The process of producing the facial planes of a polyhedron to form stellated polyhedra is well known. When applied to the icosahedron it leads to a series which is limited to eight solids if we make the condition that every face of each must be covered in forming the next of the series. If this condition is dropped a large number of combinations of the parts of these eight solids is possible. The total number of solids, limited by symmetry considerations, is 59, and they are all illustrated in The 59 Icosahedra by Coxeter, Du Val, Flather, and Petrie. The corresponding basic series of solids starting with the Rhombic Triacontahedron numbers 13, and the total number of combinations is very great.

Imagine a horizontal rectangular tray with vertical sides, and a disc lying flat in the tray and rolling around the inside of the tray in such a way that its circumference is always in contact with the edge of the tray. We shall suppose that the disc has been rolling for ever in the past, and that it will continue to roll for ever in the future. What can we say about this physical system? Our aim is to illustrate the complexity and the discontinuous nature of the system under arbitrarily small changes in the size of the tray (a feature that is typical of many dynamical systems).

Which is your favourite formula in triangle geometry? Mine is definitely the formula for the area of triangle OIH . It is well known that the perpendicular bisectors to the sides of any triangle are concurrent at a point O (centre of the circumcircle), that the angle bisectors to the vertex angles are concurrent at a point I (centre of the incircle), and that the altitudes are concurrent at a point H. If the triangle is not isosceles, then these three points are all different and uniquely determine a new triangle OIH (see Figure 1), whose area can be expressed in terms of the sides a, b, c of the original triangle. I derived such a formula 20 years ago, and later found out that it had been studied a century earlier.

Triangles with a 60° angle possess many interesting geometric properties. Here is a well-known example: In ΔABC with incentre I, let the rays meet the sides AC, AB at B1, C1 respectively. If ∠B = ∠C, then IB1 = IC1, trivially. Does the converse implication hold? No – the same condition holds when ∠A = 60° (irrespective of the relation between ∠B and ∠C). In this article we study some more properties possessed by triangles having a 60° angle.