Recall:
R^{ 2} | = 2 + T |
T^{ 2} | = 2 − V |
V^{ 2} | = 2 − R |
Substitute:
R^{ 2} = 2 + √(2 − V)
R^{ 2} = 2 + √(2 − √(2 − R))
For clarity later, change the symbol R to x:
x^{2} = 2 + √(2 − √(2 − x))
Rearrange, and square both members at the risk of generating extraneous roots:
x^{2} − 2 = √(2 − √(2 − x))
x^{4} − 4x^{2} + 4 = 2 − √(2 − x)
Again, rearrange and square both members:
x^{4} − 4x^{2} + 2 = − √(2 − x)
x^{8} + 16x^{4} + 4
− 8x^{6} + 4x^{4} − 16x^{2}
= 2 − x
Finally, rearrange to form an eighth-degree polynomial:
x^{8} − 8x^{6} + 20x^{4} − 16x^{2} + x + 2 = 0
which factors into:
(x^{3} − x^{2} − 2x + 1) (x^{4} − x^{3} − 3x^{2} + 2x + 1) (x + 2) = 0
Three of the roots are pertinent to septrights:
+ 2 cos (180° / 7) = + R |
− 2 cos (360° / 7) = − T |
+ 2 cos (540° / 7) = + V |
But four other roots, which are extraneous, come from not septrights but rather nonarights:
+ 2 cos 20° |
− 2 cos 40° |
+ 2 cos 60° = + 1 |
− 2 cos 80° |
And one root might go either way:
− 2 cos 0° = − 2
Where did those nonaright values come from? To construct nonarights we can define these:
q | = 2 cos 10° |
r | = 2 cos 20° |
s | = 2 cos 30° |
t | = 2 cos 40° |
u | = 2 cos 50° |
v | = 2 cos 60° |
w | = 2 cos 70° |
x | = 2 cos 80° |
which lead to the following formulae, which look very much like the R-T-V formulae above:
r^{2} | = 2 + t |
t^{2} | = 2 + x |
v^{2} | = 2 − v |
x^{2} | = 2 − r |
and that helps explain why they appear as extraneous roots.