Magazine article Mathematics Teaching

A Different Kind of Geometry

Magazine article Mathematics Teaching

A Different Kind of Geometry

Article excerpt

The final excerpt from Dick Tahta's unfinished manuscript A celebration of ten (see facing page) ends with the problem:

Plant 10 trees in 10 lines so that there are three trees in each line and three lines through each tree.

This problem is linked to a result within the field of projective geometry and I will offer some hints at the end of this piece, so please read no further if you are enjoying the challenge. Projective geometry was an on-going interest of Dick's. It features in the Steiner Waldorf curriculum (Whicher, 1971) and the book Geometry and the imagination (Hilbert and CohnVossen, 1952/1999). Dick's belief was that it is a topic suited to the concerns of adolescents, dealing as it does with relations and notions of infinity. Although not in the current English national curriculum, work on projective geometry would cover many elements that do appear.

In projective geometry, measurement is downplayed. There is a need to allow the existence of infinitely distant objects (for example, an infinitely distant point on a line) and the focus is on relationships not measures. As a first exercise (taken from Whicher, 1971, p. 54), I invite you to draw a projective hexagon. In plane geometry, constructing a hexagon would likely start with a circle and a centre. In projective geometry, we start with a line, which can be imagined as the horizon. Choose any three points on this line and draw a ray from each point to create a triangle. Where any two lines meet, draw a further line to the remaining point on the horizon. …

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