Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.

The stereographic projection of the sphere is a perspective projection from a point on the surface onto the opposite diametral plane. Analytical geometry provides a unified method of treating the stereographic projection and problems in spherical trigonometry. The basic result is the equation for the projection of a general circle (eq. 1). This is applied to the polar, equatorial and oblique stereographic projections, and a simple proof of Cayley’s theorem is obtained. Next, the cosine formula for a right-angled spherical triangle and the relation between angles measured along different great circles are derived. Finally, the construction used on astrolabes for determining planetary time is shown to be the stereographic projection of a problem on the sphere, which is solved.

Nice polynomials are polynomials whose coefficients, roots, and critical points are integers. The earliest papers on nice polynomials [1, 2, 3] give an explicit formula for all nice cubics. All the derivations of this formula use Pythagorean triples.

In 1999 the problem of finding, constructing, and classifying nice polynomials was added to the list of unsolved problems [4] in The American Mathematical Monthly. Other papers soon followed, including [5] and [6], with extending the results of nice polynomials to polynomials with coefficients, roots, and critical points in integral domains D. Such polynomials are called D-nice.

Tonight I shall deal with some examples of mathematical reasoning which have in common only one feature: that of being surprising, or, even, of once having been surprising. We may picture them as monsters, waiting to leap on our loose ideas about mathematics and to tear them to shreds.

I shall not concern ourselves with elementary errors. Every one of you, no doubt, can “prove” that all triangles are isosceles, and that 2 = 1 by simple algebra.

Most spirals are continuous spirals, certainly those famous in history such as the traditional Archimedean spiral (r = a + bθ) and the logarithmic spirals (r = aebθ. Here we consider ‘discrete spirals’.

Although the gradient of a scalar, and the curl and divergence of a vector can be obtained by various elementary methods in terms of curvilinear components and coordinate vectors, the corresponding results for the components of strain and the vector divergence of stress are usually obtained by specializing the formulae for covariant derivatives. The purpose of this note is to give an elementary approach to these latter results in the spirit of the theory of Cartesian tensors. Apart from standard elementary vector analysis the only prerequisite is a geometrical appreciation of the analysis of deformation. For completeness an elementary analytical derivation of the results about gradient, curl and divergence is given first by a method which, whilst certainly not new, does not appear to be very well known.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

Mathematicians are fascinated by relatios and have worked our extensive theories about them. The treatments tend to be abstract and sometimes the basic ideas are lost in the abstraction. Here we investigate some common ground between mathematical relations and down-to-earth genealogy, the study of family relations.

Family relations have been studied by mathematicians, perhaps none more playfully than Thomas P. Kirkman (the nineteenth century expert in combinatorial mathematics) in his little puzzle rhyme he sent to the Educational Times [1, p.117]:

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.