Earlier: NCLB polynomial -> Phi -> Convergence of Fibonacci NUMB3RS.
So what're the Fibonaccis again?
0 1 1 2 3 5 8 13... and like that. Fib(n+1)/Fib(n) approaches phi as n increases, where Fib(n) is the nth term, like Fib(0) = 0, Fib(1) = 1, Fib(2) = 1.
Recall our use of parentheses (curved braces) to treat objects as callables, i.e. as things we might feed arguments to and get behavior from, a function or method.
We use square brackets to indicate indexing, e.g. Fib might point into a list. Of course, behind the scenes we might convert indexing into a function, by overriding __getitem__ in Python. Click here for examples.
We also use square brackets to point into a mapping, such as a dictionary, using keys, such as character strings.
Phi is the ratio of a regular pentagon's diagonal to an edge i.e. phi = d/e. The five-fold symmetry of the pentagon associates with the icosahedron and pentagonal dodecahedron, duals of one another, and both members of the Platonic Set (see Chapter 1).
We will now bridge from five-fold symmetry (quasi crystals, viruses, geodesic spheres, fullerenes) to the lattice symmetries of Cube and Associates i.e. the remaining Platonics. This is our familiar XYZ world of uniform omnidirectional extensibility. We also address this same space with the IVM, a complementary skeletal framework (see Chapter 3).
The cube, recall, fractions into six equal half-pyramids, their apexes at the cube's center, to which we assigned volume 1/2 (see Chapter 4). Therefore six of them comprise a volume 3 cube, which embraces Kepler's Stella Octangula (see index), two intersecting tetrahedra of volume 1 each. Two of the aforementioned half-pyramids, square base to square base, comprise our Coupler, a space-filling squashed octahedron, likewise of volume 1.
Each/either of the two cube-supporting tetrahedra fraction into 24 A modules, 12 left handed and 12 right handed. Combined with left and right B modules, likewise of volume 1/24, we have the means to assemble many of our friendly XYZ/IVM lattice-embedded shapes.
Later, when we get back to 5-fold symmetry (perhaps via the Jitterbug Transformation -- depends on the hypertoon), we'll take a look at the S, E and T modules (see Chapter 10 and 'For Further Reading' in the appendix).
Much of our workstation time/energy will involve exploring the mathematical tools we might use to explore these kinds of spatial compositions and transformations. We will continue making use of Python and friends, alien languages (whatever you have time to take on -- this text cannot address your specific circumstances so please feel free to resequence our contributions into your studies as needed).
We will also make use of POV-Ray, a high quality free ray tracer. The Python + POV-Ray synergy will provide a strong platform on which to base later understanding of computer graphics and animation. "Geometric cartooning" is something we consider fun in this neck of the woods. We hope you will agree and join us in work and in play.