1. Folding Programs
Erik Demaine
CSAIL, MIT

2. Geometric Folding Algorithms
Linkages Paper Polyhedra

3. Hinged Dissection Universality [Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2008]
Theorem: For any finite set of polygons of equal area, there is a hinged dissection that can fold into any of the polygons, continuously without self-intersection
Number of pieces is pseudopolynomial (optimal)
Working on pseudopolynomial, which is best you can hope for (even without hinging)

4. Folding Meets Computing
Algorithms
Computing
Geometry
Folding Conformal Computing
Physics
Folding Algorithms
Conformal Computing

5. Mesoscale Hinged Dissection
from from from from
[Mao, Thalladi, Wolfe, Whitesides, Whitesides 2002]

6. DNA Folding
Synthetic DNA to fold into desired polygon [Rothemund — Nature 2006]
~100nm

7. Self-Assembly & Nanomanufacturing
Algorithms for programming matter into arbitrary shapes [Cheung, Demaine, Griffith, Jacobson et al. 2008]

8. Self-Assembly of Sorting Circuit

9. Self-Assembly of Sorting Circuit

10. Folding Programs
Folding assembly process is useful for more than assembly
Can reprogram an already-assembled device by “feeding” program (circuit)

11. Operating System High-level goals: Challenges: Load programs/circuits
Destroy programs/circuits
Communication channels between running programs/circuits
Challenges:
Embedding programs onto hardware
Routing “reprogramming” instructions
Feedback from program to OS (spawn)
xxx figure

12. Asynchronous Logic Automaton
2 bits = which gate (4)
3 bits for input 1 (8 neighbors including diagonal)
3 bits for input 2
Bit = 0 wire and 1 wire
Neither occupied => ready
0 wire occupied => 0 input
1 wire occupied => 1 input
Both wires occupied: unused

13. Reprogrammable Asynchronous Logic Automaton
Simple solution:
Bigger rule table, more states, more wires
Ideally:
Simple rules
Small number of additional states
Use 0+1 for reprogramming; no extra wires
Somewhere in the middle: von Neumann’s Universal Constructor (self-replicating cellular automaton)

14. One Solution Regular state (AND/OR/etc.) Reset state Forward state
0+1 switches to reset state
Reset state
Accept AND/OR/etc.; programs next regular state
Then accept N/E/W/S forwarding direction
Once programmed, auto-switch to forward state
Forward state
Send 0+1 to N/E/W/S to turn next cell to reset state
Then pass on all input bits (0/1) to next cell
0+1 switches to ready state, which ignores 0/1 inputs
Last node in chain sends back final 0+1 stream to switch ready state to regular state

15. Ongoing Work Converting data to programs
Cleaner integration of logic & programming
Hierarchical programming
Simulation
Design & develop full operating system within this framework