Verify Conway’s construction for taking cube roots

Verify Conway’s construction for taking cube roots using a compass and a straightedge with a unit distance marked on it.


Suppose we have constructed a distance k, with 0 < k < 1, with endpoints A and F. Construct the line perpendicular to AF through F, and construct the point B on this line whose distance from F is \sqrt{1-k^2}. (Note that, by the Pythagorean theorem, the distance from A to B is 1.) Now construct E on FB such that F is between E and B and the distance from E to F is \frac{1}{3}\sqrt{1-k^2}. Extend the line through A and E. Now, using the marked straightedge, construct the line passing through B whose points of intersection with AF and AE are separated by a distance of 1. Call these points C and D, and call the distances BC and AD a and b, respectively, as in the following diagram.

Finally, construct the lines through C perpendicular to AF and BE, and call the distances from C to these lines y and x, respectively, as we have labeled in green in the following diagram.

Now \triangle BFD and the triangle with vertices C and D and a side length y are similar, so that (1) \dfrac{y}{1} = \dfrac{\sqrt{1-k^2}}{1+a}. Similarly, \triangle BFD and the triangle with vertices B and C and side length x are similar, so that (2) \dfrac{x}{a} = \dfrac{b+k}{1+a}. Likewise, \triangle FAE and the triangle having vertices A and C and a side length y are similar, so that (3) \dfrac{y}{x-k} = \dfrac{\frac{1}{3}\sqrt{1-k^2}}{k} = \dfrac{\sqrt{1-k^2}}{3k}.

Finally, using the Pythagorean theorem on \triangle BDF, we have (4) (1-k^2) + (b+k)^2 = (1+a)^2. We can expand (4) and simplify to get the equation (4′) b^2+2bk = 2a+a^2.

Solving (1) and (3) for y/\sqrt{1-k^2}, we see that \dfrac{1}{1+a} = \dfrac{x-k}{3k}. Solve for x to get (5) x = k + \dfrac{3k}{1+a}. Equating with (2), we have \dfrac{3k}{1+a} + k = \dfrac{a(b+k)}{1+a}, which simplifies to (6) ab = 4k. Using (6) to eliminate b in (4′), we have a = 2k^{2/3}. (Hint: clear denominators and note that 2+a is a factor.) Similarly, eliminating a yields b = 2k^{1/3}, as desired.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: