In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria: Geometers have studied the Platonic solids for thousands of years, they are named for the ancient Greek philosopher Plato who hypothesized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, it has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes. The ancient Greeks studied the Platonic solids extensively; some sources credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing. Air is made of the octahedron. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a nonspherical solid, the hexahedron represents "earth"; these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven".
Aristotle added a fifth element, aithēr and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. Euclid mathematically described the Platonic solids in the Elements, the last book of, devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, cube and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is derived from the work of Theaetetus. In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets; the solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron and the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of, that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy, he discovered the Kepler solids. In the 20th century, attempts to link Platonic solids to the physical world were expanded to the electron shell model in chemistry by Robert Moon in a theory known as the "Moon model". For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below.
The Greek letter φ is used to represent the golden ratio 1 + √5/2 ≈ 1.6180. The coordinates for the tetrahedron and dodecahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates; these coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as or, one of two sets of 4 vertices in dual positions, as h or. Both tetrahedral positions make the compound stellated octahedron; the coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t or called a snub octahedron, as s or, seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientat
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL defined on the upper half-plane of complex numbers. It is the unique such function, holomorphic away from a simple pole at the cusp such that j = 0, j = 1728 = 12 3. Rational functions of j are modular, in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it has surprising connections to the symmetries of the Monster group. While the j-invariant can be defined purely in terms of certain infinite sums, these can be motivated by considering isomorphism classes of elliptic curves; every elliptic curve E over C is a complex torus, thus can be identified with a rank 2 lattice. This is done by identifying opposite edges of each parallelogram in the lattice. However, multiplying the lattice by a complex number, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, so we can always arrange for the lattice to be generated by 1 and some τ in H. Conversely, if we define g 2 = 60 ∑ ≠ − 4, g 3 = 140 ∑ ≠ − 6 this lattice corresponds to the elliptic curve over C defined by y2 = 4x3 − g2x − g3 via the Weierstrass elliptic functions.
The j-invariant is defined as j = 1728 g 2 3 Δ where the modular discriminant Δ is Δ = g 2 3 − 27 g 3 2 It can be shown that Δ is a modular form of weight twelve, g2 one of weight four, so that its third power is of weight twelve. Thus their quotient, therefore j, is a modular function of weight zero, in particular a holomorphic function H → C invariant under the action of SL; as explained below, j is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over C and the complex numbers. The two transformations τ → τ + 1 and τ → -τ−1 together generate the special linear group SL. Quotienting out by its centre yields the modular group, which we may identify with the projective special linear group PSL. By a suitable choice of transformation belonging to this group, τ ↦ a τ + b c τ + d, a d − b c = 1, we may reduce τ to a value giving the same value for j, lying in the fundamental region for j, which consists of values for τ satisfying the conditions | τ | ≥ 1 − 1 2 < R ≤ 1 2 − 1 2 < R < 0 ⇒ | τ | > 1 The function j when restricted to this region still takes on every value in the complex numbers C once.
In other words, for every c in C, there is a unique τ in the fundamental region such that c = j. Thus, j has the property of mapping the fundamental region to the entire complex plane; as a Riemann surface, the fundamental region has genus 0, every modular function is a rational function in j. In other words, the field of modular functions is C. T
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, this representation is unique, up to the order of the factors. For example, 1200 = 24 × 31 × 52 = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2 =... The theorem says two things for this example: first, that 1200 can be represented as a product of primes, second, that no matter how this is done, there will always be four 2s, one 3, two 5s, no other primes in the product; the requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique. This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime factorization into primes would not be unique. Book VII, propositions 30, 31 and 32, Book IX, proposition 14 of Euclid's Elements are the statement and proof of the fundamental theorem.
If two numbers by multiplying one another make some number, any prime number measure the product, it will measure one of the original numbers. Proposition 30 is referred to as Euclid's lemma, it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. Proposition 31 is proved directly by infinite descent. Any number either is measured by some prime number. Proposition 32 is derived from proposition 31, proves that the decomposition is possible. If a number be the least, measured by prime numbers, it will not be measured by any other prime number except those measuring it. Book IX, proposition 14 is derived from Book VII, proposition 30, proves that the decomposition is unique – a point critically noted by André Weil. Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.
Every positive integer n > 1 can be represented in one way as a product of prime powers: n = p 1 n 1 p 2 n 2 ⋯ p k n k = ∏ i = 1 k p i n i where p1 < p2 <... < pk are primes and the ni are positive integers. This representation is extended to all positive integers, including 1, by the convention that the empty product is equal to 1; this representation is called the canonical representation of n, or the standard form of n. For example, 999 = 33×37, 1000 = 23×53, 1001 = 7×11×13. Note that factors p0 = 1 may be inserted without changing the value of n. In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers: n = 2 n 1 3 n 2 5 n 3 7 n 4 ⋯ = ∏ i = 1 ∞ p i n i, where a finite number of the ni are positive integers, the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers; the canonical representations of the product, greatest common divisor, least common multiple of two numbers a and b can be expressed in terms of the canonical representations of a and b themselves: a ⋅ b = 2 a 1 + b 1 3 a 2 + b 2 5 a 3 + b 3 7 a 4 + b 4 ⋯ = ∏ p i a i + b i, gcd = 2 min 3 min ( a 2, b 2
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to