Accumulating Copper, One Ion at a Time

Electroforming builds thickness by depositing copper ions one at a time. After six hours at 2 amps, you’ve added roughly 22 grams of copper to your cathode—about 2×10²³ atoms. The order they arrive doesn’t matter. Atom 1000 doesn’t care whether it landed before or after atom 999. The final mass is just the sum.

This is monoidal accumulation: start with an identity (bare cathode), combine elements (copper ions) with an associative operation (addition), build a result. In code, this is a fold.

Clojure’s reduce is explicit about the accumulation:

(defn deposit [cathode ion-mass] (+ cathode ion-mass))
(def ions (repeat 1000 0.000022))  ; 1000 ions, 22 µg each
(reduce deposit 0.0 ions)
;=> 0.022

Because addition is associative, you can chunk the work. Deposit copper on three cathodes in parallel, then sum:

(def chunks (partition 333 ions))
(->> chunks
     (pmap #(reduce deposit 0.0 %))
     (reduce deposit 0.0))
;=> 0.022

Ada makes associativity mechanical. The 'Reduce attribute (Ada 2022) folds over arrays, and the compiler can parallelize if the operation is associative:

with Ada.Text_IO; use Ada.Text_IO;
procedure Electroform is
   type Micrograms is delta 0.000001 digits 10;
   Ions : array (1 .. 1000) of Micrograms := (others => 0.000022);
   function "+" (L, R : Micrograms) return Micrograms is (L + R);
   Total : Micrograms := Ions'Reduce("+", 0.0);
begin
   Put_Line ("Deposited (g):" & Total'Image);
end Electroform;

The chemist’s ammeter and the compiler’s optimizer both exploit the same fact: when combining operations commute, you can rearrange the work without changing the outcome. Electroforming three cathodes simultaneously gives you the same total mass as running them sequentially, just faster. Parallel folds split the collection, accumulate locally, then merge. Both are dividing associative work.