Ffellonics and Computation: The No-Program Baseline

Computability theory asks what can be worked out by following rules. Ffellonics asks what identical spheres build when they follow one rule of their own. These two questions look far apart, but placed side by side they show something useful about where computation comes from.

What computability theory found

In the 1930s Alan Turing, Alonzo Church and Kurt Gödel, among others, asked what a mechanical procedure could and could not do. Turing's answer was a simple machine that follows a table of rules, and one striking result: no algorithm can decide, for every possible program, whether it will ever stop. That is the Halting Problem. The Church–Turing thesis adds the idea that anything we would naturally call "effectively computable" can be done by such a machine.

The power of the Turing machine comes from one feature. Its rule table can be written to say almost anything. Change the table and you change what the machine does.

What Ffellonics offers

Ffellonics starts at the opposite end. Take identical spheres that attract on contact, with the same rule for every one of them: join where you gain the most contacts. Nothing can be rewritten. The spheres are all alike and the rule is fixed, so there is no program to change. What comes out is a short, repeating family of structures, from a pair through a triangle and tetrahedron to the close-packed lattices at the top.

So where the Turing machine is a system with enormous freedom, Ffellonics is a system with almost none. That contrast is the interesting part.

Where computation enters

There is a real research field that sits exactly between the two, called algorithmic self-assembly. In Erik Winfree's tile assembly model, small tiles carry distinct, specific bonds, and the pattern of bonds acts as a program. Such systems can, in principle, perform universal computation as they grow, and DNA tiles have been built that assemble computed patterns such as the Sierpinski triangle. (These are recalled from the literature, from Winfree's tile model and the Rothemund, Papadakis and Winfree experiments, so please check the details before quoting.)

Seen this way, Ffellonics is the zero-program limit of that field. When every tile is identical and every bond is the same, there is nothing to program and the outcomes stay simple. Add distinct tiles with specific bonds, and programs, and with them computation, become possible. A small step in this direction is a two-species version of Ffellonics, where only unlike spheres bond, which already changes which structures can form.

An honest note on predictability

It is tempting to say that a simple rule makes the outcome easy to predict. Real assembly shows more nuance.

• The outcome is probabilistic. At normal temperatures spheres jostle, so a rule gives probabilities and not a single result.

• A simple rule does not mean an easy answer. Finding the lowest-energy arrangement of many attracting spheres is believed to be a computationally hard problem, because the landscape has a huge number of local minima. (I recall results showing this for pairwise potentials, from Wille and Vennik, so verify before citing.) Nature does not solve that problem. It settles into whatever the dynamics reach, which is why real assemblies can get stuck.

So the useful lens here is how hard questions are to answer, not whether they can be answered at all.

An invitation

Here is a question that connects the two fields. Starting from the identical-sphere baseline, how much programmability is needed before assembly can compute? A little specificity, in the form of two kinds of sphere, changes the structures that can exist. A lot of it, in the form of many distinct tiles, reaches universal computation. Mapping the steps in between would show where simple physics turns into something that can compute, and Ffellonics offers the clean starting point for that map.