Abbott’s Flatland (1884) [1] is the story of the encounter between a two-dimensional creature, a Square, and a visitor from space, a Sphere. The Sphere tries to convince the Square of the reality of the third dimension, but with little success at first. Finally, after several long conversations and a few other worldly demonstrations, the Sphere succeeds in making the Square see the light (or the height?), with rather unfortunate consequences for the poor Square. Flatland became a big hit in the years and decades after it came out, and continues to enjoy a robust reputation today. Much of the fallout generated by Flatland in the more than hundred years since it was written is documented in the annotated version of the book brought out by Stewart [2].

In a September 2005 note in The Mathematics Teacher, Gordon [1] gave a variation on the method of completing the square that fraction-phobic students are likely to find easier, along with the comment: ‘This variation is so obvious it must have been discovered many times over, but I have never seen it in print and thought it would be useful to disseminate it more widely.’ The method Gordon [1] gave is to multiply by 4a, and then add b2, to both sides of ax2 + bx = −c to produce 4a2x2 + 4abx + b2 = b2 −4ac. The left side of this last equation can be factorised as (2ax + b)2, and now we have a pure quadratic whose solution is straightforward.

When exploring some triangle geometry, we stumbled upon a set of Cevian axes: lines having certain properties in common with the Euler line. These axes yielded interesting results that we had not seen before. However, we have since learnt that many of them were discovered in the 19th Century. We were also unaware that John Rigby had covered much of this ground in unpublished work in the 1990s (private communication). We are therefore reluctant to claim any of our results as new, even though we hope that some may be.

Let Q be a point on the circumcircle of triangle ABC. The reflections of Q in the three triangle sides are known to be collinear, and the line thus defined contains the orthocentre H. This fact can form the basis of a proof of the existence of the Simson line, or alternatively can be deduced from the existence of the Simson line by enlarging the Simson line by a factor of 2 from the centre Q.

Interesting algebraic and geometric results can be obtained by interleaving integer sequences term by term. To introduce this we consider how a well-known sequence can be considered as being composed of subsequences.

The standard Fibonacci sequence (Fk):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... (1)

is defined by the recurrence relation Fk = Fk - 1 + Fk - 2 for k ≥ 3 and F1 = F2 = 1.

In an article full of concrete examples, [1], Malcolm Perella asked what finite groups can be realised as groups of rational functions. When I first thought about this question, I imagined that most of the answers would be easy to locate in standard literature. Some of them are. But several have defied all my attempts to unearth them. I expect that they are hiding somewhere (as we said in our family when precious toys went astray), but they are certainly not in the books where I expected to find them! The present article attempts to answer Perella’s question by referring to appropriate sources where my search has succeeded, and supplying my own answers in other cases. Examples of the groups that arise are given in a series of exercises for readers.

The practical measurement of distance on land and sea has been of enduring interest for millennia. Here we outline a brief interaction between Francis Galton (1822-1911) and Arthur Cayley (1821-95) in which they too put the ‘geo’ back into geometry.

Galton wanted to measure everything. In the adventurous life of this many-sided Victorian scientist, quantification was the hallmark of his work. In measuring geographical distances Galton used an idea suggested by Sir George Everest (1790-1866), the Surveyor-General of India in the 1830s. In the pages of the journal of the Royal Geographical Society, Everest showed how to calculate the distance from one point to an inaccessible point. One application would be to measure the distance from a point on one side of a river to a point on the other side.

Euler made many contributions to what is now called combinatorics. Some of these arose from recreational mathematics, such as magic squares and knight's tours on a chessboard; others from his study of lotteries; and, perhaps his most important work, from the study of partitions of an integer. In what follows I shall attempt to show the breadth of his work. Euler's papers will be referred to by their Eneström numbers, such as E338; this cataloguing was carried out in the early twentieth century by the Swedish Mathematician Gustav Eneström, who was to Euler what Köchel has been to Mozart. Euler's papers can all be studied in detail on the Euler Archive, details of which are given at the end of this article.

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

In an obituary of M. L. Urquhart in [1], David Elliott quotes him as claiming that Urquhart's theorem (below) is the most elementary theorem of Euclidean Geometry ‘since it involves only the concepts of straight line and distance’.

Urquhart's theorem

Let AC and AE be two straight lines.

Let B be a point on AC, D a point on AE, and suppose that BE and CD intersect at F.

If AB + BF = AD + DF then AC + CF = AE + EF. (1)

There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2.

The most basic problem with differential equations is that of being given an equation that can be put into the form

and one pair of solution values x = x0, y = y0, from which to find either an algebraic form of its solution or the numerical value of y corresponding to a particular value of x. This can most conveniently be interpreted as finding, out of the infinity of solution curves to (1.1), an equation for the unique curve that passes through the point (x0, y0) or calculating the coordinates of one particular point on that curve.