The hypothesis that the number of possible states of every finite volume of space-time is finite.


One of the features of the universe is that it exhibits slight chirality.

That is to say that all the forces of nature (e.g. the electromagnetic, strong and gravitational forces) are symmetrical under reflection - except the weak force (as carried by the Z-boson). The result is a universe that is almost - but not quite - symmetrical under reflection.

One way in which this state of affairs could have originated is through the repeated application of more symmetrical rules - in a spatial pattern that is chiral.

One virtue of this suggestion is that very simple rules can produce behaviour that is almost symmetrical - but shows some slight chirality.

Fredkin's example automaton

Fredkin has proposed an example automaton - which illustrates the fundamental principle.

There's an implementation of that automaton here.

Fredkin describes the rules of the automaton here.

The automaton attains reversibility through the use of a rarely-employed method - known as the "guarded context" technique.

There are a number of apparent difficulties with this automaton, including:

  • Does not appear to be very dynamic. While it may well support universal computation there are no obvious signs of gliders - and the typical behaviour appears to be close to stasis.

  • Cubic symmetry. In a three-dimensional automata I would prefer a structure with F.C.C. symmetry.

However, the automaton does exhibit the same sort of "slight chirality" that we find in the universe.

Chiral time

Fredkin describes the application of a rule at differing timesteps as "chiral time".

A "chiral" object does not map onto itself by translation after reflection. Chiral time implies a series of at least three time-steps.

For example, the temporal sequence ...A, B, C, A, B, C, A, B, C... does not map onto its mirror image ...C, B, A, C, B, A, C, B, A... under translation.

Chiral time is rather common (for example) among 3D partitioning cellular automata.

Geometrical issues

The proposed automaton appears to me to be essentially cubic.

Alternate molecules are in a FCC-structure - and Fredkin describes alternate cells as "past" and "current" states - but both are necessary to determine the future state.

While alternate molecules are in a FCC-structure, that does not mean the resulting automaton is best described as having the properties and geometry associated with FCC structures.

I would consider the model to be "properly" FCC if it could be divided neatly into cells with a FCC structure.

If you divided space in the automaton into tesselating rhomboid dodecahedra - with one cell of each type in it - it would make a big mess of the automaton's neighbourhood and rule.

I consider a cell to be a single cube - and consider the automaton to have a cubic rule - with different rules applied in alternating cells.

I have been criticised for describing the structure as cubic - when the salt crystal has a FCC lattice structure.

In order to elucidate my position in more detail I have written another Java applet. This one activates in any cells in the domain of any existing active cells during each time-step.

After six time-steps (i.e. one full round) the result is as shown below:

You should be able to see that there is a preponderance of blue cells around the majority of the external surfaces - suggesting that this was the colour of the cells in the domain of the last rule applied.

To summarise the result, there is an exactly equivalent "1- step" automaton with the domain illustrated above corresponding to Fredkin's example "6-step" automaton.

While the domain is not an exact cube, it should be clear that it has cubic symmetry - and not the higher level of symmetry associated with FCC structures.

I note that neither cubic nor FCC structures are typically sufficiently symmetric for gasses based on then to obey the Navier-Stokes equation - or exhibit isotropy on a macroscopic scale.

Having spatial isotropy arise naturally from the geometry is not be a necessary property for a model of nature to have. However, spatial symmetry would almost certainly be useful - and might even be important. I don't consider its absence to be an aesthetically pleasing aspect of the model.


Chiral time provides an interesting mechanism by which the observed slight chirality of the universe could be produced in a cellular automaton by simple rules - without the use of something like an extensive look-up-table to represent the rule.

If there is good reason to believe that the universe was built in manner that encourages compactness and economy, this property may be of increased relevance.

If would probably be more likely that the universe was constructed with a budget in mind if - for example - it could be shown to be in existence under simulation.

Tim Tyler | |