# Theory of incompatible substructure problem

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## Contents |

## Schedule

- SWARM2015 paper: camera ready version submitted on 1st September, 2015.
- Submit extended version by December 2015.

## Todo

- Increase number of trials to 40.

## Parameters

- Components
- Radius of each component: 25 mm
- Thickness of each component:
**8 mm** - Radius of magnet: 1.5 mm
- Strength of magnet: N48
- Polyamide

- Container
- Radius of container: 125 mm
- Depth of container: 9 mm
- Material: Acrylic

- Shaker
- Mode: Orbital shaking
- Speed: 300 rpm
- Duration for shaking: until stable structures are formed

## Variable(s)

- Number of components used in each experiment

## Experiments

- Simulated experiments
- Probability of the formation of all substructures, and all target structures for increasing number of components

- Physical experiments
- Probability of the formation of all substructures, and all target structures for increasing number of components

### Particulars of simulation experiments (10000 trials each)

- Probability of the formation of all target structures and substructures in experiments with increasing number of components from 8 - 40

### Particulars of physical experiments (40 trials each)

- Photos of initial condition and final condition
- Assembly of one target structure
- Data to be collected: number of targets and number of incompatible substructures, time for steady state

- Assembly of two target structures
- Data to be collected: number of targets and number of incompatible substructures, time for steady state

- Assembly of three target structures
- Data to be collected: number of targets and number of incompatible substructures, time for steady state

- Assembly of four target structures
- Assembly of five target structures

- Results to show
- The simulated probability of the formation of all substructures and target structures in self-assembly experiments
- Comparison of simulated probability vs. probability achieved in physical experiments
- Reasons for similarities and/or differences between simulated probability vs. probability of physical experiments

## Open problems

Using the model of Hosokawa 1995. The system starts as a set of n components. Then the following step is repeated until no longer possible: pick a random pair of structures with total size at most the size of a target structure. Combine them. The process stops when no such pair exists, i.e. when the two smallest structures have total size more than the size of a target structure.

Consider a system of n components that form target structures of size k. Let f(n, k, i) be the probability that i target substructures form.

- Probabilities of yields.
- f(n, k, 1) + f(n, k, 2) + ...: What is the probability that at least one target structure forms?
- f(n, k, floor(n/k)): What is the probability that the maximum number of target substructures form?
- f(n, k, 1) * 1 + f(n, k, 2) * 2 + ...: What is the average number of target substructures that form?
- General closed form for f(n, k, i): what is the probability that i substructures form for some fixed i?

- Unexpected behavior.
- Prove that f(n, k, m) is non-increasing as a function of n.

## What to include in the extended version of the paper

- More (40) trials
- Details of the model
- Geometry calculations
- Master equation
- More pictures
- Video in multimedia section
- Geometry of the parts

## Meeting points

- In the case of the experiment where 24 components are used, if 2 target structures are formed (16 components consumed), then the third target structure will always form. This means that if the experiment is continued till the system reaches a steady state, then in the experiment with 24 components, there will never be a case where 2 target structures are formed.
- Doesn't this skew the probability graph? Also, how would this be reflected in the analysis?
- Obviously, this applies to the cases of 2, 4, 5 target structures (16, 32, 40 components) as well.