ash wednesday, 1998, st. francis hospital, evanston, il,
		marek lugowski, observations on computing with
		group theoretical discrete geometry -- the logo of the
		allegiance healthcare corporation of mcgaw park, il, 
		60085-6787 u.s., rev b 2/97
		------------------------------------------------------------

		the logo is a stylized equilateral triangle of heavy
		("emboldened") dots of a "graph paper" lattice of dots
		itself masked off to represent a superposition as the
		greek cross, e.g. as on the flag of switzerland:

		                       . . c
		                       . c c
		                   . . c c c . .
		                   . c c c c . .
		                   c c c c c . .
		                       . . .
		                       . . .

		observe that the lattice can be decomposed into groups of
		three points such that:    .
		                         . .

	        this triangle can be had many ways, and in fact, we don't
		insist on a trianglness -- just togetherness of three
		dots, or topological sameness in lattice space.

		for example, here are two transformations, one preserving
		and one not preserving this sameness:

		                      .
		                    . .   =goes to=>                . . .

                                      .   <>does-not-topo-go-to   .   . .
                                    . .
               
		the logo can be thought of mathematically as a field, and
		an algebra can be easily specified (e.g., via Java
		programming, for example) to carry out a transformation on
		the lattice from an arbitrary starting state to the logo
		displaying state via a sequence of flows, or nearest-neighbor
		position swaps involving the members of the discrete
		geometrical field as specified above.

		more precisely, the members, or "tiles", as introduced below,
		are autonomous agents and perform this operation in parallel,
		on ach other, without any need for supervision, always
		congregating on the outcome, even though each may well take
		a different individual route on different runs of the 
		computation, were it repeated.

		arbitrary means exactly so, without constraint:  the process
		may start anywhere -- but the outcome will always end on
		the logo, if the correct algebra is used for the nearest-
		neighbor computation.  this algebra depends only on the
		information available on simple inspection in the embedding
		hyperspace:  a smooth path from any starting point to the
		coordinates co-locating the logo, as seen in that hyperspace,
		is the entirety of the computation in the lattice space.

		a simple visualization tool for manipulating appropriate
		information in hyperspaces and subspaces geometrically
		ought to be all the programming interface one needs to make
		this computing technology useful and intuitive even to
		non-programmers.

		in point of fact, this computation can be performed
		using the already existing tile-swapping formalism dubbed
		computational metabolism (commet) in 1985 by me during my
		post-masters graduate work at the indiana university
		computer science department (technical report #200, 
                "computational metabolism").  it is further described in
  		in the reference (1).  some modifications of the model
		in the exact rule set may be needed, such as color-changing.

		a key property needed is overlapping commet's own tiles
		over one another in this tile space:

		              ----------------
		              |              |
                              |    . .       |
                          ----|------------  |
		          |   |    . c    |  |
		          |   ------------|---
		          |        c c    |
		          |               |
		          -----------------    (spread <-> for clarity)


		then various decision procedures (algorithms) can be
		devised to switch lattice sites to "on" ("c") and to
		"off" ("."), in accordance with the algebra which gives
		rise to the masked outcome we want to obtain.  

		because the behavior

		of the field is encoded into each of the tiles at the
		beginning of the computation and because they all carry
		their results as they swap places with one another, the
		overall goal is guaranteed to happen as it is nothing
		else but a sorting operation in k dimensions at once.

		any errors in this poem stem from the author not knowing
		any mathematics, being mentally ill, and writing under
		the impression of having discovered or rephrased something
		important.  heavy medications, being in love with multiple
		women and not having to go to work the next day unlike you
		stiffs are contributing factors.  corrections especially
		from mathematicians and people with free supercomputer time
		are most welome.  thank you.

		footnote:

		(1)  _artificial life, volume 1_, edited by christopher
                g. langton, the proceedings of an interdisciplinary
		workshop on the synthesis and simulation of living
		systems held september, 1987, in los alamos, new mexico,
		"computational metabolism: towards biological geometries
		for computing", marek lugowski, pp. 341-368, addison-wesley,
		1989, isbn 0-201-09346-4 (hard), 0-201-09356-1 (paper).


				Marek Lugowski
				Ash Wednesday (26 February 1998)
				Evanston, Illinois