How Math Explains the World.pdf

straightedge-and-compass construction. To this day Lindemann gets the

credit, although much of the earlier work was done by the French math-

ematician Charles Hermite. Lindemann’s proof of the transcendentality

of  is similar to Hermite’s proof that e, the base of the natural logarithm,

is transcendental. In the nineteenth century, fame was the only reward

for a mathematician—although today there are monetary prizes offered

as incentives for the solutions of major problems. Then, as now, fame

generally accrued to the person who placed the final brick in place on the

edifice, rather than those who laid the foundation.^17

Learning from Impossibility
All of the problems investigated in this chapter are great problems. A
great problem is generally relatively simple to explain, piques our curios-
ity, is difficult to resolve, and has a resolution that extends the bounds of
what we know beyond the problem itself. It makes us question whether
the assumptions we have made are sufficient to solve the problem, and
whether the tools we have are adequate for the job.
The quest to double the cube and trisect the angle led to explorations far
beyond the simple structures of line and circle that constitutes Euclidean
plane geometry. The axioms of plane geometry, as given in the first book
of Euclid’s Elements, are

1. Any two points can be joined by a straight line.


2. Any straight line segment can be extended indefinitely in a straight
   line.


3. Given any straight line segment, a circle can be drawn having the
   segment as radius and one endpoint as center.


4. All right angles are congruent.


5. (Parallel postulate) If two lines intersect a third in such a way that the
   sum of the inner angles on one side is less than two right angles,
   then the two lines inevitably must intersect each other on that side if
   extended far enough.^18

The above postulates discuss only points, lines, angles, and circles. Even

though an outline of the Elements reveals both plane and solid geometry,

the geometrical figures that are discussed are polygons and polyhedra,

circles and spheres. The duplication methods proposed by Archytas, Me-

naechmus, and Eratosthenes certainly transcend Euclidean geometry as

outlined in the Elements.

Attempts to square the circle led to a deeper analysis of the real line and

the concept of number. The resolution of the problem of constructing

Impossible Constructions 75