The binomial theorem gives a famous algebraic formula for the sum of two
numbers raised to a power. There is a corresponding geometric expression
for the volume of an n-dimensional cube with each edge broken into
two segments. Earlier in this chapter we considered squares having side
length m and area m^{2}. If we express m as a
sum of two numbers p and q, then the algebraic operation of
writing m as a sum
(p + q)^{2} | = | (p + q)(p + q) |
= | p(p + q) + q(p + q) | |
= | p^{2} + pq + qp + q^{2} |
We may combine the two terms pq and qp to obtain the familiar expression for the square of a binomial:
A glance at the diagram below makes the relationship very clear. Each
term of the expression
| |
A geometric interpretation of the square of a binomial. |
There is a similar pattern relating the volume of a cube having side length
(p + q)^{2} | = | (p + q)(p + q)^{2} |
= | p(p + q)^{2} + q(p + q)^{2} | |
= | p(p^{2} + 2pq + q^{2}) + q(p^{2} + 2pq + q^{2}) | |
= | (p^{3} + 2p^{2}q + pq^{2}) + (qp^{2} + 2pq^{2} + q^{3}) | |
= | p^{3} + 3p^{2}q + 3pq^{2} + q^{3} |
Once again, this argument can take place independent of a geometric
interpretation, but nonetheless it is useful to give one. A cube having
side length
| |
A geometric interpretation of the cube of a binomial. |
What happens to these algebraic and geometric patterns in the fourth dimension? As above, we can obtain the algebraic expression
which does not rely on any geometric argument. Nevertheless we can consider
a hypercube of side length
The sequences of coefficients that appear in the expansion of binomials are the rows in Pascal's triangle, the famous number pattern that arises in the theory of combinations. Each number in the pattern is the sum of the two numbers above it. We will see these same patterns of numbers occurring in several different guises as we analyze the structures of objects in higher dimensions.
1 | ||||||||
1 | 1 | |||||||
1 | 2 | 1 | ||||||
1 | 3 | 3 | 1 | |||||
1 | 4 | 6 | 4 | 1 |
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. |