50 Mathematical Ideas You Really Need to Know

geometry nevertheless. One model for it is the geometry on the surface of a sphere. Here the great circles (those circles that h ...

What if parallel lines do meet? ...

28 Discrete geometry Geometry is one of the oldest crafts – it literally means earth [geo] measuring [metry]. In ordinary geomet ...

The light ray y = 2x hits the point A with coordinates x = 1 and y = 2 but it will not strike the point B with coordinates x = 2 ...

A many-sided or polygonal shape Pick’s theorem gives a means for computing the area enclosed by a many- sided (or polygonal) sha ...

The Fano plane The Fano plane The Fano plane geometry was discovered at about the same time as Pick’s formula, but has nothing t ...

very symmetrical. Not so Euclidean geometry. In Euclidean geometry there are parallel lines, that is pairs of lines which never ...

schoolgirl problem (see page 167). In the theory of designing experiments the Fano plane appears as a protean example, a Steiner ...

true. Brianchon’s theorem Brianchon’s theorem was discovered much later by the French mathematician and chemist Charles-Julien B ...

29 Graphs There are two types of graphs in mathematics. At school we draw curves which show the relationship between variables x ...

Euler made an observation about a successful walk. Apart from the beginning and the end of the walk, every time a bridge is cros ...

Looking at the graph representing Königsberg, every point is of odd degree. This means that a walk crossing each bridge only onc ...

number of odd degrees would be an odd number. So it follows that x must be even. Non-planar graphs The utilities problem is an o ...

A ‘tree’ is a particular kind of graph, very different from the utitlities graph or the Königsberg graph. In the Königsberg brid ...

Cayley was able to count the number of different tree types for small numbers of points. He was able to go as far as trees with ...

30 The four-colour problem Who might have given young Tiny Tim a Christmas present of four coloured wax crayons and a blank coun ...

The western states of America Let’s look at the map of the western states of America. If only blue, green and red were available ...

the eminent Cambridge mathematician Arthur Cayley to write a paper on it in Unhappily, Cayley was forced to admit that he had f ...

A torus with two holes and in general, C = [½(7 + √(1 + 48h))]. The square brackets indicate that we only take the whole number ...

The mathematics department at the University of Indiana had a new trumpet to blow. They replaced their ‘largest discovered prime ...