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**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:

A glance at the diagram below makes the relationship very clear. Each
term of the expression *p*^{2} + *pq* + *qp*
+ *q*^{2}*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**p* + *q*

(p + q)^{3} |
= | (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**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

which does not rely on any geometric argument. Nevertheless we can consider
a hypercube of side length *p* +
*q**p*, another of side length *q*, and 8
four-dimensional 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,
four-dimensional 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 four-dimensional 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 |

Diagonals of Cubes in Different Dimensions | ||

Table of Contents | ||

Fold-Out Models of Pyramids |