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.

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.

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.

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).

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.

Fate laughs at probabilities.

—Edward Bulwer-Lytton, 1st Baron Lytton

Time spent in national pandemic ‘lockdowns’ and ‘tiers’ for most of 2020 has created an opportunity to revisit some ideas previously committed to paper, but unfinished. I now return to one of them, to continue and share the line of thought, if not to complete it.

Consider the general cubic equation:

$${x^3} + a{x^2} + bx + c = 0.$$

A change of variable: x = z + t leads to: (1)

$${z^3} + (3t + a){z^2} + (3{t^2} + 2at + b)z + ({t^3} + {at^2} + bt + c) = 0.$$

The first solution to the quartic equation is attributed to Lodovico Ferrari, a student of Geralamo Cardano. The solution was published alongside the solution of the cubic in Cardano’s book Ars Magna [1]. In this Article, we introduce a new canonical form to simplify the derivation of the roots of the equation (1)

$${z^4} + {z^3} + f{z^2} + g = 0\quad \textrm{with}\quad f,g \in \mathbb{R}.$$

The most important properties of any distribution are its moments. The first four moments, for example, can be used describe any data set fairly well. Moments can also be used for estimation.