The story of the development of non-Euclidean geometry is one of
the classic adventures in the history of mathematics. The Greek
mathematicians changed geometry from what was an important useful,
everyday, practical tool to an intellectual enterprise for the
unification of known mathematics
on a theoretical basis
. Euclid's 13 books of the Elements
contain the foundations of number theory, ratio and proportion as
well as plane and solid geometry. Later Greek scholars like
Apollonius, Eratosthenes Archimedes, Diophantus and Hiero developed
more ideas about theory and practical applications that were handed
on to the Arabs.
In their turn, Arab mathematicians took on the Greek mathematics,
but were also importantly influenced by mathematics and astronomy
coming from India, and even as far as China. For the Arabs,
practical astronomy was an important enterprise. Their religion
required them to find the exact location of Mecca, so that the
direction of Mecca from any part of their world, could be easily
found. To enable these calculations, they invented most of the
basic techniques of trigonometry, and developed Spherical Geometry.
Apart from their practical work, scholars like Al-Khowarizmi and
Omar Khayyam made major contributions to the 'pure' mathematics of
geometry and algebra. There was, in fact, quite a lot of contact
with Christian and Jewish scholars, exchanging ideas and
translating important works, in spite of periods of conflict and
pestilence. The so-called 'dark ages' in Europe were not so dark as
some have made out.
What is interesting in the story, is that for a long time, the
different developments in geometry described in this article were
motivated by purely practical aims; astronomy, pictorial
representation, and architecture. Each development had its own
theoretical basis and had relied on previous knowledge of earlier
mathematics and technology. For example, in the 10/11th century the
optics of Alhazen (Ibn al Haytham) was developed in an almost
modern scientific atmosphere that involved "criticising premises
and exercising caution in drawing conclusions and taking care in
all that we judge and criticise that we seek the truth and not be
swayed by opinions".
However, the unification of all these ideas did not even begin to
happen until the beginning of the 20th century. There were many
reasons, but for us as teachers, what we can note particularly is
the reluctance to take on new ideas, to be open to the 'what if'
possibilities, and the gradual abstraction of geometry and algebra
in the 19th century which brought about the unification of large
parts of mathematics based on ideas of structure
Now we can see the different branches of mathematics as being parts
of a whole, but importantly, no part is really separate from the
others. We describe geometry in algebraic terms (due to people like
Descartes and Galois), but we also see algebra as geometric
patterns (from the work of Poincare and Mandelbrot). There are
always opportunities in our teaching to make links between 'topics'
and break down the barriers that curriculum and text-book chapters
have created in the cause of 'efficiency'.
An excellent book on the subject is by Judith Field,
The Invention of Infinity:
Mathematics and Art in the Renaissance (OUP 1997). The
picture below is a variation of Piero's 'pavement' technique. You
can use the picture and the questions to work with pupils on
developing ideas on similarity and perspective. A copy of this
image can be found in this Word document.
How would you describe this picture?
There are a number of similar shapes here. Can you find them
and explain why they are similar? How would you describe the
relation between the shapes?
Can you extend the picture and draw another row of shapes on
the bottom? Start in the bottom right hand corner and draw a
diagonal of one of the shapes.
Extend this line across the picture. What do you notice?
Here is another exercise: