Life without Death

A cellular automaton is a type of model studied in mathematics and theoretical biology consisting of a regular grid of cells, each in one of a finite number of states, such as "on" and "off".

A pattern in the Life without Death cellular automaton consists of an infinite two-dimensional grid of cells, each of which can be in one of two states: dead or alive.

Therefore, it is P-complete to simulate patterns in the Life without Death rule, meaning it is unlikely that a parallel algorithm exists for a simulation significantly faster than that obtained by a naive parallel algorithm with one processor per cellular automaton cell and one time step per generation of the pattern.

[4][6] The chaotic patterns common in this rule may fill the plane, but they may also leave large empty rectangular regions framed by ladders, causing them to fail the density condition.

However, in 2009 Dean Hickerson found diagonally expanding patterns that eventually settle down into high-period infinite growth, solving the open problem.

Life without Death pattern that creates three ladders and shows the death of two ladders by colliding with a single cell (two different ways), the turning of a ladder and the death of a ladder by colliding with another ladder.
The number of live cells per generation of the pattern shown above demonstrating the monotonic nature of Life without Death.
An example of a fast parasitic shoot running alongside a slower ladder. When the tips of the shoot and of the ladder meet they are both destroyed creating a chaotic mess and sending two shoots back down the original ladder in the opposite direction.