Andrew Palfreyman’s article [1] reminds us of the result (1)

$${\rm{ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}\,2{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ 3 = }}\,\pi {\rm{, }}$$

Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.

The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

It is hard to understand why the problem of tracing a conic from its equation is usually subordinated to if not identified with the problem, intrinsically more difficult, of calculating axes, or why students are not encouraged to bring their knowledge of geometrical properties of the curve to the drawingboard. The essence of tracing a curve is to render the sketching of it as accurate as may be desired, and I venture to assert that for this purpose the discovery of axes and vertices is not worth a tithe of the labour which it demands. Prom the advice in some of our text-books one would imagine that the ellipse could be drawn accurately from its four vertices alone, the hyperbola from its vertices and its asymptotes, and the parabola from its axis and vertex and one or two other points. The truth is, that in any but expert hands these details are wofully insufficient; what is wanted is a multitude of points on the curve, and it matters little whether or not the vertices are included. The student finds satisfaction in drawing the curve from its equation and observing the symmetry that appears when the axes are inserted subsequently.

Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry, together with the allied subject of projective geometry, has been dethroned and in some places almost banished from the scene. While educational reform was certainly needed there is always the danger that the pendulum may swing too far the other way and that insufficient attention may be paid to geometry in its various forms. Much of the difficulty here centres round the elusive nature of the subject: What is geometry? I would like to examine this question in a very general way in the hope that this may clarify the educational reasons for teaching geometry, and for deciding what is appropriate material at different levels.

In 1952 I was invited, with teachers of mathematics from many other schools, to a reception hosted by the Cambridge University Press. The guest of honour was A. W. Siddons, and the purpose was to celebrate the fiftieth anniversary of the initiation of Godfrey and Siddons’ first book: Elementary Geometry. There can be few authors who survive to enjoy such an occasion; fewer still whose book remains in print and continues to sell well after fifty years. But this was a book which marked a turning-point in the teaching of mathematics, and set a standard for mathematical education which remains relevant as we approach the end of the century.

The real projective plane is a classical mathematical object which plays an important role in several areas of mathematics, including topology. This article gives methods for making three-dimensional models of the real projective plane out of wool. By describing these models, and how to make them, the article aims to help the reader to visualise the real projective plane more easily.

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

When the authors met at a geometry conference at Easter 1999, they found that each had discovered and investigated an elegant result (Theorem 2.1) about an octagon inscribed in a conic, Larry Evans algebraically and John Rigby synmetically. This turned out to be a rediscovery, as the result had been discussed by E. H. Lockwood in the Gazette in 1967, in an article that complements the present article. Also, almost a century earlier in 1872, the result had been posed somewhat imprecisely by T. T. Wilkinson as a problem in the Educational Times. There is a surprising connection between a special case of the octagon theorem and the golden cross-ratio.

The year 2000 was the centenary of not only Hubert’s Problems but also Dehn’s solution of the Third Problem, which was the first to be solved. The Third Problem is concerned with the Euclidean theorem that two tetrahedra with equal base and height have equal volume [5, Book XII, Proposition 5].

Today one proves this theorem by integration, showing that the volume of a tetrahedron is a third base times height. This 3-dimensional theorem is the analogue of the 2-dimensional theorem that the area of a triangle is a half base times height. But Euclid did not have access to integration, nor the real numbers, nor the product of real numbers, so he did not use the concept of an area being the product of two lengths or a volume being the product of a numerical area and a length. For instance when he wanted to state the triangle theorem he said that two copies of the triangle were ‘equal’ to a rectangle with the same base and height, where by ‘equal’ he meant here that the two triangles could be cut into pieces and reassembled into the rectangle. So Hilbert’s problem was equivalent to asking whether the same could be said about a tetrahedron. Can three copies of the tetrahedron be cut into pieces and reassembled into a rectangular prism with the same base and height? Dehn proved that it was impossible, and later Boltianskii simplified his proof. Dehn’s proof invented a real number invariant of a solid, that was invariant under cutting and reassembling, and he showed that two particular tetrahedra with the same base and height had different values of his invariant.

David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900. The first part of the lecture, a preamble to his announcement of the now-famous 23 problems, began with the words:

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].

In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem: Of two unequal angles, the larger has the shorter bisector (see [1, 2]).

Sturm passed the problem on to other mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask.

We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.