In designing this pretty puzzle, Dudeney fixed the dimensions of the cross to conform to the 5×5 grid as shown in Figure 14.1. A. E. Hill's startling solution, given in (Dudeney 1926a), is reproduced in Figure 14.2. Most attractive, it possesses 2- fold rotational symmetry. However, a certain degree of mystery has accompanied this dissection. First, no satisfactory explanation of the dissection method has been given. Second, no information seems to be available on who A. E. Hill was. Although I have had no luck in identifying Mr. Hill, I have found a reasonable explanation for the dissection method.

The dissection referred to here is that of a Greek Cross to a square in four pieces, which is found in (Lemon 1890) and reproduced in Figure 10.2. And although Eli Lemon Sheldon lived an unusual life, it appears that this dissection should not be attributed to him. Jerry Slocum (1996) reproduces a 2½ inch × 4½ inch advertising card for Scourene, a scouring soap manufactured by the Simonds Soap Company of New York City.

John Jackson (1821) posed what may be the earliest published dissection of curved figures. He converted a circular table top (a disk) to the seats of two oval stools with open handholds. His simple 8-piece dissection is shown in Figure 15.1. It has a circular cut centered on, and half the diameter of, the original disk. Two straight cuts, of diameters perpendicular to each other, complete the dissection. Half of the four inner and half of the four outer pieces reassemble to make each stool seat.

Jackson's dissection is not minimal, since Loyd (Inquirer, 1901a) gave the 6-piece dissection in Figure 15.2. The disk is first cut into the two shapes that form the Chinese monad (or Yin and Yang symbol).

The puzzle in the preceding reminiscence was a physical model of the dissection shown in Figure 1.1. The two heptagons on the right are divided by line segments into nine pieces, which then rearrange to form the 7-pointed star on the left. A geometric dissection is a cutting of one or more figures into pieces that can be rearranged to form other figures. Since the constructions are precise, this activity falls within the realm of mathematics, though it is of a decidedly recreational flavor. This is a book devoted solely to such mathematical constructions.

The non-Euclidean hyperbolic geometry of Carl Friedrich Gauss, János Bolyai, and Nikolay Lobachevsky first shocked and then revolutionized the mathematical world. The same cannot be said of the dissection work of P. Gerwien, Farkas Bolyai, and William Wallace. But their work does give us a method of dissecting any irregular polygon to any other irregular polygon of equal area, using a finite number of pieces. The methods of Lowry (1814), Wallace (1831), Bolyai (1832), and Gerwien (1833) always work, though they often produce a great abundance of pieces.

What a letdown! And although three-dimensional dissections were only briefly in that brightest of spotlights, we shall find that there have been some nice examples produced even in the shadows. We start our story a half century before dissection's fifteen minutes of fame. In an exchange of letters between Carl Friedrich Gauss and Christian Ludwig Gerling in 1844, Gauss had lamented that the only known procedure for proving the equality of the volume of a polyhedron with the volume of its mirror image was a method of exhaustion.

What was perhaps Sam Loyd's biggest goof occurred with the Smart Alec Puzzle. Loyd (Inquirer, 1901d) posed the puzzle, accompanied by a picture of a rude young man who interrupts the speaker to show off his knowledge. The attention-getting premise was that the smart aleck had assumed the puzzle to be the familiar one of cutting a mitre into four pieces of identical shape, whereas it was actually a new puzzle, that of dissecting a mitre into the fewest number of pieces that rearrange into a square. The artwork that accompanied the puzzle statement appears on the previous page. It has a charm that is irresistible.

Loyd's purported 4-piece solution, shown in Figure 23.1, was given in (Loyd, Inquirer, 1901e). And here the trouble begins, because Loyd's solution does not yield a square.

Dare we attempt to frame a fearful metaphor, using the last two stanzas of William Blake's “The Tyger,” from his Songs of Experience (1794)? Might the structure of polygons and stars be a sort of geometric meter, and the arrangement of cuts in the dissections be geometric rhyme? Do not our stars and other dissected figures luminesce in a cold mathematical stillness? Do not the best of the dissections fit together with a cunning that is diabolical? Dare we illustrate the preceding excerpt with a haloed rendition of Harry Lindgren's dissection of a {12/5} to a Latin Cross (Figure 3.1)?

As the rhyme in Blake's poem is not absolute, so also is the symmetry in lindgren's dissection not absolute. Lindgren's dissection comes close but does not possess reflection symmetry. Reflection symmetry is present when there exists a line of reference such that what is on one side of the line is exactly mirrored, or reflected, on the other side. As shown in Figure 3.2, if we split just one of the pieces into two pieces, then both the star and the cross will possess reflection symmetry about the dotted lines.

Johannes Kepler was evidently the first person to perform a unified study of how polygons can fit together to tile the plane. It appeared in Book II of Part V of Harmonice Mundi, collected in (Kepler 1940). It is impossible to know whether Kepler would have performed this study outside the framework of Harmonice Mundi. But the tilings that he investigated are so beguiling that he probably could not have resisted.

The preceding bit of fabricated exasperation, which we assign to the Belgian Paul Busschop, underscores the lack of hoopla that accompanied his introduction of a new dissection technique in the 1870s. His first dissection was labeled as empirical by Eugène Catalan (1873), although Catalan (1879) no longer retained this characterization. In his role as journal editor, Catalan removed all but the claims of two more dissections in (Busschop 1876), due to lack of space. (However Catalan was kind enough to supply Busschop's full manuscript to Édouard Lucas, who included the dissections in (Lucas 1883).)

What colossal impertinence in that young generation at Cambridge! Or was it just a colossal impatience, given the few solid dissection problems that had been posed and solved? Surely it was implacability in the face of problems that seemed so much more difficult than their two-dimensional analogues. It had been a natural step for Richmond (1943) to pose the dissection for cubes realizing 33 + 43 + 53 = 63. And considering that there were no previous dissections for any solution to x3 + y3 + z3 = w3, twelve pieces might have seemed a veritable bargain.

The dissection Cardano (1663) described is an example of the step technique, so named because of the resemblance to stair steps. In its simplest form, the step technique cuts a rectangle in a zigzag pattern, alternating horizontal cuts of one length with vertical cuts of another. By shifting the two resulting pieces by one step relative to each other, we form a different rectangle. Figure 7.1 illustrates the case for converting a (3a × 4b)-rectangle to a (4a × 3b)-rectangle.

[with apologies to Raymond Chandler]

The trail was as cold as the Minnesota tundra that Ernest Irving Freese had fled so many years before. The faded letter from C. Dudley Langford to Harry Lindgren had a 1964 date. It outlined a strategy to obtain the Los Angeles architect's magnum opus on geometric dissections – a strategy as hopeless as announcing that all the sunbaked jurisdictions in Southern California should assemble into one coherent whole. Lindgren had never persuaded Freese's widow to share the manuscript with him. And Freese's blueprints, reportedly containing some 200 plates with over 400 figures, were certainly not on file at City Hall.

Needham (1959) gave additional information about the Chinese dissection that estimates π. Anonymous (1897) and Newing (1994) provided evidence of the strain in the relationship of Loyd and Dudeney. The graphics and/or text of Loyd's Sedan Chair Puzzle, Guido Mosaics Puzzle, Mrs. Pythagoras' Puzzle, the Cross and Crescent, the Puzzle of the Red Spade, and The Smart Alec Puzzle are reproduced from (Loyd 1914).

An English translation and commentary for Kepler's polygon tilings appeared in (Field 1979). A biography of Kepler's life with a careful explication of his work can be found in (Caspar 1959), and a survey of Kepler's mathematical contributions is given in (Coxeter 1975). Hogendijk (1984) surveyed early knowledge of the structure of polygons with an odd number of sides. A comprehensive treatment of tessellations and tilings can be found in (Grünbaum and Shephard 1987). See Lines (1935), Coxeter (1963), and Wenninger (1971) for discussions of semiregular polyhedrons. Kepler (1940) first discussed the rhombic dodecahedron, which is also mentioned in (Coxeter 1963).

I rolled out some heavy machinery at the beginning of Chapter 3. Perhaps I should belatedly include a warning by Sir Geoffrey Keynes in (Blake 1967) about interpreting “The Tyger”: “It seems better, therefore, to let the poem speak for itself, the hammer strokes of the craftsman conveying to each mind some part of his meaning Careful dissection will only spoil its impact as poetry.”

The major thrust of aerodynamics in the age of the advanced propeller-driven airplane can be summarized in a word: streamlining. Research in aerodynamics in that age, generally covering the period from about 1930 to 1945, was driven by two practical concerns: (1) the need to reduce the drag on aircraft so they could fly faster and more efficiently, and (2) the problems associated with fluid compressibility as new, high performance propeller-driven aircraft began to reach speeds approaching the speed of sound. That age was characterized by intensive efforts to better understand the mechanisms of drag production, to find ways to better predict drag, to meticulously refine a configuration in order to reduce drag, and to begin to sneak up on the speed of sound without disastrous consequences. The outcome of all that effort was the trend toward more streamlining of aerodynamic bodies.

It was also during that age that the fundamental innovative ideas in aerodynamics that had been articulated at the turn of the century finally began to yield great rewards. Prandtl's boundary-layer theory, first presented in 1904, spread throughout the world of aerodynamics in the late 1920s, allowing airplane designers to make the first intelligent (and sometimes reasonably accurate) predictions of skin-friction drag. The boundary layer concept also helped to explain how flow could separate from a surface; such flow separation is the cause of form drag (pressure drag due to flow separation).

Overview

Resonant vapours as optically nonlinear media

Mutual interactions

So far, we have considered the effect of the laser on the atomic medium separately from the measurement of microscopic dynamics by the polarisation selective detection of transmitted light. In this approximation, the pump laser drives the atoms without suffering significant attenuation or polarisation changes. Conversely, the probe beam, which monitors the optical properties of the atomic medium, changes the microscopic state of the atoms only infinitesimally. This approach guarantees, e.g., that the response of the medium to the probe beam is linear. As stressed before, this assumption of one-sided interactions is always an approximation, since the conservation of energy and angular momentum make it impossible to change either of the two partial systems without compensating changes in the other part.

In 1891, the sky over Germany hosted the first successful manned, heavier-than-air flying machine - the hang glider of Otto Lilienthal (Figure 4.1). If we compare it with Cayley's 1804 glider (Figure 3.1), we can detect little fundamental difference between the two flying machines. Both have fixed wings and horizontal and vertical tail structures. Both designers were knowledgeable and concerned about the static stability of their machines, recognizing that the center of gravity should be ahead of the aerodynamically generated aerodynamic center of the vehicle (notice the ballast weight hanging from the nose of Cayley's glider and the position of Lilienthal's body). Both machines were without any mechanical means of flight control, and both were unpowered gliders. However, there was one fundamental difference: The wing of Lilienthal's glider was a rigidly curved (cambered) airfoil shape, whereas the wing of Cayley's 1804 glider was simply a flat surface.

Atoms

Historical

Early models: atoms as building blocks

The term “atom” was coined by the Greek philosopher Democritus of Abdera (460–370 B.C.), who tried to reconcile change with eternal existence. His solution to this dilemma was that matter was not indefinitely divisible, but consisted of structureless building blocks that he called atoms. According to Democritus and other proponents of this idea, the diverse aspects of matter, as we know it, are a result of different arrangements of the same building blocks in empty space (Melsen 1957; Simonyi 1990). The most important opponent of this theory was Aristotle (384–322 B.C.), and his great influence is probably the main reason that the atomic hypothesis was not widely accepted, but lay dormant for two thousand years. It reappeared only in the eighteenth century, when the emerging experimental science found convincing evidence that matter does indeed consist of elementary building blocks.