8
Fig 32
In Fig 32 Line a is part of the Lucas series.
Line b is part of the Fibonacci series.
Line c is part of the Fibonacci series offset from line b by 2 places to the right.

Add a number from Line b to one directly below in Line c and the sum of the two Fibonacci numbers is the Lucas series number directly above in Line a.
Another approach to the connexions between the two series is to consider how phi to the nth power closely approximates the nth term of the Lucas series.

(1.6181) to the 9th power approximately = the 9th term of the Lucas series, 76. [76.0411]...
In 1837 the Bravais brothers called attention to the divergence angle of 137.5°. One mathematical route to this angle is to apply the phi ratio to the circle: 360°*0.6181= 222.5°, that is the reflex angle - the difference between it and 360° being 137.5°.
But the ratio between successive terms of the Lucas series also converges on phi (so does the ratio in any number series, rational or irrational, which produces the next term by adding the two previous terms). And the numbers of the Lucas series are different from the Fibonacci numbers, so cannot be expressed in plants as points of best balance by use of the same divergence angle. How, then, to find it?
First:
 a few relationships between phi and the circle...starting with k and subtracting 1 from the divisor each time.

k: 360°/2.6181 = 137.5°
l: 360°/1.6181 = 222.5°
m: 360°/0.6181 = 582.5° (360°+ 222.5°)
n: 360°/-0.3819 (0.6181 - 1) = -942.5° (-[720° + 222.5°])
The first three examples provide 942.5° of positive rotation (137.5°+ 222.5°+ 582.5°), and the last is good for 942.5° of negative rotation. So...plenty of action...but things are pretty much where they started... still in Fibonacciland.

Which naturally focuses attention on what may be happening either side of those four examples:
Then:
 j: 360°/3.6181 = 99.5°
o: 360°/-1.3819 = -260.5° (99.5°+ 260.5°= 360°)
Lucas City!
The Lucas cyberflowers in Fig 33 show what the first fifteen iterations at 99.5° produce...
Fig 33
Where there are only two colors in a diagram the Lucas numbers occur. These are the points of best balance.
Because of the relative smallness of the 99.5° angle it is not till the 5th iteration that the starting point is crossed. This "allows" 4 to enter the Lucas series. The 137.5° angle of the Fibonacci cyberflowers causes the 4th iteration to cross, ruling out 4, and letting in 5.
Fig 34
Fig 35
Interesting also is Fig 34, a picture of a large number of iterations at 99.5°:

Not only does the ghostly cross suggest a basis of four right angles - hardly surprising - but a pair of gaps meets to form a precise right angle - surprising.

 More interest still from Fig 35 as iterations of 260.5°, having lost a section of its right angle aob early on...

then goes on to display...angle aod -137.5°...and, inside it (cool), angle aoc - 42.5°.