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A square of side length | ||

Patterns that arise in the formulas of plane geometry often correspond to
patterns in the formulas of solid geometry, and they may suggest analogous
relationships in higher dimensions. One of the simplest of these patterns
arises when we measure the length or area or volume of the basic building
blocks in any dimension -- namely a segment in a line, a square in a plane,
and a cube in space. In the line, a segment has length an if we can cover
it exactly by *m* segments of unit length. Similarly in two
dimensions, square with side length *m* can be filled exactly with
*m*^{2} unit squares. And in three dimensions, a cube having
side length *m* can be filled exactly with *m*^{3} unit
cubes. The pattern is in the exponents: in dimension *n*, the volume
of an *n*-cube having sides of length *m* is
*n*^{2}, so a four-dimensional cube having sides of length *m*
would be filled with exactly m^{4} unit four-dimensional cubes.

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A cube of side length |

But is there a geometric configuration that corresponds to this expression
*m*^{4}? It would have to be four-dimensional, and indeed it does
exist in four-dimensional space, if we interpret the word *exist* in a
different way than in ordinary speech. When we speak informally, about a
square, we think of chalkboard sketches, or much more exact renditions on
an architect's table or on a computer-aided drafting device. Yet the formal
geometric theorems pertaining to a square are about none of these physical
representations, but about the abstract idea of the square, more perfect
than anything we can construct. As the followers of Plato might put it, the
ideal square exists only in the mind of God. It is such an ideal square
that has side length exactly in covered precisely by *m*^{2} unit
squares. Similarly the volume formulas refer to perfect cubes, not to the
physical representations we see around us. So it is also that the
four-dimensional version of this algebraic expression corresponds to an
ideal object, a hypercube having side length, existing in the mind of that
same God, and filled with *m*^{4} perfect unit hypercubes. The
difference is that we are able to construct in space a model of the
*m*^{4} solid cubes, while it is not possible for us to build a
similar model of *m*^{4} hypercubes.

Volume Patterns for Pyramids | ||

Table of Contents | ||

Introduction |