o is a constructed, visual, Turing complete, language and cellular automaton with supertask properties, comprised of just one symbol. o is a synergetic hybrid of a logographical, alphabetical, natural and formal language. o lives on the interfaces between philosophy, linguistics, computer science and generative art. o is written in JavaScript, HTML, CSS and d3.js

At this moment the cells of o grow according to rule 110 for 10 generations along the unit circle. They follow the geometry of the Arbelos and join the Pappus chain using circle inversion. Each generation is contracted and mirrored on the Y-axis. o is Turing complete because it can run rule 110. Since o is Turing complete it is a type-0 language in the Chomsky hierarchy. That is, if the Church-Turing hypothesis is true, then anything that can be said can be said with o.

In my quest to find a proof of the adequacy of the Turing machine as a model of human cognition, I came across the proof by Matthew Cook (2004) of the Turing completeness of rule 110, a 1-dimensional cellular automaton similar to the Game of Life by John Horton Conway (1970). The conventional 2-dimensional representation of the evolution of a cellular automaton that is as powerful as a universal Turing machine takes up an infinite amount of space.

I came up with the idea to curl up infinity onto a circle and reduce the state and color complexity of the cellular automaton to an absolute minimum viz a 1-color, 1-dimensional cellular automaton. The result of this minimalism is a language that consists only of a single symbol—one circle—and its projections. Yet, there were two challenges. The first challenge was to find an information-preserving, unambiguous mapping capable of representing an infinite number of cells in a finite space. I found the concept of a supretask to be a suitable compression scheme. The second challenge was to find an appealing geometry for the cells. I found the pappus chain to be a suitable geometry. I came up with the idea to curl up infinity onto a circle and reduce the state and color complexity of the cellular automaton to an absolute minimum viz a 1-color, 1-dimensional cellular automaton. The result of this minimalism is a language that consists only of a single symbol—one circle—and its projections. Yet, there were two challenges. The first challenge was to find an information-preserving, unambiguous mapping capable of representing an infinite number of cells in a finite space. I found the concept of a supretask to be a suitable compression scheme. The second challenge was to find an appealing geometry for the cells. I found the pappus chain to be a suitable geometry.

Now, the cells can propagate along the invisible circumference they inhabit. Each generation leaves behind a trace that is gradually being built upon and written on by each new one. In this way we can see at a glance the power of evolution and universality in the countenance of infinity.