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.

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.

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.

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.