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fractions. In particular, I let h and k assume values 1 through 1.5. (The program code provides additional details.) Even if we place infinitely many Ford circles, none will overlap, and each will be tangential to the x axis. We can confirm this visually by magnifjmg the froth. When I showed colleagues my computer graphics of Ford circle froth, they were at first confused by the occasional instances of circles within circles. (In fact, the only other clear diagrams I had seen of Ford circles were in Rademacher’s mathematics text [11, and in it the figures were hand-drawn diagrams containing only about 13 circles.) These anomalous interior circles, which I call “Chrysler circles,” arise from the fact that my computer programs produce various h/k values that are equivalent, for example 1/2, 2/4,4/8.. . .This, in turn, produces circles with different radii but having the same x coordinate. In addition, I also plot circles at (h/k, -1/2P) to increase the symmetry of the representation and heighten its aesthetic appeal. Note that two fractions are called “adjacent”if their Ford circles are tangential . Any fraction has, in this sense, an infinitude of adjacents [2].Any circle can have an infinitude of tangential circles. Consider a godlike archer who launches an arrow at the Ford froth. Depending on where the archer aims, the outcome is different. To understand this, place the virtual archer high up-that is, select a position above the Ford froth with an appropriately large y value. To simulate the shooting of the arrow, next draw a vertical line from the location of the archer (e.g. at x = a).Trace the arrow’s straight-line trajectory as gravity pulls it down to the x axis. It turns out that if the archer’s position is at a rational point ( a is rational ), the line must pierce some Ford circle and hit the x axis exactly at the circle’s point of tangency. However, when the archer’s position is at an irrational number (a non-repeating, endless decimal value such as K = 3.1415 . . .), it cannot pass directly to the x axis from a Ford circle. In other words, the arrow must kavr every circle it enters. However , as I mentioned previously,every circle it leaves is completely surrounded by a chain of adjacents. Therefore the archer’s arrow travelling along x = a must enter another circle. This is true for all the Ford circles it pierces. Therefore,when the godlike archer is located at an irrational point, the archer’sarrow must pass through an infinity of circles! This all relates to the fact that even though there are an infinite number of rational and irrational numbers, the infinite number of irrationals is, in some sense, greater than the infinite number of rationals. Three-dimensional froth renditions using spheres instead of circles, although not shown in this abstract , also have considerable aesthetic appeal. References 1. H. Rademacher, Ifighn-Mathmnaticsfrom an Elrmentary P o d o f V i m (Boston.MA: Birkhauser, 1983). 2. L.R. Ford, “Fractions,”Anim’canMalherfuztir.! Monthly45 (1938) pp. 5 8 M 1 . PROJECTIONS FROM ACTUAL,ITY Quentin Williams, 10 Nevi1 Road, Bishopston, Bristol BS7 YEQ, England. Received 18May 1995.AccepdJM publication 6.r RogerR Mulina. Paintings are often wrongly described as “photographically realistic.”Indeed, they more usually impose on the viewer quite different conditions of accommodation and assimilation than do photographs. Those copied from fixed photographs usually look inert and embarrassingly inaccurate, and few that are copied from any source show genuine photographic qualities. Perhaps a painting can appear photographically realistic only when the aesthetic values of painting are absent. In any event, a photographically realistic appearance has best been achieved when a painting is copied from a real image projected by a camera obscura or camera lucida. An extremely small number of master painters, using this method, achieved a unique form of painting that produced the equivalent of color photographs, 300 years before the advent of color photography. tical purposes, in using the camera obscura. Apart from the now famous light spots he felt obliged to copy verbatim from the ground-glass screen, and...

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