It is a trite but true remark to say that there are two fundamental random processes, one being the Poisson process and the other being Brownian motion. They are each fundamental in two ways; first, they do seem to describe much of the natural world very well, and second, the student is well-advised to study and master these two before going on to look at harder and more recondite processes. This note gives a brief glimpse of some aspects of the Poisson process, which is technically by far the more tractable of the two.

I have a soft spot for geometric problems which lead to diophantine equations, that is, equations for which only positive integer solutions are sought. I have not seen the following naturally occurring problem discussed anywhere.

Problem A

A closed rectangular box has its volume numerically equal to its total surface area. If the length, breadth and height of the box are all positive integers, find all possible sets of dimensions of the box.

The solution of this pretty problem is completely elementary and can be fully understood by those with a basic knowledge of algebra.

We remark that to solve any three-dimensional problem it is often useful to step down a dimension and to consider the corresponding problem for plane figures.

Some remarkable new series for π, in which the convergence is exponentially rapid, have been discovered by Bailey et al. [1, 2] using machine subroutines; see also Wagon [3]. An example is equation (17) below. Although the authors also provide algebraic proofs, these seem at first sight to be very special. The purpose of this note is to provide a more general demonstration and to deduce other instances of such series. It will be shown that all the results follow naturally from one or other of two simple lemmas.

The Farey sequence of order n(Fn) is the sequence of all reduced fractions between 0 and 1, whose denominator does not exceed n, arranged in increasing order of magnitude.

For example, F6 is .

The Fibonacci sequence is the sequence for which u1 = 1, u2 = 1 and un = un-1 + un-2,namely, 1, 1, 2, 3, 5, 8, 13, … , un

At first sight, there is little connection between Farey’s fractions and Fibonacci’s integers. The purpose of this note is to show and explore such a connection and hence derive some properties of the Fibonacci sequence directly from previously proved properties of the Farey sequences. The former properties are well-known, but their rather unusual derivation from Farey properties may have some interest.

This note describes a possible approach to the direct calculation of both derivatives and integrals of power functions using rather simple algebra and limits that could be easily accessible to students beginning a calculus course. The calculations provide a powerful example of the fundamental theorem, and could be used to allow students to discover the fundamental theorem from the example. The main calculation, which is not found in many calculus textbooks and is due to Descartes, is described by Maor [2]. While this approach may seem like the popular American ‘reform’ calculus, one may find a somewhat similar approach (presenting integral calculus from first principles) in Tom Apostol’s classic and lovely text [1].

A new and general proof of theorem [1] is presented, which accommodates directed angles and covers all possible configurations of line and conic. A sign convention for the use of directed angles in this setting is developed, and examples are given showing how this fundamental theorem can be used to establish, much more directly and succinctly, the many tangent/focus properties of conies.

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).

For a natural number n ≥ 3 and a positive real number ξ let Pn (ξ) denote the area of the regular n-gon with side of length ξ and let P(ABC … Z) be the area of a convex polygon ABC … Z.

In 1996 Josip Kovačević observed [1] that in every triangle ABC with sides a, b and c and with the angle A equal to π/3 radians the following relation holds:

Similarly, if the angle A is equal to 2π/3 radians then

On the other hand, when the angle A is π/2 radians, then Pythagoras’ theorem implies which could be rewritten as so

because P(ABC) = bc/2. The last observation is the starting point for Veljan’s results in [2] and [3] that

when the angle A is π - 2π/n radians and that

when the angle A is 2π/n radians.