We define a Fibonacci fraction circle to be a circle passing through an infinite number of points whose coordinates are of the form $\left( {{{{F_k}} \over {{F_m}}},{{{F_n}} \over {{F_m}}}} \right)$ , where the F’s are Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …). For instance, the circle (1)

$${x^2} + {\left( {y + {1 \over 2}} \right)^2} = {5 \over 4}$$

$\left( {{1 \over 1},{0 \over 1}} \right),\,\left( {{1 \over 2},{1 \over 2}} \right),\,\left( {{1 \over 5},{3 \over 5}} \right),\,\left( {{1 \over {13}},{8 \over {13}}} \right),\left( {{1 \over {34}},{{21} \over {34}}} \right),\,...,$

A friend of mine [1] mentioned a problem to me, which he was told had an interesting solution involving an unexpected square root. I have not seen this problem described elsewhere, so I have carried out my own analysis, which I will present here. In fact the solution involves not only square roots, but also higher roots … and a logarithm.

There are visual components of mathematics that are vital to the learning of the subject and should be a focus for its teaching. One aspect is the impressive capacity of the brain to visualise, an important element in the discovery process of some prominent mathematicians and scientists. The pedagogical importance of diagrams in mathematics is considered, with examples showing their role in problem-solving, elucidation, persuasion and decision making, concluding with some particularly stimulating and revealing diagrams for use in the classroom.

We deal with an extension of the six-point circle theorem for the quadrilateral [1] when the Van Aubel configuration is generalised as in [2] and [3]: similar parallelograms are constructed, all internally or all externally, on the sides of a given quadrilateral.

We denote the real logarithm of a positive number a by ln a, so that ax = exp (x ln a), and we shall discuss what is known about the real solutions x of the equation (1)

$$x = {a^x},\;\,a > 0.$$

A golden right triangle of first type is a right triangle such that the shortest side is the golden section of the hypotenuse as defined in [1], [2]. If we consider a golden rectangle and one of its diagonals, we obtain a right triangle with the shortest side being the golden section of the other side; this right triangle will be named golden right triangle of second type.

This Article will find formulae for approximating the volume of a solid of revolution, analogous with the Trapezium rule and Simpson's rule [1] for approximating an area. It will also consider the order of these approximations.

Many players know that the secret of winning the game of nim (and other “impartial” combinatorial games) is to write the sizes of the game’s piles in base 2 and then add them together without carry. The proof of this well-known procedure (described below) is both straightforward and convincing. Nonetheless, the procedure still appears magical, as though a rabbit has been pulled out of a hat. Astute students (and frustrated professors) often ask why the winning strategy for such games involves base 2, and not some other base. After all, nothing about the game of nim itself – the game rules, the configuration of the tokens, etc. – provides any hints about the origin of base 2 in this setting. Minimal insight is offered by most published proofs, which themselves tend to either appear almost wizardly in nature (i.e. assume the base-2 method and show that it miraculously solves the problem) or employ combinatorial arguments that supply little abstract intuition (at least to the authors of this article).

The business of making mathematical approximations in the physical sciences has a long and noble history. For example, in the earliest days of pyramid construction in ancient Egypt it was necessary to approximate lengths required in construction, especially when they involved irrational numbers. Similarly, surveyors in early Greece seeking to lay out profiles of right-angle triangles or circles on the ground invariably ended up making approximations regarding measurements of required lengths, as indeed is the case today. Practitioners have always faced the problem of having to decide when parameters in theory have been met satisfactorily in the practice of measurement. Further, before the advent of hand-held calculators, students in schools in the UK would have been very familiar with the approximation 22/7 for the transcendental number π, obtained perhaps by comparing (as this author did) the measured circumferences of many laboriously drawn circles of different sizes with their diameters. Despite the advent of sophisticated calculating devices and facilities, such as computers and spreadsheets, the practice of making approximations is still much in evidence in theoretical work in fields associated with physical phenomena. Such approximations often result in formulae that are easy to use and remember, and moreover can produce theoretical results that support directly, or otherwise, results from measurements. In this respect, the practical mathematician does not have to seek results to many decimal places when measurement facilities allow for accuracy to only a few. The purpose of this Article is to illustrate this point by discussing an example drawn from the realms of antenna theory, relating to the performance of a dipole antenna. It is not the purpose here to delve into the derivation of dipole theory, but to extract the relevant information and show how useful mathematical approximations can be employed to simplify a relationship between parameters of interest to an antenna engineer. To this end, it will first be necessary to introduce some antenna concepts that might be new to the reader.

A bicentric quadrilateral is a convex quadrilateral that can have both an incircle (it is tangential) and a circumcircle (it is cyclic), see Figure 1. We know of only a dozen characterisations of bicentric quadrilaterals published before. In all of them the starting point is either a tangential or a cyclic quadrilateral, which then must satisfy some condition in order also to be of the other type. Before we proceed to prove seven new such necessary and sufficient conditions for bicentric quadrilaterals, we review one characterisation and one property of tangential quadrilaterals that we will apply later.

If a circle rolls without slipping around an equal fixed circle, then a point carried by the rolling circle traces out a limaçon of Pascal. (This is Etienne Pascal, father of Blaise. The word limaçon is derived from the Latin limax, a snail.) If the fixed and rolling circles have radius 1, and the point P carried by the rolling wheel is distant a from its centre, then for a > 1 the limaçon has an inner and an outer loop, joining up at a node. For a = 1 it has a cusp, and is then a cardioid, so-called because it is heart-shaped. See Figure 1, where we have plotted the cases a = $${3 \over 4}$$ , a = 1 and a = $${3 \over 2}$$ .

The product over the integers 1 to N is written as N!, is called N factorial, and is defined as: (1)

$${\rm{N! = 1 \times 2 \times 3 \times }} \ldots {\rm{ \times (N - 1) \times N}}{\rm{.}}$$

Euler’s striking proof of the fact that the infinite sum

${1 \over {{1^2}}}\, + \,{1 \over {{2^2}}}\, + \,{1 \over {{3^2}}} + \,...\, + \,{1 \over {{n^2}}} + \,...\, = {{{\pi ^2}} \over 6}$

$\zeta (s)\zeta = \sum\limits_{n = 1}^\infty {{1 \over {{n^s}}}} $

$\zeta (2)$

$\sum\limits_{n = 0}^\infty {{1 \over {{{(2n + 1)}^2}}} = {{{\pi ^2}} \over 8}} .$

It was known before Archimedes (287-212 BC) that the circumference of a circle was proportional to its diameter and that the area was proportional to the square of its radius. It was Archimedes who first supplied a rigorous proof that these two proportionality constants were the same, now called π [1]. He started with inscribed and circumscribed hexagons and increased the number of sides from 6 up to 96 by successively doubling it. His result was not a single value. In fact he generated five intervals each of which contained π. He calculated a lower bound from the inscribed polygon and an upper bound from the circumscribed polygon of 96 sides. This gave him the interval () or (3.140845, 3.142857), which is less accurate than the interval bounded by half-perimeters of the inscribed and circumscribed 96-gons, which is (3.141031, 3.142714).

It is tempting to accept bets when the outcome has a positive expectation favouring the bettor. We examine situations where this is not enough of a criterion as a basis for betting. The argument we present shows that the probability of winning the bets in the long run has to be qualified in a certain way beyond positive expectation. We introduce an alternative index as a criterion to recommend to bettors.

Let us define a Fibonacci-Lucas hyperbola as a hyperbola passing through an infinite number of points of the form (Fm, Ln), where the Fm are distinct Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21,…, where F0 = 0), and the Ln are distinct Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29,…,. where L0 = 2). The simplest examples are 5x2 - y2 = 4, which contains the points (Fk, Lk) with odd subscripts, e.g. (1, 1), (2, 4), (5, 11), and 5x2 - y2 = -4, which contains the points with even subscripts, e.g. (0, 2), (1, 3), (3, 7); (see [1, 2]). These follow immediately from the identity (1)

$$L_n^2 - 5F_n^2 = 4{( - 1)^n}.$$

The correspondence between the discrete and the continuous is a fascinating theme in mathematics. The Euler-Maclaurin sum formula, discovered independently and almost contemporaneously by Leonhard Euler (1707–1783) and Colin Maclaurin (1698–1746), in the early 1730s, relates the sum of the values of a function at the integers in the interval [a, b] with its integral over [a, b]. It thus equates a discrete sum with a continuous sum (integral) of a related function.

On his website dedicated to questions and investigations arising out of dynamic geometry technology, Michael de Villiers has a series called Geometry Loci Doodling [1]. These are locus problems connected to the centroids of cyclic quadrilaterals – ‘centroids’ in the plural, for there are three different kinds of centroid depending whether one understands the quadrilateral in terms of its vertices, perimeter or area. The corresponding centroids are the point-mass centroid, the perimeter-centroid, and the lamina-centroid. In each case, de Villiers keeps three vertices of the quadrilateral fixed on the circumcircle, and then traces the locus of the different centroids as the fourth point moves round the circle. In this paper, I shall take a brief look at the point-mass centroid and then a lingering view of the lamina-centroid.