The purpose of this note is to draw attention to two infinite families of antiprisms that differ slightly from the usual Keplerian kind. These figures have surely been noticed earlier by geometers and philomorphs, but I was unable to track down a reference to them. My attention was drawn to these figures by a Christmas gift that my family received recently. The gift was a cake in a box of a rather unusual shape (see Figure 1). On examining the box, I saw that it could be characterized variously as a deformed antiprism, an unfolded pyramid or a truncated prism. The box had two square faces parallel to each other, with the line joining the centres of the squares being perpendicular to the plane of either square. The squares were rotated 45° relative to each other, exactly as in a square antiprism, but one square had only half the area of the other. The separation of the squares was such that, of the eight triangles linking them, four were identical equilateral triangles while the rest were identical isosceles triangles.

In his letter in the March 1995 issue of the Gazette (79, p. 123), John Sharp points out that the correct viewing position for a picture is with the eye opposite the central vanishing point, that is, on the normal to the picture plane at the point of meeting of all lines in the scene represented that are perpendicular to the picture plane. In order to obtain a true view of the scene depicted, it is also necessary that the eye be placed at the correct distance from the picture.

The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles. For that reason it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view.

Any position of the human body (or of any other jointed mechanism) can be described by giving the angles of joints. The number of angles needed for an unambiguous description is the number of degrees of freedom of the mechanism. For example, the position of a hinge joint is described by just one angle: a hinge allows only one degree of freedom. The human knee is a hinge. The ankle, however, allows rotation about two axes – you can tilt your foot toes up or toes down, and you can also rock it sideways so that the sole faces inwards towards the other foot – so it gives two degrees of freedom. The hip is a ball and socket joint allowing rotation about any axis through the centre of the ball, but any position can be described by just three angles (measured, for example, in three planes at right angles to each other), so it allows three degrees of freedom. In total, there are six degrees of freedom in each leg, making twelve in all, and suggesting that we need twelve equations of motion to describe walking. If we took account of the flexibility of the foot and the movements of the arms, we would need more.

Solid shapes which pack together to fill space cover a large and varied range. I recently found five such polyhedra which I have not seen described elsewhere. The names which I have used for them here are rather ferocious I am afraid, but that just happens to be the way the names of polyhedra work.

The prime numbers are often called the building blocks of number theory, a classic case of a ‘sine qua non’. If the corpus of the theory of numbers is looked upon as an architectural pile then the primes will be found amongst its foundations, and amongst its walls and buttresses, and indeed amongst the array of pinnacles and turrets which burst forth from it and stand proud, alone and magnificent. One such pinnacle had been discussed and planned for a hundred years before it was finally constructed in 1896 by two men working totally independently of each other (collapse of stout analogy). With this achievement, the prime numbers, those familiar yet raw beasts of mathematics, in one sense had been tamed. The long journey, which had begun with the tentative definitions of the ancient Greeks, was completed.

Given the critical dependence of our civilisation on mathematics, it is paradoxical that it lacks any general acclaim or popularity, and usually invokes fear and mysticism. Perhaps this is because, for most of us, the process of acquiring mathematical skills is somewhat protracted and painful. It is often perceived as a nightmare topic, which engenders a defence mechanism of open ignorance that is often lauded. In truth, most people can become reasonably competent. However, the time and effort needed to acquire the formal rigour for even a modest capability is often seen as excessive by modern standards.

The Brocard points P and P' of the triangle ABC, illustrated in Figure 1, are such that ∠PAB = ∠PBC = ∠PCA = ω and ∠P'BA = ∠P'CB = ∠P'AC = ω. Problem 78.A in the March 1994 edition of the Mathematical Gazette required a proof that the maximum length of PP' is R/2, where R is the circumradius of triangle ABC. The solution given by various solvers in the November 1994 edition of the Gazette is essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924. In the present paper various properties of the triangle and its Brocard points are established in terms of the lengths of the sides of the triangle. Some of the results may be new, but the methods used encompass ‘Mathematics Ancient and Modern’ in the guises of homogeneous areal coordinates and computer algebra!

Everyone knows that the condition for the quadratic to have two equal roots, i.e. to have a repeated root, is that its discriminant should be zero. We should remark, at the outset, that we are concerned only with ordinary polynomials whose coefficients are complex numbers. Indeed, little is lost if a reader assumes that all our polynomials are real, i.e. have real numbers for all their coefficients, though their complex roots must be considered as well as their real ones.

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:

1. (1) Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.

2. (2) Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.

We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].

The classical theorems of Ceva and Menelaus make assertions about the value of certain products of ratios of lengths in configurations in the affine plane. We shall use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc. Several results of this kind are known. Apart from the classical results there are, for example, Ceva's and Menelaus' Theorems for n-gons, Hoehn's Theorem for pentagrams [1], and the Selftransitivity Theorem of [2].