Diamond Bracket Forms and How to Count to Two; 8 of 10

From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188

7. How to Count to Two

Here is the “First Brownian Modulator”:

_________________________________________

___________________________________     |

_____________________________     |     |

_______________________     |     |     |

_________________     |     |     |     |

_____________   |     |     |     |     |

_________   |   |     |     |     |     |

_____   |   |   |     |     |     |     |

    | | | | | | |     |  |  |   | |  |  |

| Z |_|_|_| | | | Z   |__|__|___| |  |  |

|     | |_____|_|________|  |        |  |

|     |       | |___________|________|  |

|     |_______|_____________|           |

|_____________|_________________________|

It is equivalent to the bracket-form system:

Z  =   input

B  =   [ KZ ]

C  =   [ BD ]

D  =   [ BE ]

E  =   [ DF ]

F  =   [ HZ ]

G  =   [ FE ]

H  =   [ KC ]

K  =   [ HG ]

G. Spencer-Brown, in his Laws of Form, claims that this circuit “counts to two”; i.e. when Z oscillates twice between marked and unmarked, K oscillates once.  Soon we will diffract this circuit, to reveal a circular “rotor” circuit.

Here is a “Naive Modulator”:

a   =    [ hz ]

b   =    [ gz ]

c   =    [ ad ]

d   =    [ bc ]  

e   =    [ c [z] ]          

f    =   [ d [z] ]

g   =    [ eh ]

h   =    [ fg ]

And here is the “Kauffman Modulator”:

a  =  [bjz]

b  =  [aiz]

c  =  [ad]

d  =  [bc]    =    z/2 ;   the half-period oscillator

i  =  [bd]

j  =  [ac]