Degrees of Infinity | ThatsMaths. Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing. Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right). The concept of infinity has exercised the greatest minds throughout the history of human thought. Set Theory Cantor was the inventor of set theory, which is a fundamental foundation of modern mathematics. Bijection between two sets of cardinality 4. But suppose the sets are infinite. Cantor carried these ideas much further, showing in particular that the set of all the real numbers (or all the points on a line) have a degree of infinity, or cardinality, greater than the counting numbers.
Infinities without limit Aleph-zero, the cardinality of the natural numbers and the smallest transfinite number. Strange but True: Infinity Comes in Different Sizes. In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond! " The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.
That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others. Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals. " In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. The American government's growth has been severed and we need to take a cue from biology to revive it : C_S_T.
Can you represent PI in a finite number of digits in any number system? : askscience. Banach–Tarski paradox. Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".
Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices. It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another. Banach and Tarski publication and . 7. 7 (seven; /ˈsɛvən/) is the natural number following 6 and preceding 8. Mathematics In fact, if one sorts the digits in the number 142,857 in ascending order, 124578, it is possible to know from which of the digits the decimal part of the number is going to begin with. The remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start.
For example, 628 ÷ 7 = 89 5/7; here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case, 628 ÷ 7 = 89.714285. Graph of the probability distribution of the sum of 2 six-sided dice Basic calculations Evolution of the glyph On the seven-segment displays of pocket calculators and digital watches, 7 is the number with the most common glyph variation (1, 6 and 9 also have variant glyphs).
Automotive and transportation Classical world Classical antiquity Commerce and business Film Place Time. Ralph Abraham. Ralph H. Abraham (b. July 4, 1936, Burlington, Vermont) is an American mathematician. He has been a member of the mathematics department at the University of California, Santa Cruz since 1968. Life and work Ralph Abraham earned his Ph.D. from the University of Michigan in 1960, and held positions at UC Santa Cruz, Berkeley, Columbia, and Princeton. He has also held visiting positions in Amsterdam, Paris, Warwick, Barcelona, Basel, and Florence. He founded the Visual Math Institute at UC Santa Cruz in 1975, at that time it was called the "Visual Mathematics Project". Abraham has been involved in the development of dynamical systems theory in the 1960s and 1970s.
Another interest of Abraham's concerns alternative ways of expressing mathematics, for example visually or aurally. Works Publications Film DMT: the Spirit Molecule, as himself Further reading Thomas A. References External links Henri Poincaré. French mathematician, physicist, engineer, and philosopher of science Jules Henri Poincaré (, ; French: [ɑ̃ʁi pwɛ̃kaʁe] ( listen); 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. The Poincaré group used in physics and mathematics was named after him. Life Education First scientific achievements Career Work
Bell’s math showed that quantum weirdness rang true. First of two parts There’s just enough time left in 2014 to sneak in one more scientific anniversary, and it just might be the most noteworthy of them all. Fifty years ago last month, John Stewart Bell transformed forever the human race’s grasp on the mystery of quantum physics. He proved a theorem establishing the depth of quantum weirdness, deflating the hopes of Einstein and others that the sanity of traditional physics could be restored. “Bell’s theorem has deeply influenced our perception and understanding of physics, and arguably ranks among the most profound scientific discoveries ever made,” Nicolas Brunner and colleagues write in a recent issue of Reviews of Modern Physics.
Before Bell, physicists’ grip on the quantum was severely limited. Weirdness was well established, but not very well explained. Einstein didn’t buy it, insisting that underlying the quantum fuzziness there must exist a solid reality, even if it was inaccessible to human eyes and equations. Koch snowflake. Fractal and mathematical curve The first seven iterations in animation Zooming into the Koch curve First four iterations Sixth iteration The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described.
The Koch snowflake can be built up iteratively, in a sequence of stages. Times the area of the original triangle, while the perimeters of the successive stages increase without bound. Construction The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: divide the line segment into three segments of equal length.draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.remove the line segment that is the base of the triangle from step 2.
The first iteration of this process produces the outline of a hexagram. A fractal rough surface built from multiple Koch curve iterations since An. Luca Pacioli. Fra Luca Bartolomeo de Pacioli (sometimes Paccioli or Paciolo; c. 1447–1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and a seminal contributor to the field now known as accounting. He is referred to as "The Father of Accounting and Bookkeeping" and he was the first person to publish a work on the double-entry system of book-keeping. He was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany.  Life Luca Pacioli was born between 1446 and 1448 in Sansepolcro (Tuscany) where he received an abbaco education. This was education in the vernacular (i.e. the local tongue) rather than Latin and focused on the knowledge required of merchants. He moved to Venice around 1464, where he continued his own education while working as a tutor to the three sons of a merchant.
It was during this period that he wrote his first book, a treatise on arithmetic for the boys he was tutoring. Mathematics Chess See also Johannes Kepler. German astronomer and mathematician (1571–1630) Johannes Kepler (; German: [joˈhanəs ˈkɛplɐ, -nɛs -] ( listen); 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae.
These works also provided one of the foundations for Newton's theory of universal gravitation. Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein. Early life Childhood (1571–1590) Graz (1594–1600) Octahedron. In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
An octahedron is the three-dimensional case of the more general concept of a cross polytope. Regular octahedron Dimensions If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is while the midradius, which touches the middle of each edge, is Orthogonal projections The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face.
Cartesian coordinates An octahedron with edge length sqrt(2) can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then Area and volume Dual Nets Stellation. In geometry, stellation is the process of extending a polygon (in two dimensions), polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure.
Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus or stella, which means "star". Kepler's definition In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, a regular compound of two tetrahedra.
Stellating polygons See also Stellated octahedron. The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's Divina Proportione, 1509. It is the simplest of five regular polyhedral compounds. It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. Construction The stellated octahedron can be constructed in several ways: Related concepts A compound of two spherical tetrahedra can be constructed, as illustrated.
The two tetrahedra of the compound view of the stellated octahedron are "desmic", meaning that (when interpreted as a line in projective space) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. In popular culture Breaking up the indivisible to observe the implausible—particles with a fractional charge | Ars Technica. It was 1909 when Robert Millikan and Harvey Fletcher carried out their famous oil drop experiment in which they determined that the smallest unit of charge possible was 1.592x10-19 Coulombs, a value we now refer to as e, the fundamental charge (the modern accepted value is 1.602176565(35)x10-19 C). It is the magnitude of the negative charge carried by the electron, as well as the positive charge of a proton. It is also the smallest unit of charge that any stable, independent particle can possibly have—no particles can have -3/4e charge, nor can they carry +2.8e of charge—barring technicalities.
A paper published in this week's edition of Science examines in detail one of the technical loopholes to the preceding statement. We have spent a large amount of time breaking up hadrons to our heart's content, resulting in a spew of quarks, bosons, and other fundamental particles. But there may be a way to split up something that looks a lot like an electron. Bell’s math showed that quantum weirdness rang true.
Reynolds number. Dimensionless quantity used to help predict fluid flow patterns The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place. A vortex street around a cylinder. This can occur around cylinders and spheres, for any fluid, cylinder size and fluid speed, provided that it has a Reynolds number between roughly 40 and 1000. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing.
The concept was introduced by George Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds (1842–1912), who popularized its use in 1883. Definition The Reynolds number is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities. With respect to laminar and turbulent flow regimes: The Reynolds number is defined as where: History or. Størmer number. Leucocoprinus birnbaumii, aka Lepiota lutea, the yellow houseplant or house plant soil mushroom, Tom Volk's Fungus of the Month for February 2002, Why is there Fibonacci Sequence in Mandelbrot Set? 7 The Fibonacci Sequence. Mandelbrot set.
"sacred" geometry. Citizen Maths - Course. The unexpected math behind Van Gogh's "Starry Night" - Natalya St. Clair. Orthogonal matrix. Orthogonality. Prime Curios!: 9901. Numerology. Gödel numbering. Kurt Gödel. Spirasolaris. Fibonacci number. Sacred geometry. Gödel's incompleteness theorems. What on Earth is a Logarithm? For V. Montcello.