The reader will be familiar with the golden rectangle, i.e. that rectangle which leaves another rectangle of the same shape when the largest possible square is cut from it (Fig. 1). In this article the effect of successively cutting squares from rectangles of other shapes will be investigated.

The concept of the direct product is widely used in the theory of groups, algebras, vector spaces and allied topics, but its use as a manipulative tool in matrix algebra is less common.

The present note shows how the direct product of matrices arises in elementary work, derives some of its more important properties, and gives some examples of its use.

We live in a golden age regarding the opportunities to explore Euclidean geometry. The access to dynamic geometry computer programs for everyone has made it very easy to study complex configurations in a way that has never been possible before. This is especially apparent in the study of triangle geometry, where the flow of new problems, properties, and papers is probably the highest it has ever been in the history of mathematics. Even though it has increased a bit in recent years, the interest in quadrilateral geometry is significantly lower. Why are triangles so much more popular than quadrilaterals? In fact, we think it would be more logical if the situation were reversed, since there are so many classes of quadrilaterals to explore. This is the primary reason we think that quadrilaterals are a lot more interesting to study than triangles.

The use of reflecting surfaces to unscramble distorted pictures has been apparent over the past three centuries, at least, in an art form known as anamorphic (ana = back again, morphe = form) art. Of course, working in the opposite sense, the mirrors in fun parlours give distorted images from an undistorted (!) object. A famous portrait of “Bonnie Prince Charlie” in the West Highland Museum, Fort William, Scotland, appears as a series of incomplete concentric circular streaks of colour, but when a cylindrical reflecting surface (the anamorphoscope) is placed at the centre of the painting a recognisable image is seen on looking into it. Niceron and DuBreuil both wrote treatises in the seventeenth century on the techniques of producing anamorphograms, and Gardner has given a recent discussion of the general topic. DuBreuil described a method for generating a distorted object (drawing) that will be transformed into an undistorted image by the use of a conical reflecting surface whose base sits on the plane of the object. The drawings are very easily done by following simple rules of geometry.

The mid-points of the sides of any quadrilateral form a parallelogram. Repeated applications of this inscription process will lead to two families of similar and similarly situated figures. It is proved below that the general n-sided polygon lying in space of h dimensions gives rise to similar sequences in the limit. If the initial polygon is not re-entrant convergence is rapid, as will be seen from Figures 1 and 5. The parallelogram is replaced by the affine transformation of a regular polygon lying in a plane.

The following paragraphs have been assembled in consequence of my reading Dr. Cundy’s note on 25-point geometry. Towards the end of it, apparently mindful of the adjunction of a “ line at infinity ” to the Euclidean plane, he adjoins a line to the 25-point plane and so obtains a geometry of 31 points. Here I reverse this procedure : I start with the 31-point geometry and thereafter assign to one of its 31 lines the rôle of the “ line at infinity ”. This seems more in the spirit of Cayley's dictum at the end of his Sixth Memoir on Quantics, that “ descriptive geometry is all geometry ” and metrical geometry only a part thereof.

In 1569 Gerhardt Mercator (1512-1594) constructed the chart now called Mercator’s Map. On it the meridians and parallels are sets of parallel straight lines at right angles, a course on the Earth’s surface always cutting the meridians at the same angle is a straight line cutting the meridians on the map at the same angle, and the shape of any very small area on the map is the same as the shape of the corresponding small area upon the Earth. This chart can be made by gradually increasing the distances between the degrees of latitude in advancing from the equator towards the poles. But though Mercator did, in fact, construct such a map, it is not known on what principle he proceeded, and it is conjectured that he obtained his division of the meridians by observing on a globe on which rhumb-lines had been drawn the intersection of these lines with the different meridians.

It seems likely that the teaching of statistics in schools will become more common and will be the responsibility of teachers of mathematics. Many of the pupils will not be mathematical specialists.

By analysing the forces acting on an oar during the power stroke it is possible to estimate the mean velocity given to a racing shell. Use is made of oarlock force data, obtained experimentally by the Australian Institute of Sport, to obtain accurate predictions in the case of a racing pair. Similar conclusions would hold for boats with any number of oars. The effect of altering the angle between oar shaft and blade is investigated, leading to a suggestion for a design change to improve oar efficiency. Predictions are made of the resulting improvement in boat position in a 2000-metre race.