Chapter 4: Geometry

4.7.7 CLASSICAL CONSTRUCTIONS
The ancient Greeks used straightedges and compasses to nd the solutions to numerical
problems. For example, they found square roots by constructing the geometric
mean of two segments. Three famous problems that have been proved intractable by
this method are:
1. The trisection of an arbitrary angle.
2. The squaring of the circle (the construction of a square whose area is equal to
that of a given circle).
3. The doubling of the cube (the construction of a cube with double the volume
of a given cube).
A regular polygon inscribed in the unit circle can be constructed by straightedge
and compass alone if, and only if, Ò has the form Ò ¾ Ô ½ Ô ¾ Ô , where is a
nonnegative integer and the Ô are zero or more distinct Fermat primes (primes of
the form ¾ ¾Ñ ·½). The only known Fermat primes are for 3, 5, 17, 257, and 65537,
corresponding to Ñ ¼ ½ ¾ ¿ . Thus, regular polygons can be constructed for
Ò ¿,4,5,6,8,10,12,15,16,17,20,24,...,257,....
4.8 COORDINATE SYSTEMS IN SPACE
4.8.0.1 Conventions
When we talk about “the point with coordinates ´Ü Ý Þµ” or “the surface with equation
´Ü Ý Þµ”, we always mean Cartesian coordinates. If a formula involves another
type of coordinates, this fact will be stated explicitly. Note that Section 4.1.2
has information on substitutions and transformations relevant to the three-dimensional
case.
4.8.1 CARTESIAN COORDINATES IN SPACE
In Cartesian coordinates (or rectangular coordinates), a point È is referred to by
three real numbers, indicating the positions of the perpendicular projections from the
point to three x ed, perpendicular, graduated lines, called the axes. If the coordinates
are denoted Ü Ý Þ, in that order, the axes are called the Ü-axis, etc., and we write
È ´Ü Ý Þµ. Often the Ü-axis is imagined to be horizontal and pointing roughly
toward the viewer (out of the page), the Ý-axis also horizontal and pointing more
or less to the right, and the Þ-axis vertical, pointing up. The system is called righthanded
if it can be rotated so the three axes are in this position. Figure 4.30 shows a
right-handed system. The point Ü ¼, Ý ¼, Þ ¼is the origin, where the three
axes intersect.
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