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"How many spheres of radius r may simultaneously be tangent to a fixed sphere of the same radius?" This question goes back at least to the year 1694, when it was considered by Isaac Newton and his contemporary David Gregory. Johann Kepler had already shown in 1611 [8] that twelve outer spheres could be arranged around a central sphere, and now Gregory claimed that a thirteenth could be added. Newton disagreed, but neither man proved his claim, and it was not until 1874 that a proof was found, substantiating Newton's conjecture. (R. Hoppe's original proof is described in [2]; more recently, proofs have been found by Giinter [7] and by Schiitte and van der Waerden [10]. Perhaps the most elegant proof known is the one given by John Leech [9]. For a history of the problem, see [6].) It's worth noting that the 4-dimensional version of this problem is still unsolved; no one knows how to arrange more than twenty-four hyperspheres around a central hypersphere, but neither has it been shown that a twenty-fifth cannot be added. In higher dimensions the situation is even less well understood, with the remarkable exception of dimensions 8 and 24, for which the maximum contact numbers are precisely known [1]. The source of difficulty in the original Gregory-Newton problem (and the reason it took 180 years to be solved) is that there is almost room for a thirteenth sphere; the twelve spheres of Kepler can be pushed and pulled in all sorts of ways, and it's credible that some sort of fiddling could create a space big enough to accommodate an extra sphere. This note will deal with a related question, in the spirit of Erno Rubik: if we label the twelve spheres and roll them over the surface of the inner sphere at will, what permutations are achievable? The answer is surprising, and the proof requires only the rudiments of analytic geometry and group theory. One fringe benefit of this enterprise is that it leads to a natural coordinatization of the vertices of the regular icosahedron. An equivalent formulation of the problem is gotten by considering only the centers of the spheres: we imagine twelve vertices, free to move on a sphere of fixed radius R = 2r about a fixed origin but subject to the constraint that no two vertices may ever be closer together than R. We will find it convenient to switch back and forth between the two formulations. For definiteness, put r = 1, 1 = 1. To begin, we must prescribe an initial configuration for the outer spheres. One possibility that springs to mind is to arrange the twelve vertices to form a regular icosahedron. As we'll see, any two distinct vertices of the icosahedron inscribed in the unit sphere are at least

The formula for the Eucidean distance between two points in the plane leads, early in the study of analytic geometry, to the definitions of the classical conic sections-the circle, ellipse, and hyperbola-in terms of distances to fixed points. Additionally, the parabola can be defined in terms of distances to a fixed point and a fixed line. After modest additional study in the area, the student becomes aware that the ellipse and hyperbola have comparable " point-line" definitions. It is this point-line approach to the conic sections I wish to pursue here-not in the context of Eucidean distance but using, instead, the taxicab metric defined in the coordinate plane by

When one considers the prodigious prolificity of theorems established by Apollonius and others upon the cone and the conic sections, it may be surprising to observe that certain aspects still remain unexplored. Although Apollonius (3rd century B. C.), a master of the synthetic method in geometry, established 387 propositions on the cone during his forty-year life time, his results have been extended by many others even down to modern times. In the latter part of this paper the author wishes to contribute one new development on the cone, but first proposes to invite the reader's attention to some earlier aspects of the cone's illustrious career. The discovery of the conic sections is attributed to Menaechmus in the 4th century B. C. He employed conics to solve the famous Delian problem -the Duplication of the Cube. Apollonius investigated the non-focal properties of the conic, its conjugate diameters, asymptotes, and the harmonic property of the pole and polar for the case in which the pole lies outside the curve. Not until the 17th century were any further major advances made on conics. The latter theorem of Apollonius was then generalized by the genius of Desargues. Among the lost works attributed to Euclid were four Books on Conics, the last one however not being completed by him. It seems that Apollonius took over the completion of these and added four more of his own. He defined the basic cone as one having an oblique axis and set on a circular base. This was called a scalene cone. He proved that all sections parallel to the base are also circular; and that there is another parallel set of circular sections, "sub-contrary" to these. In the latter part of this paper the author will invoke modern analytic geometry methods to determine the dihedral angle between these two sub-contrary sets of circular sections. Apollonius derived the equivalent of the focus-directrix properties of the sections of this general cone, obtaining results equivalent to the modern analytic definitions of the conics. However, he made no reference to a directrix nor did he evince any knowledge thereof. Neither did he utilize a focus for the parabola. He showed that the conic has the same property with reference to any diameter as it has with reference to its axis (a special diameter). He expressed the fundamental property of each conic by equations between areas of associated rectangles. These relations were equivalent to the Cartesian equation of the conic. However, it remained for Pappas to state the modern locus definition for the conics. The latter first introduced the concept of eccentricity. Apollonius assigned the appropriate names to the sections, parabola (equivalent), ellipse (falls short) and hyperbola (exceeds). Straight lines and circles were then known as "plane loci", whereas conics were spoken of as "solid loci".

Let's call a solution of (1) in integers an Almost Pythagorean Triple (APT). In analytic geometry (1) represents a hyperboloid of revolution of one sheet, a doubly ruled surface. The tangent plane at a point cuts this surface in two straight lines called rulings. There are two sets of rulings, and each line of one set meets no other line of that set, but meets every line of the other set except one line, to which it is parallel. The transformation which sends the point (x, y, z) into (-x, -y, z) sends each ruling into a parallel ruling of the other set. We shall show that the integral solutions of (1) lie on such rulings, and that we can use this fact to find all APT's. With a table of squares you can make a list of APT's. The first six are: (5,5,7), (4,7,8), (8,9,12), (7,11,13), (11,13,17), (10,15,18). You may notice that integers with last digit 2,3,7,8 repeatedly appear as the z-coordinate of an APT. Those ending in 2 or 7 form an arithmetic progression, as do the corresponding x and y coordinates. Likewise with those ending in 3 or 8. These arithmetic progressions give us the formulas

You recall with what delight you first discovered that many geometrical and physical situations can be analyzed, and problems solved, by means of diflerential and integral calculus. Fortunately this pleasure of discovery can be continued, for many parts of more advanced mathematics and physics use the concepts of calculus as basic tools. This is particularly true in the study of properties of curves. and surfaces in space. Just as in plane analytic geometry we ordinarily use equal scales on two directed axes at right angles to each other in locating points (x,y), in solid analytic geometry we use equal scales on three concurrent directed axes which are at right angles each to each in locating points (x,y,z). It is clear that if two of these directions are fixed, then there are exactly two possible directions for the third, as in Figures 1 and 2. In the first of these figures, an ordinary right-hand

Thle individual wh1o encounters hyperbolic geometry for the first time in such a book as the one by Wolfe [31 has the stimu-lating experience of developing the analogue of a substantial part of euclidean geometry uising the same essential spirit and methods as those of Euclid. T'his is followed by a development of hyperbolic trigonometry which provides the reader with all the rudimentary data for solving triangle problems in the hyperbolic plane. IIowever, only a few texts, like t'he one of Sommerville [2], make an attempt to dleveloe a hyperbolic analytic geometry in a way which parallels thie usual freshmaan course in analytics, and the student hlas only to attem,r)t as simple a problem as the determination of the eqtuation of the altitude of an arbitrary triangle to realize the inadequacy of his equipment. Other texts, such as that by Coxeter [I] approach hiyperbolic geometry from the viewpoint of pro,jective geometry. To a student familiar with projective geometry it is then a relatively simple matter to think of the hyperbolic plane as embedded in the projective d1arne and develop its analytic geometry in the usual projective coordinates. Fronm the beginner's point of view it seems an unniatural way of getting analytic results for the hyperbolic plane. Tre aim of this paper is to present a brief development of hyperbolic analytic geometry following the usual procedures of analytics quite familiar to students. Only a single use is made of calculus, and thlis could probably have been avoided. From the discussion follows naturally thie idea of introducing a different coordinate system wh-tic'h amounts in fact to embedding in the proj ective plane. Tnus the ,)rocedure of establishing hyperbolic geometry as a subgeometry of projective geometry is obtained naturally from within in an elementary way, With the exception of the last section, which utilizes a little projective geometry and the already noted single use of calculus, nothing is assumee(I beyond analytic geometry and an introduction to nyperbolic g,eometry. rhe question of applicability of the results and methods to binocular sensory space [4] remains to be examined.

Algebra as a tool has been used in problems of geometry. But we are always far behind in using a new tool when discovered. There is little hesitation in using determinants in elementary algebra. I don' t see why we should hesitate to use matrices in freshmen analytic geometry. I believe it brings a new interest and also it eliminates some tedious arithmetic. 1. Definition: We are used to writing a point of the plane as P = (x, y). Let us ignore " ," and write it as (x y). We call (x y) a row matrix corres