Oscillators
The periodic patterns of Game of Life are either oscillators or gliders. Oscillators are patterns that return to the initial state after a number of generations called the period of the oscillator.Gliders also return to an initial state but moves a certain distance in every cycle.
Oscillators and gliders are particularly interesting in connection with genetic studies. The periodicity means that the genetic flow can be described by means of quadratic cycle matrices that define a kind of Markov chains. A good almost complete collection of oscillators can be find in Mark D. Niemic's Life Pages
Types
The periodic patterns are characterized by their periods and their cell types. In the type box the periodicity is first indicated (Ex: Per. 3).The cell type notation (see below under Cell types and Cycle matrix) I,P and S are used to characterize the other property (Ex: IS-osc. or PS-glider)
Cell types
The cell types are described in Mathematical aspects of this site.Short definitions:
Stator cells (I-cells) are alive during the whole oscillator cycle.
Casing stator cells get no surviving offspring.
Bushing stator cells get surviving offspring and hence influence the rotor cells.
Rotor cells disappear during a cycle but return to the initial position.
Primary rotor cells (P-cells) get durable offspring.
Secondary rotor cells (S-cells) do not get durable offspring.
The following colours are used:
| Casing stator |  |
| Bushing stator |  |
| Primary rotor |  |
| Secondary rotor |  |
Authors
Contributing authors so far are:- Gunnar Johnsson (GJ), Dep. of Mathematics, KTH.
- Amir Saghedin, E03, KTH.
- Magnus Wideberg, E00, KTH.
- Erik Rigtorp, E04, KTH.
- Kalle Ilves, E04, KTH.
- Fredrik Löf, E04, KTH.
- Zhou Peng, E04, KTH.
- Amir Saghedin, E03, KTH.
Cycle matrix
The cycle matrix A describes the transition of colour indices among the cells during a full cycle of the oscillator. If the colour indices at generation n are put into a vector vn (where the component order corresponds to the ordering of the cells) then the vector vn+1 = Avn (common matrix-vector multiplication) represents the colours of the cells after a cycle. (Think of index 1 as black and index 0 as white).Note that A has row sum = 1.
The cycle matrix has a normal form ( see Math aspects) where
certain blocks, I,P,G,H and S, are identified. It turns out that the diagonal blocks can be identified with the cell types:
I with stator cells, P with primary rotors and S with secondary rotors.
If one of these blocks is nonzero, the corresponding cells exist in the oscillator.
The oscillators studied so far are either IS-oscillators (toad, jam, barberpoles etc.) or PS-oscillators (glider, tumbler etc.).
There are certainly IPS-oscillators out there, but no one has so far been genetically studied.
Open question: Which is the smallest IPS-oscillator?
Conjecture: Lonely bee (period: 9, initial number of cells: 34).
Determinant
In case the determinant of the cycle matrix A is nonzero, the inverse of A, A-1 , exists. This means interestingly that as long as A is known, the initial state of any chain of colour indices can be recovered.More precisely: If vo is the vector of initial colour indices and vn the corresponding vector after n cycles, we have
vn= An vo and hence
vo= (A-1)n vn .
Limit matrix
The limit matrix B is the limit of the matrix sequence A, A2, ...,An,... as n tends to infinity.
The vector Bvo can therefore be interpreted as the asymptotic colour index vector. If e.g. the initial vector has index 1 (black cell) in the jth position and index 0 otherwise (white cells), the jth column of B gives information on the asymptotic colours from that particular initial vector. The kth index in the jth column is the asymptotic index for cell nr k from that initial state.
In the case of smaller PS-oscillators (e.g. the glider) the limit matrix turns out to be of Π-type, i.e. a matrix with identical rows and hence with singlevalued columns. This reflects the fact that an initial black primary rotor cell of the glider will gradually colour the other cells evenly with indices converging to the single value of the column corresponding to the initial black cell.
An other interesting feature is the genetic impact of a cell. This is a measure of the total genetic influence
from a certain cell. The genetic impact of cell nr j, gj, is defined as the sum of the jth column of B (not counting the component 1 in diagonal position that belong to stator cells).
Note that only bushing stator cells and primary rotor cells have nonzero genetic impact.
For a PS-oscillator it is easier to read off the relative genetic impact, rj which is
proportional to gj but normed to get the total sum = 1.The single value of the jth column of B is equal to rj
since B has row sum = 1.
The total sum of all gj:s is equal to the number of rotor cells.
This reflects the fact that if all cells with nonzero genetic impact are black initially, all rotor
cells are asymptotically coloured black.