Spencer-Brown's CounterA simple analysis by Jeff James
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.
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
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