The Fen country

The Fen country is a large area of Eastern England covering much of the counties of Cambridgeshire, Lincolnshire and Norfolk, which for centuries was flat and swampy. Between 1622 and 1656 large scale drainage works were carried out by a Dutch engineer, Sir Cornelius Vermuyden, thus reclaiming large areas of land for agriculture. Before the Fens were drained, Ely was an island, known as ‘Eel Island’ because of the abundance of eels.    

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Ely and some history

Ely began as a religious community founded by St Etheldreda, queen of Northumbria, who built an abbey there about 670 AD. Later in its history, Ely was used as a hide-out by Hereward the Wake, the Angle-Saxon patriot who led the resistance against William the Conqueror until 1071.

The present cathedral was begun in 1083 by Simeon, the first Norman abbot; the eastern part was completed 26 years later. The western part was completed by 1190.

Some feeling of these ancient beginnings can be experienced today. If Ely is approached during autumn towards the end of the day when the mists are rising, the cathedral appears as a ghostly silhouette, and it is easy to relive those early times in imagination. Again, in summer Ely stands as an island in the midst of waves of rippling, wind-swept grain.

 Link  http://www.cathedral.ely.anglican.org/
































The cathedral octagon

The cathedral is the third largest in England, and is a prominent landmark, visible from far and wide. When the Norman tower collapsed in 1322, the cathedral was not given a spire, but a unique stone octagon, surmounted by an octagonal timber ‘lantern’. This octagon is the only structure of its kind in England, and is a creation of great beauty and mathematical interest.      

Drawing the octagon  

Suppose you were the architect designing the octagonal tower. You might decide that the octagon could be inscribed in a circle. But how would you construct it? What size would the various angles be?

 Investigate In the diagram, angles a, b and c are marked. Find the size of each, the number of equal angles of each size, and the relationship between a, b and c. What do you find for regular polygons with different numbers of sides?
























The symmetries of the octagon

You will notice that we have drawn four diagonals (diameters) of the octagon in this sketch.

 Investigate Discuss the ways in which the octagon can be moved so that the moved octagon exactly covers the original position. You might consider rotations about the centre, or reflections in the diagonal lines. How many different transformations can you find? Four? Eight? Sixteen?

Now look at this photograph of the Ely octagon. Can you see some extra lines of symmetry? Do these give rise to some new transformations which leave the octagon invariant?

 Investigate See if you can identify the symmetries of regular polygons having different numbers of edges.          

 






















The octagram

This diagram shows a mathematical attempt to construct a creative ‘lantern’ to fit inside the octagon. The bold 8-sided shape is called a regular octagram.

 Investigate What size are the various angles of the octagram? What is the ‘exterior angle’ through which one must turn to get from one edge to the next? Can you relate this to the number of edges?

 Investigate Check out other n-grams. What does a pentagram look like? a hexagram? How do these figures relate to the divisors of the number n?

 Project  In what ways do various polygons and other mathematical shapes appear in buildings? Answer this from observation in your own town, and from books on architecture in your library.

Further reading ...

Barcham, P. (1990), ‘Star polygons’, The Australian Mathematics Teacher 46/4, 18 – 19.

Coxeter, H.S.M. (1969), Introduction to geometry, Wiley.