The Algebra of Switches

1938 · Logic · The companion to Plate III

The relay adder Shannon published in 1938, worked out step by step.

A Demonstration is a plate’s companion print: the reasoning behind the discovery, worked out on a sheet of its own.

Paper ground, a warm cream, on Hahnemühle Photo Rag, 308 gsm cotton

The plate it accompanies

Plate III, The Algebra of Switches: At twenty-one, Shannon showed that switches can do logic, and add. See it

What it shows

In 1937, Claude Shannon showed that circuits of switches obey the algebra of logic George Boole founded. With that algebra he designed a network of relays that adds in base two, and Plate III draws the network adding one and one.

Writing 1 for a closed switch and 0 for an open one, the sheet tries every case. Two switches in series multiply their values, two contacts of one relay in series act as one, which is Boole’s law x² = x, and any table of 0s and 1s can be wired as a network.

In base two, each column adds a digit from each number and the carry from the column before. The column’s carry and sum depend only on how many of those three digits are 1, so Shannon merged the branches of his network that give the same count and removed every contact it did not need.

Set to add one and one, the adder on the plate lights the carry lamp but not the sum lamp, so the lamps read 10, which is two in base two.

Close up, at about the size it prints at Medium

Step 5 of the sheet: rows of discs for the carries, the two numbers and their sum, under the column values 8, 4, 2 and 1. — Step 5. Six plus seven in base two, added one column at a time, with the 1s of the first number in blue and those of the second in yellow. The bottom row reads 13.

Step 7: the plate’s network of relays at a smaller size, with a red path from the supply rail to the lit carry lamp. — Step 7. The plate’s adder, set to add one and one. Current runs along the red path through the blue and yellow relays to the carry lamp, and the sum lamp stays dark.

A table of the eight cases of three inputs, with the number of 1s, the carry and the sum for each. — All eight cases, from 0 + 0 + 0 to 1 + 1 + 1, with the carry and the sum the adder gives for each.

Hang them together

Plate III

Demonstration III

Demonstration III is the companion to Plate III, The Algebra of Switches, which shows Shannon’s relay adder adding one and one. Hung side by side in the same size and frame, the two cost 10% less bought together.

Plate III and Demonstration III, Medium, 42 × 59 cm each, in black frames.

The sheet

Shannon’s algebra of switches (A Symbolic Analysis of Relay and Switching Circuits, 1938), from one relay to his adder

1Two switches in series
2Two switches in parallel
3One relay’s two contacts, and Boole’s law
4Any table of 0s and 1s, wired
5Adding in base two
6Counting the 1s
7The plate’s adder, adding one and one
†And a question to take further: why the networks still add in Shannon’s own notation, where 0 meant a closed circuit

Sizes and prices

Size	Unframed	Framed	Print
Medium42 × 59 cm	£95$145	£225$295	420 × 594 mm
Large59 × 84 cm	£160$215	£390$515	594 × 841 mm
Extra large84 × 119 cm	£285$385	Unframed only	841 × 1189 mm

Paper: Hahnemühle Photo Rag, 308 gsm cotton, printed as a giclée by theprintspace.
Printed in: London for the UK, Brooklyn for the US. Unframed prints travel rolled in a tube.
Framed medium: A2 frame, glazed with 3 mm acrylic.
Framed large: A1 frame, glazed with 3 mm acrylic.
Extra large: unframed only. A0 is a standard frame size.
Frames: solid wood, finished in black, white or oak veneer, made in Germany. Each comes ready to hang.
Edition: open, with a certificate. File version v1.