Toothed Teeth

dentes cum dentes...

It's time that we start discussing software, but that means you'll have to start do some maths to follow along.  We'll attempt to keep the maths real by using only compass and straight-edge, however even these simple constructs lead to some interesting Archimedean wormholes.  I'll demonstrate a few drawing machines along the way as we philosophise about Archimedes' Circles and Geometric Series.

I've been distracted Lindenmayer Systems lately, but I'll returning to some simple geometric constructs again soon.  Sometimes, I spend a lot of time playing with an idea, translating and morphing the form again and again.  L-systems are highly language restricted, but that's why we have compilers, higher order to turtles; Sophus Lie and the Lie Algebra named after him.

Notes:
-Poincaré Hyperbolic Disks.  As it turns out that Astronomical observations of planetary object, made from Earth take on a distinctively hyperbolic form. 
-knots, Not Knot.
-Inca method of calculation.  Modify 'quipu' into rope timing belts and construct slide-ruler/Lehmer-Sieve with exponential, trigonometric and multiplication functions.
-L-system Algebra and Transformations; rotations/translation.
-astrolabe, slide-ruler and stick computers
-Euclid's fifth postulate (Ελάφια Ευκλείδης, έχετε τέτοιες όμορφες ιδέες!)

to be continued...

...some time earlier... 

Manual drive.

Two layers are driven in opposite directions at twice the input frequency resulting in a differential frequency between them of four times the input frequency.  High frequency clocks are useful for sampling the more slowly clocked memory or display discs.  It has all to do with the fundamental quantities of information.

Even the teeth have teeth.

You might not be surprised that Information Theory has named one of those quantities, "Self-information".  Claude Shannon (1916-2001) had bit say about this subject.  It's interesting to note that Shannon worked with Vannevar Bush (1890-1974) on a mechanical analogue computer called a differential analyser whilst at MIT.

Vannevar's differential analyser.  (source: http://www.computerhistory.org)

Bush's differential analyser was a hairy beast of network upon network of complicated compound gears and rollers.  Sound familiar?  One day after analysing this device, Shannon got up from his seat, ran across the room and jumped out the window (I made that up).   From then on he dedicated his life's work to Symbolic Logic, Boolean Algebra and Digital Computers (That bit is true).  In 1937 wrote his master's thesis on, "A Symbolic Analysis of Relay and Switching Circuits."

Claude Shannon jumping from Vannevar Bush's Laboratory.

There's no jumping out the window for us.  The turtles are piled up to the window sill.  The fire is for the "surprisal" factor.