## Why isn't A1 paper 595 mm × 841 mm?

In those troubled times of pandemic, it is good to be focusing on existential questions. And here is one that has been causing me to lose the little bit of hair I have left: why isn’t the A1 paper size 595 mm by 841 mm, or A5 149 mm by 210 mm?

The ISO standard defines the A series as follows:

Size Format
0 841 × 1189
1 594 × 841
2 420 × 594
3 297 × 420
4 210 × 297
5 148 × 210
6 105 × 148
7 74 × 105
8 52 × 74
9 37 × 52
10 26 × 37

You’ll notice that the height of a given size is the width of the previous size; a paper size is defined by “halving the preceding paper size across the larger dimension.” The base size is A0, whose area is 1m² and ratio 1:√2.

The formula given by Wikipedia gives for size i is:

$$\alpha_A \times (2^{-\frac{i+1}{2}}) \qquad \mathrm{by} \qquad \alpha_A \times (2^{-\frac{i}{2}})$$

In Clojure this becomes:

(defn iso-aseries-size [size]
{:size (str "A" size)
:width (* alpha-a (Math/pow 2 (- (/ (inc size) 2))) 1000.0)
:height (* alpha-a (Math/pow 2 (- (/ size 2))) 1000.0)})


with the following:

(defn fourth-root [n]
(Math/sqrt (Math/sqrt n)))

(def alpha-a (fourth-root 2))


Trying our formula, we get for the following paper sizes from A0 to A10:

pdfdata.paper> (map #(iso-aseries-size %) (range 11))
;; => ({:size "A0", :width 840.8964152537146, :height 1189.2071150027211}
{:size "A1", :width 594.6035575013606, :height 840.8964152537146}
{:size "A2", :width 420.4482076268573, :height 594.6035575013606}
{:size "A3", :width 297.3017787506803, :height 420.4482076268573}
{:size "A4", :width 210.22410381342866, :height 297.3017787506803}
{:size "A5", :width 148.65088937534014, :height 210.22410381342866}
{:size "A6", :width 105.11205190671433, :height 148.65088937534014}
{:size "A7", :width 74.32544468767007, :height 105.11205190671433}
{:size "A8", :width 52.556025953357164, :height 74.32544468767007}
{:size "A9", :width 37.162722343835036, :height 52.556025953357164}
{:size "A10", :width 26.278012976678582, :height 37.162722343835036})


If we round correctly, we get A5 595 mm × 841 mm, and other sizes also look wrong:

pdfdata.paper> (map #(iso-aseries-size %) (range 11))
;; => ({:size "A0", :width 841, :height 1189}
{:size "A1", :width 595, :height 841}
{:size "A2", :width 420, :height 595}
{:size "A3", :width 297, :height 420}
{:size "A4", :width 210, :height 297}
{:size "A5", :width 149, :height 210}
{:size "A6", :width 105, :height 149}
{:size "A7", :width 74, :height 105}
{:size "A8", :width 53, :height 74}
{:size "A9", :width 37, :height 53}
{:size "A10", :width 26, :height 37})


OK. So digging a bit more, another site gives this other formula

For size n,

• if n is even, the dimensions are:

• if n is odd, the dimensions are:

It’s not difficult to see this formula is actually identical to the previous one ($2^\{frac{1}{4} = \sqrt4{2}$), but for the fun of writing in Clojure, here it is:

(defn iso-aseries-size [size]
(if (even? size)
{:size (str "A" size)
:width (/ 1000.0 (* (Math/pow 2.0 (/ size 2)) (fourth-root 2)))
:height (/ (* 1000.0 (fourth-root 2)) (Math/pow 2 (/ size 2)))}
{:size (str "A" size)
:width (/ (* 1000.0 (fourth-root 2)) (Math/pow 2 (/ (inc size) 2)))
:height (/ 1000.0 (* (Math/pow 2.0 (/ (dec size) 2)) (fourth-root 2)))}))


So how do we correctly calculate the dimensions? We just go back to the basic principles: set the width of the previous size as the height of the new size, calculate the area of a given size by halving the area of the previous size, round down, and divive by height to get the new width:

(defn iso-aseries-size [size]
(if (= size 0)
{:size (str "A" size)
:height (round (* 1000.0 alpha-a))
:width (round (* 1000.0 (/ 1.0 alpha-a)))}
(let [prev (iso-aseries-size (dec size))
area (* (:width prev) (:height prev))
new-area (floor (/ area 2.0))]
{:size (str "A" size)
:height (:width prev)
:width (floor (/ new-area (:width prev)))})))


which returns the correct dimensions:

pdfdata.paper> (map #(iso-aseries-size %) (range 11))
;; => ({:size "A0", :height 1189, :width 841}
{:size "A1", :height 841, :width 594}
{:size "A2", :height 594, :width 420}
{:size "A3", :height 420, :width 297}
{:size "A4", :height 297, :width 210}
{:size "A5", :height 210, :width 148}
{:size "A6", :height 148, :width 105}
{:size "A7", :height 105, :width 74}
{:size "A8", :height 74, :width 52}
{:size "A9", :height 52, :width 37}
{:size "A10", :height 37, :width 26})


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