The Geometry of the Binomial Theorem
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 ndimensional 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 corresponds to the geometric operation of breaking a
segment of length m into two segments of length p and
q. If we break a vertical edge of a square having side length
m into two segments and draw a horizontal line through the break
point, we decompose the square into two rectangles. Then doing the same
with a horizontal edge produces a decomposition of the square into four
pieces: a square of side length p, a second square of side length
q, and two rectangles having side lengths p and q. The
four parts of this geometric decomposition correspond to the four terms of
the expansion of the square of a binomial. The algebraic problem of finding
the square of expression p + q is
equivalent to the geometric problem of finding the area of a square having
side length p + q. The rules of
algebra enable us to multiply out the square of a binomial, without having
to appeal to a geometric diagram:
(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:
(p + q)^{2} =
p^{2} + 2pq + q^{2}
A glance at the diagram below makes the relationship very clear. Each
term of the expression p^{2} + pq + qp
+ q^{2} gives the area of
one of the four pieces that form the square. The areas of the pieces add up
to the area of the original square, namely (p + q)^{2}.

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 in threespace
to the third power of the binomial p + q. The algebraic procedure is direct
enough:
(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 p + q can
be decomposed into eight pieces: a cube of side length p, another of
side length q, and six square prisms, three having height p
and square base of side length q, and three having height q
and square base of side length p.

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
(p + q^{4})
= p^{4} + 4p^{3}q
+ 6p^{2}q^{2}
+ 4pq^{3} + q^{4}
which does not rely on any geometric argument. Nevertheless we can consider
a hypercube of side length p +
q and describe a geometric decomposition analogous to the
threedimensional case. This decomposition will have 16 pieces: a hypercube
of side length p, another of side length q, and 8
fourdimensional prisms with cubical bases, 4 having height p and
cubical base of side length q, and 4 having height q and
cubical base of side length p. In addition there are 6 new objects,
fourdimensional prisms having at each vertex two edges of length p
and two of length q. Thus the geometric pattern generalizes even though we
cannot construct in our space a fourdimensional model that will illustrate
the decomposition.
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 
[an error occurred while processing this directive]