Spencer-Brown's Counter

In Laws of Form, Spencer-Brown shows a counter he designed with his brother for counting railroad cars. His diagrammatics do not reveal the structure and symmetry of the counter, leading to disbelief as to its validity. Here I redraw his counter as a graph, and show how it indeed does divide by two.

Spencer-Brown's counter as a graph

In this graph, node can be treated as a NOR gate. The input to the counter is at the lower-left and the output is at the lower-right. The input feeds nodes on opposite sides of the circle, while its negation feeds nodes out of phase with the input.

Evaluation is stylized in the graphic rules below. A node turns white if only if at least one of its inputs is black.

All unmarked to marked

Any marked to unmarked

The four phases of the counter can be seen below to rotate a triangular constellation of black nodes around the circle. The states progress clockwise, as does the constellation within them. Each state represents a settling of the circuit given a new input.

Phases of the counter