For a while now I am thinking about Mutilated Multilayer Gear system, and it's ability to express logic.
The basic idea is that we can take a regular "gear" with 6 or 8 teeth, and remove some of the teeth (this is called mutilated gear), then we can use several layers of such mutilated gears to add logic robustness.
Here is an example:
MMG_3dview.gif (587.83 KiB) Viewed 14653 times
I've build an emulator of Grid of such Gears, but have no time to continue the project to emulate fully the CGOL logic (or at least rule 110).
Check out this code and this repository.
Anyway I've built a binary adder, and made a nice presentation explaining how this system works.
Check out this page with the presentation.
P.S. The larger context of this project is to provide mechanical brain alternative to electronics.
1. One idea is if the gears would be very small - like several atoms size, the system could even compete with electronics.
2. Another option is a case where electronics is unavailable. For example a von-neuman probe, and for less sci-fi we could build heat resistant brain, that could function in high temperature, like digging robots that can get deeper into the earth crust.
P.S. A more visual example of the MMG grid:
Multi_Layer_Gear_Grid.gif (6.8 MiB) Viewed 14642 times
I was thinking a little bit about physical implementation of this system. It would obviously be better stored inside a cuboid rather than on plane.
So I will need a new notion: MultiStage Mutilated Multilayer Gear System. To be precise each such "vertical rod" is built from stacks of MMGs, and each stage is independent of any other stage. That is lets say we have 6 levels of gears inside a single rod, it could be that:
Layer 0-2 and Layer 3-5 are rotating independently while inside each layer tripplet they are rotating together.
And it could be Laeyr 0-5 all rotating together as one rigid body.
And it could be Layers 0-5 each is independent single layer mutilated gear.
Generally speaking in SMMG Staged Mutilated Multilayered Gear system we have in each 2d point a rod with mutilated single gears G1, G2, ...., Gn
S.t. they can be separated into rigid MMG_k = {Gi,...Gi+j} for j in MMG_k's start and end. And unifying all MMG_k for j 1 to m will cover all Gi without intersections.
Now with this construction we can prove that it can be folded:
Theorem: SMMG of size (2W)xHxD can be folded into SMMG of size (W+S)XHx(2D+S) where S is number of stages, keeping the same functionality.
Proof:
Assuming we have G0, with |G0| = (2W)xHxD
now lets take the two SMMG of size WxHxD, one next to other (left of the fold and right of the fold)
we can add another "fixed gear" on the edge that will "simulate neighbour's rotation".
That is if we make two multi stage pieces of size WxHxD one on top of the other, and "stitch them by the crease using "copy rods" which are fully rigid MMG, that for two neighbouring MMGs in the original that are after separation are on two different stages, will have "full rotation copy" of the bottom, by adding another layer "full copy gear" on the bottom, and the same teeth as the 180 degree rotated bottom to fit the 180 degree rotated top. You can say we made one rigid rod that will simulate "touch of rotation", by fully copying or "connecting" the behaviour of the rod as of the bottom, but transmiting its state deterministically and fully to the top. Thus providing full functionality as if the rod didn't existed and it was the original system.
But if we have S stages, each stage generates a unique rod with the respective separated MMGs logic. So actually if we have S stages, each stage need to be transmitted individually with separated rod individually. So two stage system will become four stage with extra two rods to attach each stage to itself.
All this is true up to direction of rotation. But as those two stages are completely separated reversing the direction will keep the same logic and functionality.
QED
Explanation of significance: if have MMG grid of dimentions 100x100x3 we can compress it into a cube of size 52x51x15 using this theorem. Then by padding the original we can make it after another two folds 28x32x68 with 16 stages and now there is no way to use the theorem to compress the cuboid even more.
So I was thinking that if we already doing a multi stage system, then all grid wires intersections can be solved with another level instead of say transforming the signal through two xor gates.
Also multistage is physically a little bit harder to produce as we need to make sure each MMG is moving separately in each stage, but production a side - this seems to be like a "magic box" that can be reduced in width and height significantly in cost of additional depth
I imagine a miniature gear box with physical representation of SMMG, and an engines attached to this box, and when you activate the engine, tens of thousands of gears are rotating at once, implementing CGOL logic, and on top of this box there is a black and white glass half transparent glass, thought which you can see the gear rotating, and which iterates CGOL logic flipping every 1024 rotations by CGOL logic.
P.S - This box will need to have on the other side a special "setup stage buttons". By pressing the reset button - the CGOL state is erased, and by pressing each of the CGOL grid buttons, it flips the initial state to 1. After that - pressing plat button, will start CGOL simulation again.
I've attached a very bad attempt to give a feel for this "CGOL gear box"
I have made an example of wire with delays. That is we can pass a signal along a gear system instantly using regular full gears, but sometimes we want to delay it by several ticks. For example for AND gate we will need to setup signal at time T and T + 1 and only then rotate the gear and continue the signal and otherwise the signal should die out, and the whole system reset. All this is possible only when our wires can adjust timing of the signal.
Here is an example, the delay in this case is 4 + custom delay mod 8. Iteration is rotation of a grid by 360/8.
