be constructed with only compass and straightedge are the special case where n = 0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In 1895, James Pierpont studied the same class of regular polygons.
Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, the primes N for which Pierpont's construction works are the Pierpont primes. However, Pierpont did not describe the form of the composite numbers with 3-smooth totients; as Gleason showed, these numbers are the ones of the form 2m3nρ given above. The smallest prime, not a Pierpont prime is 11. All other regular N-gons with 3 ≤ N ≤ 21 can be constructed with compass and trisector. A Pierpont prime of the second kind is a prime number of the form 2u3v − 1; these numbers are 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73
Andrew M. Gleason
Andrew Mattei Gleason was an American mathematician who as a young World War II naval officer broke German and Japanese military codes over the succeeding sixty years made fundamental contributions to varied areas of mathematics, including the solution of Hilbert's fifth problem, was a leader in reform and innovation in mathematics teaching at all levels. Gleason's theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him. Gleason's entire academic career was at Harvard University, from which he retired in 1992, his numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and Harvard Society of Fellows, presidency of the American Mathematical Society. He continued to advise the United States government on cryptographic security, the Commonwealth of Massachusetts on mathematics education for children until the end of his life. Gleason won the Newcomb Cleveland Prize in 1952 and the Gung–Hu Distinguished Service Award of the American Mathematical Society in 1996.
He was a member of the National Academy of Sciences and of the American Philosophical Society, held the Hollis Chair of Mathematics and Natural Philosophy at Harvard. He was fond of saying that mathematical proofs "really aren't there to convince you that something is true—they're there to show you why it is true." The Notices of the American Mathematical Society called him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship and service in equal measure." Gleason was born in Fresno, the youngest of three children. His older brother Henry Jr. became a linguist. He grew up in Bronxville, New York, where his father was the curator of the New York Botanical Garden. After attending Berkeley High School he graduated from Roosevelt High School in Yonkers, winning a scholarship to Yale University. Though Gleason's mathematics education had gone only so far as some self-taught calculus, Yale mathematician William Raymond Longley urged him to try a course in mechanics intended for juniors.
So I learned first year calculus and second year calculus and became the consultant to one end of the whole Old Campus... I used to do all the homework for all the sections of. I got plenty of practice in doing elementary calculus problems. I don't think there exists a problem—the classical kind of pseudo reality problem which first and second-year students are given—that I haven't seen. One month he enrolled in a differential equations course as well; when Einar Hille temporarily replaced the regular instructor, Gleason found Hille's style "unbelievably different... He had a view of mathematics, just vastly different... That was a important experience for me. So after that I took a lot of courses from Hille" including, in his sophomore year, graduate-level real analysis. "Starting with that course with Hille, I began to have some sense of what mathematics is about."While at Yale he competed three times in the founded William Lowell Putnam Mathematical Competition, always placing among the top five entrants in the country.
After the Japanese attacked Pearl Harbor during his senior year, Gleason applied for a commission in the US Navy, on graduation joined the team working to break Japanese naval codes. He collaborated with British researchers attacking the German Enigma cipher. In 1946, at the recommendation of Navy colleague Donald Howard Menzel, Gleason was appointed a Junior Fellow at Harvard. An early goal of the Junior Fellows program was to allow young scholars showing extraordinary promise to sidestep the lengthy PhD process, he returned to Harvard in the fall of 1952, soon after published the most important of his results on Hilbert's fifth problem. Harvard awarded him tenure the following year. In January 1959 he married Jean Berko. Berko, a psycholinguist, worked for many years at Boston University, they had three daughters. In 1969 Gleason took the Hollis Chair of Mathematics and Natural Philosophy, the oldest scientific endowed professorship in the US, he retired from Harvard in 1992 but remained active in service to Harvard and to mathematics: in particular, promoting the Harvard Calculus Reform Project and working with the Massachusetts Board of Education.
He died in 2008 from complications following surgery. Gleason said he "always enjoyed helping other people with math"—a colleague said he "regarded teaching mathematics—like doing mathematics—as both important and genuinely fun." At fourteen, during his brief attendance at Berkeley High School, he found himself not only bored with first-semester geometry, but helping other students with their homework—including those taking the second half of the course, which he soon began auditing. At Harvard he "regularly taught at every level", including administratively burdensome multisection c
In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d in which a is nonzero. Setting f = 0 produces a cubic equation of the form a x 3 + b x 2 + c x + d = 0; the solutions of this equation are called roots of the polynomial f. If all of the coefficients a, b, c, d of the cubic equation are real numbers it has at least one real root. All of the roots of the cubic equation can be found algebraically; the roots can be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms such as Newton's method; the coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3; the solutions of the cubic equation do not belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are non-rational complex numbers. Cubic equations were known to the ancient Babylonians, Chinese and Egyptians.
Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did; the problem of doubling the cube involves the simplest and oldest studied cubic equation, one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task, now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century. In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. Hippocrates and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations.
Some others like T. L. Heath, who translated all Archimedes' works, putting forward evidence that Archimedes solved cubic equations using intersections of two conics, but discussed the conditions where the roots are 0, 1 or 2. In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, two of them with q = 0. In the 11th century, the Persian poet-mathematician, Omar Khayyam, made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions, he found a geometric solution. In his work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: x3 + 12x = 6x2 + 35. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī, wrote the Al-Muʿādalāt, which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions, he used what would be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions, he understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations. In his book Flos, Leonardo de Pisa known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x = 20. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40, which has a relative error of about 10−9.
In the early 16th century, the Italian mathematician Scipione del Ferro found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it. In 1530, Niccolò Tartaglia received two problems in cubic equations from Zuanne da Coi and announced that he could solve them, he was soon challenged by Fiore. Each contestant had to put up a certain amoun
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be either star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed; these properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle; that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon has an inscribed circle or incircle, tangent to every side at the midpoint, thus a regular polygon is a tangential polygon. A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon; the symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4...
It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd all axes pass through a vertex and the midpoint of the opposite side. All regular simple polygons are convex; those having the same number of sides are similar. An n-sided convex regular polygon is denoted by its Schläfli symbol. For n < 3, we have two degenerate cases: Monogon Degenerate in ordinary space. Digon. In certain contexts all the polygons considered. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described as triangle, pentagon, etc. For a regular convex n-gon, each interior angle has a measure of: × 180 degrees, or equivalently 180 n degrees; as the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides the internal angle is 179.964°.
As the number of sides increase, the internal angle can come close to 180°, the shape of the polygon approaches that of a circle. However the polygon can never become a circle; the value of the internal angle can never become equal to 180°, as the circumference would become a straight line. For this reason, a circle is not a polygon with an infinite number of sides. For n > 2, the number of diagonals is 1 2 n. The diagonals divide the polygon into 1, 4, 11, 24, … pieces OEIS: A007678. For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from an arbitrary point in the plane to the vertices, we have ∑ i = 1 n d i 4 n + 3 R 4 = 2. For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem; this is a generalization of Viviani's theorem for the n. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by R = s 2 sin = a cos