First principle. In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption. In mathematics, first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and fitting parameters. First principles in formal logic[edit] In a formal logical system, that is, a set of propositions that are consistent with one another, it is probable that some of the statements can be deduced from one another.
A first principle is one that cannot be deduced from any other. Aristotle's contribution[edit] Terence Irwin writes: Descartes[edit] Profoundly influenced by Euclid, Descartes was a rationalist who invented the foundationalist system of philosophy. In physics[edit] Notes[edit] See also[edit] External links[edit] Euclid's Elements. Pigeonhole principle. An image of pigeons in holes. Here there are n = 10 pigeons in m = 9 holes. Since 10 is greater than 9, the pigeonhole principle says that at least one hole has more than one pigeon.
In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads (see below). The first formalization of the idea is believed to have been made by Peter Gustav Lejeune Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle"). Examples[edit] Softball team[edit] Sock-picking[edit] Hand-shaking[edit] Hair-counting[edit] Theory of everything. A theory of everything (ToE) or final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe.[1]:6 Finding a ToE is one of the major unsolved problems in physics.
Over the past few centuries, two theoretical frameworks have been developed that, as a whole, most closely resemble a ToE. The two theories upon which all modern physics rests are general relativity (GR) and quantum field theory (QFT). GR is a theoretical framework that only focuses on the force of gravity for understanding the universe in regions of both large-scale and high-mass: stars, galaxies, clusters of galaxies, etc. On the other hand, QFT is a theoretical framework that only focuses on three non-gravitational forces for understanding the universe in regions of both small scale and low mass: sub-atomic particles, atoms, molecules, etc. Historical antecedents[edit] [edit] Theoretical physics. Chaos theory.
A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted.
Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions. In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. Where , and , is: . Chaos game. Animation of chaos game method The "chaos game" method plots points in random order all over the attractor. This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail. With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained.
See also[edit] Chaos theory References[edit] Jump up ^ Barnsley, Michael (1993). External links[edit] IFS Fractal fern and Sierpinski triangle - JAVA applet. X - LONDON AT MIDSUMMER. I believe it is supposed to require a good deal of courage to confess that one has spent the month of August in London; and I will therefore, taking the bull by the horns, plead guilty at the very outset to this dishonourable weakness. I might attempt some ingenious extenuation of it. I might say that my remaining in town had been the most unexpected necessity or the merest inadvertence; I might pretend I liked it—that I had done it, in fact, for the love of the thing; I might claim that you don't really know the charms of London until on one of the dog-days you have imprinted your boot-sole in the slumbering dust of Belgravia, or, gazing along the empty vista of the Drive, in Hyde Park, have beheld, for almost the first time in England, a landscape without figures.
Golden spiral. Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.[1] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. Formula[edit] The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:[2] or with e being the base of Natural Logarithms, a being an arbitrary positive real constant, and b such that when θ is a right angle (a quarter turn in either direction): Therefore, b is given by The numerical value of b depends on whether the right angle is measured as 90 degrees or as for θ in degrees; Golden rectangle.
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is is approximately 1.618. Construction[edit] A method to construct a golden rectangle. A golden rectangle can be constructed with only straightedge and compass by 4 simple steps: Construct a simple squareDraw a line from the midpoint of one side of the square to an opposite cornerUse that line as the radius to draw an arc that defines the height of the rectangleComplete the golden rectangle Relation to regular polygons and polyhedra[edit] Three golden rectangles in an icosahedron An alternative construction of the golden rectangle uses three polygons circumscribed by congruent circles: a regular decagon, hexagon, and pentagon.
The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle. Applications[edit] See also[edit] References[edit] Jump up ^ Euclid, Book XIII, Proposition 10.Jump up ^ Burger, Edward B.; Starbird, Michael P. (2005). External links[edit] Logarithmic spiral. A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". Definition[edit] In polar coordinates the logarithmic curve can be written as[1] or with being the base of natural logarithms, and and being arbitrary positive real constants. In parametric form, the curve is with real numbers The spiral has the property that the angle between the tangent and radial line at the point is constant. The derivative of is proportional to the parameter . ) the spiral becomes a circle of radius .
Approaches infinity ( → 0) the spiral tends toward a straight half-line. Is called the pitch. Spira mirabilis and Jacob Bernoulli[edit] Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Properties[edit] Starting at a point goes toward is finite. . , where to the origin. Dimension (mathematics and physics) A diagram showing the first four spatial dimensions. 1-D: Two points A and B can be connected to a line, giving a new line segment AB. 2-D: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD. 3-D: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-D: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
In physical terms, dimension refers to the constituent structure of all space (cf. volume) and its position in time (perceived as a scalar dimension along the t-axis), as well as the spatial constitution of objects within—structures that correlate with both particle and field conceptions, interact according to relative properties of mass—and are fundamentally mathematical in description. The concept of dimension is not restricted to physical objects. A tesseract is an example of a four-dimensional object. Superstring theory. 'Superstring theory' is a shorthand for supersymmetric string theory because unlike bosonic string theory, it is the version of string theory that incorporates fermions and supersymmetry. Since the second superstring revolution the five superstring theories are regarded as different limits of a single theory tentatively called M-theory, or simply string theory.
Background[edit] The deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, galaxies, super clusters), with quantum mechanics, which describes the other three fundamental forces acting on the atomic scale. The development of a quantum field theory of a force invariably results in infinite (and therefore useless) probabilities. Evidence[edit] Superstring theory is based on supersymmetry. Extra dimensions[edit] See also: Why does consistency require 10 dimensions? Number of superstring theories[edit] The five superstring interactions. Tesseract. A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope".[1] The tesseract is the four-dimensional hypercube, or 4-cube. According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες ("four rays"), referring to the four lines from each vertex to other vertices.[2] In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract.
" Some people[citation needed] have called the same figure a tetracube, and also simply a hypercube (although a tetracube can also mean a polycube made of four cubes, and the term hypercube is also used with dimensions greater than 4). Geometry[edit] Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. A tesseract is bounded by eight hyperplanes (xi = ±1). See also[edit] Convex regular polychoron. The tesseract is one of 6 convex regular polychora In mathematics, a convex regular polychoron is a polychoron (4-polytope) that is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions). These polychora were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.
Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher-dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no exact three-dimensional equivalent. Properties[edit] Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog of Euler's polyhedral formula: where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.). Visualizations[edit] See also[edit] References[edit]
Schläfli–Hess polychoron. In four-dimensional geometry, Schläfli–Hess polychora are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes). They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra. History[edit] Four of them were found by Ludwig Schläfli (the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell) while the other six were skipped because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2).
Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder. Names[edit] Symmetry[edit] Table of elements[edit] 4th dimension. Four-dimensional space. In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing).
Spacetime is not a Euclidean space. History[edit] An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. One of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension?
In 1908, Hermann Minkowski presented a paper[8] consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.[9] But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. Vectors[edit] Pi. The golden ratio. Logarithmic spiral. Golden rectangle. Pi. Golden ratio.
Dimensions. Convex regular polychoron. Tesseract. Dimension (mathematics and physics) Golden spiral. Logarithmic spiral.