3D Turtles is an exploration into combining art, mathematics, and blockchain! So, what is a 3D turtle? First imagine orienting a turtle in 3 dimensional space. We then tell the turtle to move according to a set of rules — "movement rules" if you will.
These rules depend mainly on 2 fractions. Lets take 13/99 and 21/99 for example. 13/99 = 0.131313... and 21/99 = 0.21212121.... The rules are as follows:
For each time step that the turtle moves, determine the direction to move the turtle towards by looking at the current digit of 13/99 and 21/99. So, for the first time step, we look at the first digits of 0.131313... and 0.212121... to get "1" and "2". Then, for the second time step, we use "3" and "1" to generate the movement direction.
Why is this interesting?
- There are many visually stunning patterns which can be generated. See our curated list of shapes.
- There is complexity observed in the shapes drawn reveals some deep fundamental truths about our universe: incredible complexity can evolve from the simplest rules.
- The math behind whether a drawn patter "closes" (i.e. does not go on forever) or not is surprising and quite nice.
For details, interesting math, and fun open problems, see the paper. If you are interested in playing around and generating patterns, see the demo site. For the codebase see the Github Repository.
3D Turtles was inspired by "Plotting Pi and Searching for Mona Lisa", a video released by the fantastic YouTube channel, Numberphile. Essentially, the video explores using Turtle graphic to explore visualizations of rational numbers (e.g. 1/13) and irrational numbers (e.g. Pi).
Not all minted shapes are necessarily unique. Minting simple duplicate shapes (ones found very easily, see page 11 of the paper) is not allowed by the smart contract. Still, duplicate shapes can be minted by finding fractions which lead to the turtle drawing the same shape (again, see page 11 of the paper).