history of mathematics - National STEM Centre

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How does your
knowledge of
permutations help you to
deduce that there must
be six turning operators
when there are four
vectors?
148
Figure / 1.5
1 By analogy with the equations considered in two dimensions, find all the
equations satisfied by the vectors and the turning operations in three dimensions.
Notice that the result of 7 x k cannot be defined as a vector, since the operator 7 can
only act on vectors in the plane containing i and j to produce other vectors in the
same plane. In general 7 x k will produce a new object. Similar comments apply to
.7 x i and K x j.
2 Deduce the following.
/=y 2 =A-=-l
IxJ=-JxI=K
Activity / /. 15 Reflecting on algebra
These three results are the defining relations of quaternions, the name given to the
three turning operators /, J and K, which were invented by Hamilton.
Clifford's algebra in three dimensions is formed from the basic building blocks: the
scalars; vectors; bi vectors or the turning operators ixj, ixk, jxk; and the
trivector i x j x k. All other distances, steps and turns can be produced from these
basic building blocks.
So far, all your extensions of Clifford's original approach have been to spaces which
you can visualise. In two and three dimensions, you can draw diagrams to represent
the building blocks. Just because you cannot visualise, or draw pictures in,
dimensions any higher than three does not mean that there are no more Clifford
algebras to discover.
You can produce a four-dimensional algebra by generalising the three-dimensional
Clifford algebra. Instead of using three basic vectors, you can use four. There will
then be six turning operators, and so on.
A complete generalisation of the Clifford algebra is obtained when you do not
specify a particular number of dimensions - and hence the number of basic steps -
instead you specify how to construct the algebra for n basic vectors. The method of
construction would work satisfactorily for any value ofn, 1, 2, 3, 27, 102, or
whatever you choose.
1 Find an example from Books 1 to 5, or from your wider experience, where the
elements of a collection of objects other than numbers are being combined by two
different operations, rather like addition and multiplication of real numbers,