We investigate the pairing of theorems about parabolas through a dual transformation. Theorems and constructions concerning a parabola in a two-dimensional space can be in one-to-one correspondence with theorems and constructions concerning a parabola in the two-dimensional dual space. These theorems are called dual theorems.

Peter Neumann was a major figure in UK mathematics during a career spanning over 50 years. His parents, Bernhard and Hanna Neumann, were group theorists, so it was perhaps inevitable that Peter would become a specialist in the same area. A notable occurrence, and a sign of things to come, was the publication in 1962 of a joint paper with his parents which gave rise to the concept of an NNN group. There can be few instances in mathematics of three members of the same family co-authoring a paper. His parents spent many years in Australia and Peter maintained strong links with that country throughout his own professional life.

The mathematical investigations of natural phenomena in the seventeenth century led to the inventions of calculus and probability. While we know the works of eminent natural philosophers and mathematicians such as Isaac Newton (1643-1727), we know little about the learned women who made important contributions in the seventeenth century. This article features Princess Elisabeth of Bohemia (1618-1680), whose intellectual ability and curiosity left a unique mark in the history of mathematics. While some of her family members were deeply involved in politics, Elisabeth led an independent, scholarly life, and she was a close correspondent of René Descartes (1596-1650) and Gottfried Leibniz (1646-1716).

According to reports in the media, there is a dearth of practical examples that students of mathematics en route to their qualification can feast upon, at either sixth form level or an undergraduate level. Despite these alleged shortages, it is this author’s opinion that there are numerous examples that can be drawn from the literature and it is the purpose of this article to address the issue by providing examples from the realms of antenna reflector theory and the use therein of conic sections. Some students will be familiar with conic sections and others might not, but the numerous instances of their manifestation in the real world would suggest that they are a force to be reckoned with, and this is certainly true from a mathematical perspective.

Following Euler, we denote the side lengths and angles of a triangle ABC by a, b, c, A, B, C in the standard order. Any line segment joining a vertex of ABC to any point on the opposite side line will be called a cevian, and a cevian AA′ of length t will be called long, strictly long, or balanced according as t ≥ a, t > a or t = a. If A′ lies strictly between B and C, AA′ is called an internal cevian. This convention regarding cevians is not universal, and it is, for example, in a heavy contrast with that in [1, p. 73].

A tournament is a series of games (matches) among competitors for an overall prize. Beyond sporting events, tournament structure can relate to elections, applicants competing for an employment position, etc. Tournament structure also exists within certain species of birds and mammals where dominance relations develop, forming a tournament-like pecking structure [1]. The format of the tournament depends on the tournament’s objective giving consideration to time, financial, geographical, and other constraints. The two most common formats where players compete pairwise are the single elimination (knockout, sudden death) tournament and the round robin tournament. Our discussion includes both. A third popular format is that of a contest, where all players perform simultaneously only once. Examples of this type include most golf tournaments as well as track and field athletic events. A detailed analysis of contests as they relate to topics discussed here is given in [2].

Here we give an account of two appealingly simple techniques to generate monotonic sequences that were developed by the author and Grahame Bennett. Sadly, Bennett died in 2016 after a lifetime of distinguished contributions in the field of inequalities. This article is dedicated to him.

Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

In a previous article [1] we examined the concept of blood relations in a family tree and we used mathematics to highlight and simplify the descriptions involved, such as child, parent, great-uncle, second cousin, third cousin once removed.

The world is amply supplied with probability teasers. These can mystify and intoxicate. Here are some classics.

This Article is about the many complicated tasks that one mathematician had to carry out, and the barriers he had to overcome in order to publish one very important book in the history of applied mathematics.

This article begins with a new look at earlier work on this topic and continues with the presentation of a graphical technique for the determination of the months of a year for which the 13th is a Friday.

It might be thought that the likelihood of the 13 th day of a month being a Friday is the same as that of the 13th being any other day, but this is not so. The full day/date repeat cycle is 400 years, this being the interval between century years which are leap years. The outcome of an extraordinary counting exercise some 50 years ago by a 13-year-old Eton schoolboy, S. R. Baxter [1, 2], was to show that, over this period, the 13th of a month will be a Friday at least once more than any other day. Of necessity, Baxter’s calculation was intricate and an independent confirmation was undertaken as a matter of interest. The 400-year period beginning 1 March 2014 was considered. The work is described in four stages.

Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.

An important first step in understanding or solving a problem can be the selection of coordinates. Insight can be gained from finding invariants within a class of coordinate systems. For example, an important feature of rectangular coordinates is that the Euclidean distance between two points is an invariant of a change to another rectangular system by a rigid motion, which consists of translations, rotations and reflections. Indeed, the form of the distance function is an invariant. In calculus courses, students learn about polar coordinates, so that useful curves can be simply expressed and more easily studied.

It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.

The ‘midpoint’ approximation to the integral $$\int_0^1 f $$ is

$${M_n}\left( f \right) = {1 \over n}\sum\limits_{r = 1}^n f \left( {{{2r - 1} \over {2n}}} \right)$$

A geometric dissection is a cutting of a geometric figure (or a finite set of figures) into pieces that we can rearrange to form another geometric figure (or finite set of figures). If our figures are required to be polygons, then there is always a dissection that has just a finite number of pieces. This was established by John Lowry [1], William Wallace [2], Farkas Bolyai [3], and Karl Gerwien [4]. The American Sam Loyd [5] and the Englishman Henry Ernest Dudeney [6, 7] emphasised the goal of minimising the number of pieces that resulted from such a standard dissection.