In solid constructions, we allow for the
use of a (possibly only hypothetical) conic drawing tool. Given a
constructible point A, a constructible line l, and a
constructible length e, we are permitted to
draw the conic section with focus A, directrix l, and
eccentricity e. The constructible points in this context
are then the points where a pair of (distinct) constructible lines,
circles, or conics intersect.
In particular, we can draw the parabola y = x2.
Consider the circle that is centered at (a,b) and goes
through the origin. This circle has the equation
(x - a)2
+ (y - b)2 = a2 + b2.
If a point (x, y), other than the origin, also lies on
the parabola , y = x2 then
(x - a)2
+ (y - b)2 = a2 + b2,
x2 - 2ax
+ x4 - 2bx2 = 0,
| x3 + (1 - 2b)x - 2a = 0. |
(1)
|
The regular 7-gon has a solid construction, a fact that was known to
Archimedes. To construct it, we must construct x = e2pi/7, which satisfies the equation
x7 - 1 =
(x - 1)(x6 + x5 + x4
+ x3 + x2 + x + 1) = 0.
Let w = x + x-1 = 2cos(2p/7). Then
w3 = x3
+ 3x + 3x-1 + x-3,
w2 = x2
+ 2 +x-2,
so
w3+ w2
- 2w - 1 = 0.
To exploit (1), we complete the cube. That is, we make the
substitution z = w + 1/3, so that
(z - 1/3)3 +
(z - 1/3)2 - 2(z - 1/3) - 1 = 0,
which simplifies to
z3- (7/3)z
- 7/27 = 0.
We therefore choose (a,b) = (7/54,5/3) and proceed as
follows (following along in the figure): We construct the circle
centered at (7/54,5/3) that goes through the origin. We drop a
perpendicular from a point of intersection of this circle with the
parabola y = x2, to find the point (w +
1/3,0). We find the point (w/2,0) and the
perpendicular to the x-axis through this point. This
perpendicular intersects the unit circle at two of the seven points of
a regular
7-gon. We use those points to find the rest of the vertices.