The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenly-spaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball..
French curves are plastic (or wooden) templates having an edge composed of several different curves. French curves are used in drafting (or were before computer-aided design) to draw smooth curves of almost any desired curvature in mechanical drawings. Several typical French curves are illustrated above.While an undergraduate at MIT, Feynman (1997, p. 23) used a French curve to illustrate the fallacy of learning without understanding. When he pointed out to his colleagues in a mechanical drawing class the "amazing" fact that the tangent at the lowest (or highest) point on the curve was horizontal, none of his classmates realized that this was trivially true, since the derivative (tangent) at an extremum (lowest or highest point) of any curve is zero (horizontal), as they had already learned in calculus class...
A linkage with six rods which draws the inverse of a given curve. When a pencil is placed at , the inverse is drawn at (or vice versa). If a seventh rod (dashed) is added (with an additional pivot), is kept on a circle and the locus traced out by is a straight line. It therefore converts circular motion to linear motion without sliding, and was discovered in 1864. Another linkage which performs this feat using hinged squares had been published by Sarrus in 1853 but ignored. Coxeter (1969, p. 428) shows that
A linkage invented in 1630 by Christoph Scheiner for making a scaled copy of a given figure. The linkage is pivoted at ; hinges are denoted . By placing a pencil at (or ), a dilated image is obtained at (or ).
A linkage which draws the inverse of a given curve. It can also convert circular to linear motion. The rods satisfy and , and , , and remain collinear while is kept parallel to . This condition holds automatically if .Coxeter (1969, p. 428) shows that if , then
A mechanical device consisting of a sliding portion and a fixed case, each marked with logarithmic axes. By lining up the ticks, it is possible to do multiplication by taking advantage of the additive property of logarithms. More complicated slide rules also allow the extraction of roots and computation of trigonometric functions.According to Steinhaus (1999, p. 301), the principle of the slide rule was first enumerated by E. Gunter in 1623, and in 1671, S. Partridge constructed an instrument similar to the modern slide rule. The Oughtred Society, a group of slide rule collectors, claims that W. Oughtred invented the first slide rule in 1622.The slide rule was an indispensable tool for scientists and engineers through the 1960s, but the development of the desk calculator (and subsequently pocket calculator) rendered slide rules largely obsolete beginning in the early 1970s...
Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..
Napier's bones, also called Napier's rods, are numbered rods which can be used to perform multiplication of any number by a number 2-9. By placing "bones" corresponding to the multiplier on the left side and the bones corresponding to the digits of the multiplicand next to it to the right, and product can be read off simply by adding pairs of numbers (with appropriate carries as needed) in the row determined by the multiplier. This process was published by Napier in 1617 an a book titled Rabdologia, so the process is also called rabdology.There are ten bones corresponding to the digits 0-9, and a special eleventh bone that is used the represent the multiplier. The multiplier bone is simply a list of the digits 1-9 arranged vertically downward. The remainder of the bones each have a digit written in the top square, with the multiplication table for that digits written downward, with the digits split by a diagonal line going from the lower left..
A puzzle involving disentangling a set of rings from a looped double rod, originally used by French peasants to lock chests (Steinhaus 1999). The word "baguenaudier" means "time-waster" in French, and the puzzle is also called the Chinese rings or Devil's needle puzzle. ("Bague" also means "ring," but this appears to be an etymological coincidence. Interestingly, the bladder-senna tree is also known as "baguenaudier" in French.) Culin (1965) attributes the puzzle to Chinese general Hung Ming (A.D. 181-234), who gave it to his wife as a present to occupy her while he was away at the wars.The solution of the baguenaudier is intimately related to the theory of Graycodes.The minimum number of moves needed for rings is(1)(2)where is the ceiling function, giving 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, ... (OEIS A000975). The generating function for these numbers is(3)They are also given by..
A graphical plot with abscissa given by the number of consecutive numbers constituting a single period and ordinate given by the correlation ratio . The equation of the periodogram iswhere each of the terms of the sequence consists of a simple periodic part of period , together with a part which does not involve this periodicity , so is the standard deviation of the s, is the standard deviation of the s, and is the number of periods covered by the observations.
The finite volume method is a numerical method for solving partial differential equations that calculates the values of the conserved variables averaged across the volume. One advantage of the finite volume method over finite difference methods is that it does not require a structured mesh (although a structured mesh can also be used). Furthermore, the finite volume method is preferable to other methods as a result of the fact that boundary conditions can be applied noninvasively. This is true because the values of the conserved variables are located within the volume element, and not at nodes or surfaces. Finite volume methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks.Hyman et al. (1992) have derived local, accurate, reliable, and efficient finite volume methods that mimic symmetry, conservation, stability, and the duality relationships between the gradient,..
Consider a network of resistors so that may be connected in series or parallel with , may be connected in series or parallel with the network consisting of and , and so on. The resistance of two resistors in series is given by(1)and of two resistors in parallel by(2)The possible values for two resistors with resistances and are therefore(3)for three resistances , , and are(4)and so on. These are obviously all rational numbers, and the numbers of distinct arrangements for , 2, ..., are 1, 2, 8, 46, 332, 2874, ... (OEIS A005840), which also arises in a completely different context (Stanley 1991).If the values are restricted to , then there are possible resistances for 1- resistors, ranging from a minimum of to a maximum of . Amazingly, the largest denominators for , 2, ... are 1, 2, 3, 5, 8, 13, 21, ..., which are immediately recognizable as the Fibonacci numbers (OEIS A000045). The following table gives the values possible for small .possible resistances11234If..
Let be the resistance distance matrix of a connected graph on nodes. Then Foster's theorems state thatwhere is the edge set of , andwhere the latter sum runs over all pairs of adjacent edges and is the vertex degree of the vertex common to those edges (Palacios 2001).