There are some simple known cases of self-replicating patterns in 2D from "[Conway's Game of Life][1]," a set of rules for calculating successive game states of an infinite grid of square cells, each of which can take one of two states at any given moment. It is a cellular automaton, just like the environment for the Von Neumann universal constructors mentioned in Cort Ammon's answer. [1]: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini which creates a copy of itself while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders which oscillate between two stable configurations made of Chapman-Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is in fact the first spaceship constructed in the Game of Life which is neither orthogonal nor purely diagonal (these are called knightships).

On November 23, 2013, Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.

One challenge for life like ours, with membranes and a liquid basis, is that no "tunnels" through membranes can exist without compromising the structural integrity. However, there are alternate ways of getting things across a membrane: gates or "zippers" that can close and open, or vesicle formation followed by dissolution of the vesicle membrane.

Regarding evolution, I don't even know how life evolved from non-life in the 3D world! There's so much unclear about the early events of abiogenesis that I don't think you'd need to explain it in your world, either. After life is established, it seems natural to think that it would evolve over time.

There are some simple known cases of self-replicating patterns in 2D from "Conway's Game of Life," a set of rules for calculating successive game states of an infinite grid of square cells, each of which can take one of two states at any given moment. It is a cellular automaton, just like the environment for the Von Neumann universal constructors mentioned in Cort Ammon's answer.

Wikipedia says the following on the subject:

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini which creates a copy of itself while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders which oscillate between two stable configurations made of Chapman-Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is in fact the first spaceship constructed in the Game of Life which is neither orthogonal nor purely diagonal (these are called knightships).

On November 23, 2013, Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.

One challenge for life like ours, with membranes and a liquid basis, is that no "tunnels" through membranes can exist without compromising the structural integrity. However, there are alternate ways of getting things across a membrane: gates or "zippers" that can close and open, or vesicle formation followed by dissolution of the vesicle membrane.

Regarding evolution, I don't even know how life evolved from non-life in the 3D world! There's so much unclear about the early events of abiogenesis that I don't think you'd need to explain it in your world, either. After life is established, it seems natural to think that it would evolve over time.

There are some simple known cases of self-replicating patterns in 2D from "[Conway's Game of Life][1]," a set of rules for calculating successive game states of an infinite grid of squares, each of which can take one of two states at any given moment. [1]: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini which creates a copy of itself while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders which oscillate between two stable configurations made of Chapman-Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is in fact the first spaceship constructed in the Game of Life which is neither orthogonal nor purely diagonal (these are called knightships).

On November 23, 2013, Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.

One challenge for life like ours, with a liquid basis, is that no "tunnels" through materials or membranes can exist without compromising the structural integrity. However, there are alternate ways of getting things across a membrane: gates or "zippers" that can close and open, or vesicle formation followed by dissolution of the vesicle membrane.

Regarding evolution, I don't even know how life evolved from non-life in the 3D world! There's so much unclear about the early events of abiogenesis that I don't think you'd need to explain it in your world, either. After life is established, it seems natural to think that it would evolve over time.

There are some simple known cases of self-replicating patterns in 2D from "[Conway's Game of Life][1]," a set of rules for calculating successive game states of an infinite grid of square cells, each of which can take one of two states at any given moment. It is a cellular automaton, just like the environment for the Von Neumann universal constructors mentioned in Cort Ammon's answer. [1]: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini which creates a copy of itself while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders which oscillate between two stable configurations made of Chapman-Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is in fact the first spaceship constructed in the Game of Life which is neither orthogonal nor purely diagonal (these are called knightships).

On November 23, 2013, Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.

One challenge for life like ours, with membranes and a liquid basis, is that no "tunnels" through membranes can exist without compromising the structural integrity. However, there are alternate ways of getting things across a membrane: gates or "zippers" that can close and open, or vesicle formation followed by dissolution of the vesicle membrane.

Regarding evolution, I don't even know how life evolved from non-life in the 3D world! There's so much unclear about the early events of abiogenesis that I don't think you'd need to explain it in your world, either. After life is established, it seems natural to think that it would evolve over time.