Tilings have appeared in human activities since prehistoric times. The mathematical theory of tilings contains a rich supply of interesting and sometimes surprising facts as well as many challenging problems. A vast literature exists on the subject of tiling, and almost every imaginable variant of the question ‘How can a space be tiled by replicas of a set?’ has been discussed. Figure 1 shows a tiling of a portion of the plane where each tile is of the same size and shape and has a fractal boundary.

The activities described in this paper are highly addictive. They can be injurious to work, mental health, personal relationships, serious research and the future of the rain forest. The authors can accept no responsibility for the subsequent behaviour of people who read this paper.

Geometric dissection theory is connected with ‘cutting up’ plane and solid objects and reassembling the pieces to form other objects. It is a very ancient and venerable mathematical activity going back to Euclid (circa 300 BC) and perhaps beyond. In Euclid’s Elements we find a square dissected into four pieces to illustrate the algebraic identity (a + b)2 = a2 + b2 + 2ab.

Choosing seven mathematical theorems or ideas to take to a desert island is more difficult than I first thought. In an earlier article (with Colin Fletcher) I discussed other people’s choices. Now comes the time to bite the bullet and make my own. I was reminded that the choices do not have to be ‘great’ or ‘state-of-the-art’ or involve any crystal ball gazing. They do not have to be among the hundreds of fundamental theorems that abound in mathematics all they are bound to be is personal.

In this article, we discuss a combinatorial problem arising from the National Lottery. The question considered is the following: Given a works (or any other) National Lottery syndicate of say n persons, how small can n be and still allow to ensure that at least one person in the syndicate has a winning ticket? Of course we will only be able to ensure that the required ticket wins the minimum prize of £10. However, we will show that a system to achieve this end can be constructed using some remarkable combinatorics which are well-known parts of modern mathematics. In fact, the system we propose for this problem has n = 290 and at least two tickets will win £10 prizes. It will be perfectly possible that the system will result in some members of the syndicate winning a fourth, third, second or even first prize. Although the combinatorics are quite interesting, it is not possible to ‘beat’ the National Lottery using our system. We have applied our system to the actual results of the National Lottery since its inception. Our experimental evidence shows that even using our seemingly efficient scheme, payout averages only about 28% of expenditure (showing an average loss of about 72%). However, in a typical draw, our system would actually have produced 5 or so ‘winning’ tickets for the syndicate.

The well-known non-terminating decimal expansion has the property that, if the first half of the recurring part, 142, is added to the second, 857, the result is 999. Surprisingly, this is not a ‘one-off’ phenomenon. For example,

The reader is invited to test and . Further examples will be found in John R. Silvester’s article. We shall examine when and why the above property of recurring decimals occurs.

There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.

A recurring topic of discussion is the perceived difficulty, or awkwardness, of establishing the fundamental rules of vector algebra by rigorous geometrical arguments (see, for example, [1]). The following exposition is an attempt to show that, for those acquainted with the rudiments of linear algebra, the essentially simple theory of vector algebra can be clarified by presentation in the context of quaternion algebra, whose elegance and usefulness may still be somewhat undervalued. We make no explicit use of trigonometry, but of course some underlying geometrical ideas are mentioned.

Over sixty years ago, a schoolmaster by the name of A. P. Rollett set a problem for a lower fifth form class which was unwittingly based upon the famous Archimedean problem of the Arbelos. This caused his pupils substantial difficulties and eventually it prompted Rollett to send a Note to the Mathematical Gazette seeking the mathematical insights of the readership regarding this question. He received very helpful responses from over twenty readers whose numbers included many notable mathematicians of the day such as E. A. Maxwell, E. H. Neville and R. S. G. Rutherford etc. Solutions were also received from two members of the clergy, (Canon) D. B. Eperson and the Rt Revd the Bishop of Kootenay, which may suggest that this problem was of sufficient intractability as to be deserving of divine inspiration! But this was not the end of it because, during the next twentythree years, there followed a series of very interesting related notes and articles continuing up to 1960 all of which made reference to Rollett’s initial homework problem of 1937.

A sign similar to Figure 1 has become increasingly common in recent years; many, including the author, carry this ancient Christian symbol on their cars. Probably few know that it is a simple mathematical curve. It is of course questionable whether Apollonius knew it, though he could have done, and even more questionable that early Christians knew their Apollonius. But they certainly knew some form of the symbol. Nowadays it is often crudely reduced to two circular arcs - making steering difficult for the poor fish.

The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.

For any triangle, and any point on its circumcircle, the feet of the perpendiculars from the point to the sides of the triangle are collinear. The line through them is the Simson (pedal, Wallace, or Wallace-Simson) line. If two circles have radii in ratio 1:3, and the smaller rolls within the larger, a fixed point on the circumference of the smaller describes a deltoid (or tricuspid hypocycloid) curve (Euler, 1745). The envelope of the Simson lines of any triangle is a deltoid.

Given chords OA1A2, OB1B2 and OC1C2 of a conic with segment lengths ai = OAi, bi = OBi, ci = OCi (for i = 1, 2) this article will establish a metrical relationship between ai,bi,ci, and the sines of the angular directions of the chords.

We suggest a way to make traditional geometry more interesting for students. Investigating the connection between classical geometry and modern chaos theory seems to be a suitable way of motivating them. Excursions into complex number iterations and interesting Mandelbrot sets offer many methods of discovering and exploring.

In H. Zeitler considers the complex function The classic case n = 2 was investigated by Mandelbrot. We are discussing a more general case where the exponent n is a rational number.

In the years 1603–1867, known as the Edo period, when Japan isolated itself from the western world, the country developed its own style of mathematics, especially geometry. Results and theorems of traditional Japanese mathematics, known as ‘Wasan’, were usually stated in the form of problems; these were originally displayed on wooden boards (‘sangaku’) hung in shrines and temples, but many later appeared in books, either handwritten with a brush or printed from wood blocks. (See [1, 2, 3] for more details.) Solutions to the problems were not given, but numerical answers were sometimes provided. The problems were written in a form of language called ‘Kanbun’, based on Chinese, which cannot be readily understood by modern Japanese readers even though written Japanese makes extensive use of Chinese characters.