No Place Like Null Space

K[X]

Tsunami the Cat, in a broken Polynomial Ring.

In any era of time, Aryabhata would have been called a great mathematician and astronomer.  Aryabhata lived in India from 476-550 and made so many contributions to science and mathematics, he might seem to have come from the future; elliptical planetary orbits (beating Kepler to it by 1000 years), number series, a series approximation of π squares and cubes, trigonometry, and even the null of the place-value number system we know today.  Although, he didn't use a symbol for null coefficients.  Instead, he simply used a space.

Pointy eared Vulcan.

Aryabhata also demonstrated that π is irrational and created one of the first known sine tables, which is the subject of this post.  Traditionally, Indian mathematicians have represented the sine (jya) of an angle as a length of a line segment, rather than the ratio of site opposite to an angle to the length of the hypotenuse (opposite/hypotenuse).  It's all a matter of perspective and assigning some symbols.  Which quickly brings me to the point, some problems are more easily solved in certain domains than others.

The digital computer hardware of today is particularly well suited for bit-wise operations such as shifts and adds.  This why computer scientists expel so much effort transforming all other mathematical functions into this domain.  This is especially the case for computers without multiply functions, as is common in CPU's implement on an FPGA.

In 1959, Jack E. Volder first described an algorithm that could compute trigonometric functions as a series of additions, bit-shifts, and subtractions, and table lookups.  He was working on a navigational computer and wanted to replace the analogue angular position resolver with a digital encoder.   This algorithm has now been further extended to support hyperbolic functions, logarithms and exponential functions, square roots, multiplication and division.

Volder's algorithm.

I'm not going to beat you over the head with a lot of maths.  I'm saving that for the paper.  You can lookup the CORDIC algorithm and marvel at its use of trigonometric functions, rotational matrices and series.  However, what I would like you to consider is the following...

There exists an inverse to the CORDIC algorithm, where trigonometric functions and tables can be transformed into bit-wise operations.

Well, division, tables and trigonometric functions are stone-age simple with pegs and gears.  Do you see where I'm going with this?

Mister Turtle, we meet again.

My initial implementation of what I'm a calling an "Inverse CORDIC Mechanism" isn't a purely analogue (if that's even the right word, it's more like circular).  There's a few bits of binary here and there for control and data load/store, but I think you'll agree that being able to build up compound logic from a few gears and levers is rather clever.   I'll try to share as much as possible without going into the software, too much.  All that waxing and waning about "Whether 'tis nobler in the mind to suffer" is all a bit circular. 

"With this regard their currents turn awry,
And lose the name of action.-- Soft you now!" --Hamlet 87-88

***Stay tuned for next turtle filled episode where "The Philosopher" will be joining us as a guest blogger and we'll reveal the next version of Grey Goo the card game.