Subject: Delian Problem
Date: Wed, 14 Nov 2001 12:30:13 +0000
From: Roger Wibberley

Duplication of the Cube
Roger Wibberley

Proposition

The marks which lie along DB delimit the twelve mean proportionals between D and B. As such the successive magnitudes they encompass are equivalent to the successive powers of the twelfth root of 2 (signified as x).

Observations

  1. The 12 triangles whose bases lie along BD are similar since all corresponding sides are parallel.

  2. It must be shown that these same 12 triangles are disposed such that any adjacent pair has the same proportional relationship as any other adjacent pair.

  3. If such proportionality can be shown to apply to any corresponding portions of these triangles, then it will also have been shown to apply to all corresponding elements.

  4. It is axiomatic to the Ray Theorem that when diverging straight lines emanating from a common point are intersected by parallel lines, the corresponding segments into which such parallel lines are cut by these diverging lines will bear a common proportional relationship.

Proof

According to the Ray Theorem,

b:c = d:e = f:g = h:i = j:k = l:m = n:o = p:q = r:s = t:u = v:w.

Since a and b are opposite sides of the same parallelogram, a must equal b; therefore a:c = b:c. Since also d=c, and d:e=b:c, c:e=a:c. But f also = e. Since f:g=d:e, and e:g = f:g, therefore, e:g=c:e. Similarly h=g, j=i, l=k, n=m, p=o, r=q, t=s, and v=u.

Thus e:g (=f:g) = c:e; g:i (=h:i) = e:g; i:k (=h:i) = g:i; k:m (=l:m) = i:k; m:o (=n:m) = k:m; o:q (=p:q) = m:o; q:s (=r:s) = o:q; s:u (=t:u) = q:s; and u:w (=v:w) = s:u.

Since any adjacent pair of the corresponding sides a, c, e, g, i, k, m, o, q, s, u and w will consequently bear a common proportionality with any other adjacent pair, so do all other similarly corresponding elements (including the triangles themselves).

It is thus the case that adjacent lengths marked by points along DB (being the bases of these triangles) similarly bear a common proportional relationship with all other adjacent lengths similarly marked. This being so, these points must delineate the 12 mean proportionals along DB, whose values increase successively by the twelfth root of 2.

It would follow that the position marked x4 is an accurate construction for the cube root of 2 (being the twelfth root of 2 raised to the fourth power).

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