Holgate Lectures - titles offered Dr John Silvester

These talks are primarily aimed at 6th formers but can be adapted if requested to make them suitable for year 11 pupils.

All talks require a data projector and (preferably) a whiteboard or OHP as well.

Lecture 1: Primes and Polygons

Which regular polygons can be constructed with ruler and compasses? Most people can manage an equilateral triangle, and a regular hexagon; and a square isn’t too difficult. A regular pentagon is decidedly trickier, but possible; however a regular heptagon is impossible. Gauss, while still a teenager, was the first to discover how to construct a regular heptadecagon, and he also sorted out precisely which regular polygons can be constructed. This has to do with Fermat numbers that are prime; and Euler managed to show that not all Fermat numbers are in fact prime.

All of this will be illustrated with Geometer’s Sketchpad, together with just a hint of the underlying algebra, via Maple.

Lecture 2: Computers: can we trust them?

I shall give various examples of how computers can mislead us: (i) simple-minded arithmetic calculations that give surprising or even grossly incorrect results; (ii) a deliberate deception where a beguiling diagram leads to an obviously false result; and (iii) a genuine mistake:
a theorem stated on a prominent mathematical web site that looks highly plausible but is in fact false, and the web search for the true version of this theorem.

Lecture 3: All tilings great and small

We find all possible regular and semi-regular tilings of the plane by a computer search, and illustrate them and the relations between them with Geometer’s Sketchpad. We look at Penrose’s non-periodic tilings, Voderberg tilings, and spiral and other tilings with rotational symmetry. We also look at problems of tiling a square with dominoes, and with trominoes.

Lecture 4: Geometry Ancient and Modern

This is a trip through geometrical methods from ancient Greece until (almost) recent times, illustrated by several proofs of the same theorem using methods from Euclid, from coordinate geometry, and from the theory of cubic curves.

Lecture 5: Many cheerful facts about the square of the hypotenuse

Some highly visual proofs of Pythagoras' theorem, together with problems based on it, and some related theorems and curious facts.

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