If you visit the Manchester Museum of Science and Technology take a look at the sculpture in front of the museum. There are two large stainless steel objects, a sphere and an inverted cone. On the ground around the cone are variously shaped patterns of black and brown. However, when the patterns are viewed as reflected from the cone they take on the portraits of four famous scientists connected with the city of Manchester (see figure 1 on the previous page). This is an example of anamorphosis, or anamorphic art. Anamorphosis refers to an image or drawing which is distorted so that it can only be recognized when viewed in a particular way, or using a special device (like the cone in the above example). The word is derived from the Greek anamorphoun - to transform.

One of the consequences of Descartes’ new approach to geometry (1637) was an increased interest in instruments for drawing conic sections. Conic section drawers, which is the name I shall give to instruments for drawing conic sections in a continuous movement, had been described earlier, but the central role of conic sections in the geometry of Descartes made them by about 1640 into a new topic of research. This article sketches the background to this growing interest and then describes some of the instruments designed by the Dutch mathematician Van Schooten (1615/6 – 1660). It concludes with an application of this material in mathematics teaching.

Recent years have seen a marked increase in the tempo of the hunt for digits of the decimal expansion of π as shown in the table below.

In recent years we have been teaching mathematical analysis using the computer to give experiences to help students understand the concepts [1]. It has proved to be a valuable aid in many ways, but the method depends on giving a suitably rich environment enabling students to experience the full range of possibilities. A great weakness of the use of regular computer languages is that they hold numbers in memory as a finite expansion in base two, so, effectively, all numbers in such a system are rational. This meant that we were unable to demonstrate any ideas that depend on the difference between rationals and irrationals; for instance, we could not plot graphs of the kind

If a square pyramid is cut obliquely by a plane as shown in figure 1, a pentagonal cross-section PQSTR is generated. The “Beijing theorem” states that the only case in which this pentagon can be regular is that in which the pyramid is half an octahedron and (except for a complication to be discussed later) the coordinates of the points P, Q, R, S and T are uniquely determined.

July 1990. The Pop Maths Roadshow comes to town. “Chaos” Rules OK.

28 July. Receive invitation to post-Pop Maths party in September. One of the conditions of eligibility to attend is to tackle four problems listed on the invitation. Surely only mathematicians do this sort of thing? I glance at the problems. Two of them I know how to tackle, and a third I haven’t a clue how to begin; but it is the first problem that intrigues me most.

It has been my practice for many years to introduce the concept of a vector space to students by looking at families of magic squares and triangles. Consider, for example, all possible magic triangles like that shown in figure 1(a), where each circle is to be filled by a real number so that the totals along each edge of die triangle are constant. Then scalar multiples of such triangles, together with sums of them, will also be magic. Once four numbers are put into the circles, such as a, b, c, and d in figure 1(b) then the other two entries are uniquely determined. Because the other four numbers can always be freely chosen, there are four degrees of freedom.

The candidates have sat the examination. The examiners have marked the scripts. Everyone has gone on holiday … except the chief examiners, for whom one more task remains. Before the Board’s computer can start to churn out the results slips, they must decide the formula for converting the marks into grades.

When there is just a single paper, and no course work, as with many AS examinations, this is straightforward enough. The chief examiner, making a judgement of the quality of the scripts against grade descriptions, backed up by past experience, will propose a set of grade boundaries – that is, the lowest marks which should achieve each of the available grades – and the conversion is then automatic. For example, if the grade boundaries are A 76, B 68, C 59, D 50, E 41 and N 32, then candidates with marks of 61, 99 and 30 will be graded C, A and U respectively.

While musing on the graceful sylvan scene
Depicted on a classic Grecian urn,
John Keats declared that “truth” and “beauty” mean
The same, and here this statement’s worth we learn.

“Thou still unravish’d bride of quietness!
Thou foster-child of Silence and slow Time”,
Wrote Keats [1], the um’s calm beauty to express,
While he, by permutations of the rhyme,
Did generate its group of symmetries,
Which is dihedral and of order six.
The um and its inversion therefore match
And all its patterns replicate in threes:
So triplet dancers demonstrate their tricks
While silent pipers play a triple catch.