In his article on the flexagon family [1], R. F. Wheeler uses a mapping scheme which illustrates the behaviour of a flexagon. I shall introduce an alternative scheme and use it to expound a method of constructing and folding the net of any flexagon.

The cover picture shows a 22° halo seen in Preston on Sunday 28 October 2001 at 10 am. In contrast with a rainbow, the red band is on the inside rather than the outside, the colour bands are much thinner, and the arc is centred round the sun instead of being opposite to the sun. Also, it is not raining. Haloes occur on still sunny days when there is plenty of cirrus cloud. It is possible to see a full circle depending on how high the sun is in the sky. This never happens with a rainbow except when seen from an aeroplane. Sometimes the colours are particularly vivid at the 3 o’clock and 9 o’clock points. These vivid areas are called parhelia or mock suns. Sometimes the parhelia are all that can be seen.

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.

David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and

Pagni observed, as have others, including Mason et al. [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t}, which are well known to have the sum of their cubes equal to the square of their sum.

We define a Watt quadrilateral to be a quadrilateral with a pair of opposite sides of equal length. See Figure 1.

2. Linkages

The Watt linkage (Figure 2) has equal-length cranks AD and BC, A and B fixed, and coupler bar CD. It was devised by James Watt about 1784 to constrain the steam-engine piston rod, connected at E, the midpoint of CD, to approximate straight-line motion over a limited range.

Boolean class algebra is a special case of lattice theory and it is well known that the propositional calculus is Boolean. It is also well known that there are sixteen and only sixteen truth functionally distinct relationships between two elementary propositions in the pronositional calculus.