Curious quaternions November 2004 John Baez is a mathematical physicist at the University of California, Riverside. He specialises in quantum gravity and n-categories, but describes himself as "interested in many other things too." His homepage is one of the most well-known maths/physics sites on the web, with his column, This Week's Finds in Mathematical Physics, particularly popular. The birth of complex numbers Like many concepts in mathematics, complex numbers first popped up far from their main current area of application. The birth of the complex The particular challenge that brought mathematicians to consider what they soon called "imaginary numbers" involved attempts to solve equations of the type ax2+bx+c=0. To solve such an equation, you must take square roots, "and sometimes, if you're not careful, you take the square root of a negative number, and you've probably been repeatedly told never ever to do that. The complex number with real part 1 and imaginary part 2 where and and up by . is the same as .
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. Properties 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 The following table shows some 2-dimensional projections of these polychora. See also References External links
Quaternions and spatial rotation Using quaternion rotations A Euclidean vector such as (2, 3, 4) or (ax, ay, az) can be rewritten as 2 i + 3 j + 4 k or ax i + ay j + az k, where i, j, k are unit vectors representing the three Cartesian axes. A rotation through an angle of θ around the axis defined by a unit vector is represented by a quaternion using an extension of Euler's formula: The rotation is clockwise if our line of sight points in the same direction as u→. It can be shown that this rotation can be applied to an ordinary vector in 3-dimensional space, considered as a quaternion with a real coordinate equal to zero, by evaluating the conjugation of p by q: using the Hamilton product, where p′ = (px′, py′, pz′) is the new position vector of the point after the rotation. In this instance, q is a unit quaternion and It follows that conjugation by the product of two quaternions is the composition of conjugations by these quaternions. which is the same as rotating (conjugating) by q and then by p. . Example and Let
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 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. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R. 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? Little, if anything, is gained by representing the fourth Euclidean dimension as time. Vectors This can be written in terms of the four standard basis vectors (e1, e2, e3, e4), given by Geometry
Sexagesimal Origin It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn. A traditional counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full. According to Otto Neugebauer, the origins of the sixty-count was through a count of three twenties. Usage Babylonian mathematics Numbers larger than 59 were indicated by multiple symbol blocks of this form in place value notation. ) to represent zero, but only in the medial positions, and not on the right-hand side of the number, as we do in numbers like 13,200. Other historical usages Notation Modern usage Fractions
Many-worlds interpretation The quantum-mechanical "Schrödinger's cat" paradox according to the many-worlds interpretation. In this interpretation, every event is a branch point; the cat is both alive and dead, even before the box is opened, but the "alive" and "dead" cats are in different branches of the universe, both of which are equally real, but which do not interact with each other. The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. The original relative state formulation is due to Hugh Everett in 1957. Later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and 1970s. The decoherence approaches to interpreting quantum theory have been further explored and developed, becoming quite popular. Before many-worlds, reality had always been viewed as a single unfolding history. Outline Wojciech H.
Duodecimal The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. Origin Languages using duodecimal number systems are uncommon. Places Comparison to other numeral systems A dozenal multiplication table
The Landscape multiverse Quaternary numeral system It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties. Relation to other positional number systems Relation to binary As with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, quaternary does not enjoy the same status. By analogy with bit, a quaternary digit is sometimes called a crumb. Occurrence in human languages Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16 (not 10). Hilbert curves Genetics
Eternal inflation Eternal inflation is predicted by many different models of cosmic inflation. MIT professor Alan H. Guth proposed an inflation model involving a "false vacuum" phase with positive vacuum energy. Parts of the Universe in that phase inflate, and only occasionally decay to lower-energy, non-inflating phases or the ground state. In chaotic inflation, proposed by physicist Andrei Linde, the peaks in the evolution of a scalar field (determining the energy of the vacuum) correspond to regions of rapid inflation which dominate. Alan Guth's 2007 paper, "Eternal inflation and its implications", details what is now known on the subject, and demonstrates that this particular flavor of inflationary universe theory is relatively current, or is still considered viable, more than 20 years after its inception.  Inflation and the multiverse Both Linde and Guth believe that inflationary models of the early universe most likely lead to a multiverse but more proof is required. History
Level IV: Ultimate ensemble