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:
x2 = 2 + √(2 − √(2 − x))
Rearrange, and square both members at the risk of generating extraneous roots:
x2 − 2 = √(2 − √(2 − x))
x4 − 4x2 + 4 = 2 − √(2 − x)
Again, rearrange and square both members:
x4 − 4x2 + 2 = − √(2 − x)
x8 + 16x4 + 4
− 8x6 + 4x4 − 16x2
= 2 − x
Finally, rearrange to form an eighth-degree polynomial:
x8 − 8x6 + 20x4 − 16x2 + x + 2 = 0
which factors into:
(x3 − x2 − 2x + 1) (x4 − x3 − 3x2 + 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:
| r2 | = 2 + t |
| t2 | = 2 + x |
| v2 | = 2 − v |
| x2 | = 2 − r |
and that helps explain why they appear as extraneous roots.