Exponential Assembly Top Down Nano

NanotechNews.com writes: "The article describes a new milestone in the Top Down nanotechnology process: "Exponential assembly is a manufacturing architecture starting with a single tiny robotic arm on a surface. This first robotic arm makes a second robotic arm on a facing surface by picking up miniature parts ? carefully laid out in advance in exactly the right locations so the tiny robotic arm can find them ? and assembling them. This is an exponential growth rate, hence the name exponential assembly." Standard MEMS, the largest independent high-volume manufacturer of Micro Electro Mechanical Systems and Zyvex created a partnership, the second article available here. This partnership could lead to a better assembling technology in MEMS and the Top-Down Nanotechnology and Nanolithography."

dns style?

what i want to know is, ...does each arm have the knowledge - independant of each other or something else - of how to complete it's job.

My .02,

Hitchhiker's Guide was there first

Anybody else remember that scene in Mostly Harmless where Ford Prefect breaks down the door to the head editor's office?

There are little nanotech bots in the doorframe whose sole purpose in life is to wait until this happens. Then they crawl out of the frame, assemble each other into larger bots, rebuild the door, disassemble each other, crawl back into frame, and wait...

Anyhow, I know some people working with MEMS. Very cool stuff.

Richard 'God' Feynman

Elsewhere on Xyzzy's site you can find the original nanotech (and, indeed, MEMS to nano) talk; "There's Plenty of Room at the Bottom". If you've read anything about Feynman, you're already a fan. If you haven't, this [zyvex.com] is a good place to start.

How does this work?

Okay, picture a 2-D surface upon which these parts are laid. Now there is one arm in the middle of this. Let's say that each arm produces another arm in t=1. Now there are 2 arms, then 4, etc... Hence the O(2^n) sorta growth rate.

Here is my question. Assuming the arms are stationary, it is reasonable to assume that they can only build an arm adjacent to itself (and if they move, moving would be a O(n) process).... This means that for any grid area n^2, there are(n+2)^2 adjacent squares.

Because of this fact, I don't see how these things can achieve any more than a O(n^2) growth rate, because the adjacent resources available to these bots would be O(n^2).

Anyone know how these buggers get around this limitation?

Re:How does this work?

First, I think you are correct in pointing out that exponential growth cannot strictly apply for very long before the system becomes "starved" for resources. This is true in natural populations, and we expect the same should apply here.

However, your order estimates are incorrect.

This means that for any grid area n^2, there are(n+2)^2 adjacent squares.

Yes, and the rate of growth is determined by the difference between these two, which is O (n), not O(n^2).

Because of this fact, I don't see how these things can achieve any more than a O(n^2) growth rate, because the adjacent resources available to these bots would be O(n^2).

In fact, it is O (n). To easily visualize this, imagine the system in 1D for a moment. After the first unit assembles its nearest neighbors, each additional unit builds the next unit at the end of the line of units. This leads to a constant rate of growth. In 2D, the rate of growth is determined by the rate of change of the area, not length, which leads to O (n). In 3D, the rate of growth is determined by the rate of change of the volume, which leads to O (n^2). The result in each case is easy to visualize : it is limited by the boundary of the N-D volume the units have already filled.

All this said, I think this whole discussion doesn't emphasize that even an O (n) growth rate can be vastly enormous for large enough n. The main problem is that it appears much slower to start up than a truly exponential process, which could be realized for a longer duration if the newly built units were "mixed" randomly into the grid. This could be the case if each unit could be assigned to move some random distance under its own power, after each building cycle. Eventually, however, the exponential rate of growth will turn over when the system's capacity is reached.