The Uses of Irrationality John D Barrow

Paper Sizes

The Square Root of Two 1 1 22 Does 2 = P/Q with P,Q integers ?? with P,Q integers ?? Assume Assume 2 = P/Q and P,Q integers with no common divisor P 2 = 2Q 2 so P 2 is even and P must be even as well (because even = even x even or even x odd) So P = 2N and Q 2 = ½ x 4N 2 = 2N 2 So Q 2 and Q are both even as well. Therefore P and Q have a common divisor 2. This contradicts our original hypothesis – which is therefore false. 2 cannot be written as a rational fraction, P/Q, with P,Q integers: it is called an ‘irrational’ number Euclid Book 10, but known to the Pythagoreans 2 =  99/70

r Height: width = r/1 Height: width = 2/r r 2 rotate r/1 = 2/r  r 2 = 2 and r = 2 Nice Irrational Aspect Ratios

And so on….. If you cut format A(N) paper parallel to its shorter side into two equal pieces of paper, these will have format A(N+1) All sizes rounded to the nearest millimetre

International Standard Paper Sizes

Tolerances ±1.5 mm (0.06 in) for dimensions up to 150 mm (5.9 in) ±1.5 mm (0.06 in) for dimensions up to 150 mm (5.9 in) ±2 mm (0.08 in) for lengths in the range 150 to 600 mm (5.9 to 23.6 in) ±2 mm (0.08 in) for lengths in the range 150 to 600 mm (5.9 to 23.6 in) ±3 mm (0.12 in) for any dimension above 600 mm (23.6 in) ±3 mm (0.12 in) for any dimension above 600 mm (23.6 in)

The Lichtenberg Ratio Georg Christoph Lichtenberg wrote to Johann Beckmann on 25 th October 1786 about the advantages of a 2 paper-size ratio Georg Christoph Lichtenberg wrote to Johann Beckmann on 25 th October 1786 about the advantages of a 2 paper-size ratio ‘Love is blind but marriage restores its sight’

Size Height x Width (mm) Height x Width (in) 4A02378 x 1682 mm93.6 x 66.2 in 2A01682 x 1189 mm66.2 x 46.8 in A01189 x 841 mm46.8 x 33.1 in A1841 x 594 mm33.1 x 23.4 in A2594 x 420 mm23.4 x 16.5 in A3420 x 297 mm16.5 x 11.7 in A4297 x 210 mm11.7 x 8.3 in A5210 x 148 mm8.3 x 5.8 in A6148 x 105 mm5.8 x 4.1 in A7105 x 74 mm4.1 x. 2.9 in A874 x 52 mm2.9 x 2.0 in A952 x 37 mm2.0 x 1.5 in A1037 x 26 mm1.5 x 1.0 in A-series Paper Sizes

B-series Paper Sizes Length and width of B(n) are the geometric mean size of A(n) and A(n-1): B(n) = [A(n) x A(n-1)] eg size of B1 is (A1 x A0) size Beware Japanese standard B paper sizes! Japanese A series has the usual 2 scaling but Japanese B series is defined by the arithmetic mean not the geometric mean. This introduces other magnification scalings and is not used internationally.

Size Height x Width (mm) Height x Width (in) B x 1000 mm 55.7 x 39.4 in B x 707 mm 39.4 x 27.8 in B2 707 x 500 mm 27.8 x 19.7 in B3 500 x 353 mm 19.7 x 13.9 in B4 353 x 250 mm 13.9 x 9.8 in B5 250 x 176 mm 9.8 x 6.9 in B6 176 x 125 mm 6.9 x 4.9 in B7 125 x 88 mm 4.9 x. 3.5 in B8 88 x 62 mm 3.5 x 2.4 in B9 62 x 44 mm 2.4 x 1.7 in B10 44 x 31 mm 1.7 x 1.2 in B-series Paper Sizes

The Deep Magic of Xerox Machines All A series paper enlargements and reductions are by factors of 2 = 1.41 = 141% for enlargements and 1/2 = 0.71 = 71% for reductions 71 %, 84%, 119%, 141% 1/2,1/2. 2, 2 A3  A4, B4  A4, A4  B4, A4  A3 B5  A4, A5  A4

Photos of Xerox Machine Control Panels ‘…looks just like his dad’

from t o A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A0 100%71%50%35%25%18%12.5%8.8%6.2%4.4%3.1% A1 141%100%71%50%35%25%18%12.5%8.8%6.2%4.4% A2 200%141%100%71%50%35%25%18%12.5%8.8%6.2% A3 283%200%141%100%71%50%35%25%18%12.5%8.8% A4 400%283%200%141%100%71%50%35%25%18%12.5% A5 566%400%283%200%141%100%71%50%35%25%18% A6 800%566%400%283%200%141%100%71%50%35%25% A7 1131%800%566%400%283%200%141%100%71%50%35% A8 1600%1131%800%566%400%283%200%141%100%71%50% A9 2263%1600%1131%800%566%400%283%200%141%100%71% A %2263%1600%1131%800%566%400%283%200%141%100% Go Forth and Multiply 0.71 x 0.71 = 0.504, 0.71 x = , x 0.71 = /0.71 = 1.408, 1.408/0.71 = 1.983, 1.983/0.71 = 2.793, 2.793/0.71 = 3.968

Newspapers Broadsheet 29½ ” x 23½” (750 x 600 mm) -- depth x width) Tabloid (or ‘Compact’) 17” x 11” (430 x 280 mm) Berliner 18.5” x 12.5” (470 x 315 mm)

C4 envelope A4 letter fits easily inside unfolded C5 envelope A4 letter fits easily inside Folded in half C6 envelopeSize Height x Width (mm) Height x Width (in) C x 917 mm 51.5 x 36.1 in C1 917 x 648 mm 36.1 x 25.5 in C2 648 x 458 mm 25.5 x 18.0 in C3 458 x 324 mm 18.0 x 12.8 in C4 324 x 229 mm 12.8 x 9.0 in C5 229 x 162 mm 9.0 x 6.4 in C6 162 x 114 mm 6.4 x 4.5 in C7 114 x 81 mm 4.5 x. 3.2 in C8 81 x 57 mm 3.2 x 2.2 in C9 57 x 40 mm 2.2 x 1.6 in C10 40 x 28 mm 1.6 x 1.1 in C-series Paper Sizes C(n) = [A(n)xB(n)] A4: 297 x 210 mm

A0, A1 technical drawings, posters A1, A2 flip charts A2, A3 drawings, diagrams, large tables A4 letters, magazines, forms, catalogues, laser printer and copying machine output A5 note pads A6 European Toilet paper(!), postcards B5, A5, B6, A6 books C4, C5, C6 envelopes for A4 letters: unfolded (C4), folded once (C5), folded twice (C6) B4, A3 newspapers, supported by most copying machines in addition to A4 B8, A8 playing cards Uses

Forma t Width Width (metres) Height Height (metres) A(n)2 −1/4−n/2 2 1/4−n/2 B(n)2 −n/2 2 1/2−n/2 C(n)2 −1/8−n/2 2 3/8−n/2 Some Handy Formulae for Paper Tigers

Quantum Gravitational Paper! A233 has an area m 2  ( m) 2  Gh/c 3 = 1 Planck area unit S = k B (surface area)/(Planck area) Bekenstein-Hawking Entropy Breakdown of classical and quantum picture of space!

Areas and Paper Weights A0 has area 1 sq m A0 has area 1 sq m A4 has area 1/2 4 = 1/16 sq m A4 has area 1/2 4 = 1/16 sq m Common paper quality is 5 gm Common paper quality is 5 gm per page for A4 C4 envelope weighs less than 20 gm C4 envelope weighs less than 20 gm You can put 16 A4 pages in the envelope before it weighs You can put 16 A4 pages in the envelope before it weighs (16 x 5) + 20 = 100gm (16 x 5) + 20 = 100gm Good for calculating weight Good for calculating weight of stacked papers of stacked papers

Technical Drawing Pen Nibs Standard sizes: 2.00mm, 1.40mm, 1.00mm, 0.70mm, 0.50mm 0.35mm, 0.25 mm, 0.18mm, 0.13mm They all differ by a factor of approx 2 = Four colour-coded standards: 0.25, 0.35, 0.50, 0.70 mm Draw with 0.35 mm pen on A3 paper and reduce to A4 You can draw on the copy with a 0.25mm pen. Stencil templates have similar scaling 5mm high letters have thickness 0.5mm (brown nib) in A0 Copy to A1 and text is 3.5mm high and 0.35 mm thick (yellow nib)

Real Irrationality: American Paper Sizes USA, Canada and Mexico are the only three major countries that don’t use the International standard A, B and C series paper sizes Letter” (216 × 279 mm), “Legal” (216 × 356 mm), “Executive” (190 × 254 mm), “Ledger/Tabloid” (279 × 432 mm) US photocopiers usually have two or more paper trays. Enlarging of a “Letter” page onto “Legal” paper will cut off margins! Some copiers offer the larger “Ledger” layout, but it also has a different aspect ratio and changes the margins during magnification or reduction. Hopelessly inefficient and inconvenient!

 2 -  -1 =0 = ½ {1 + 5} = … 1/ =  - 1 = … The Golden Ratio /1 = 1/(-1)

Euclid’s Definition A CB c 300 BC  1 = AC/CB = AB/AC = (  + 1)/   2 -  -1 =0 = The real number that is farthest from any rational number

(1) (1) Two good approximations   (5/6) and   7/5e Accurate to 1.2 x and 1.6 x Rational approximations are 1, 1 + 1/1, 1 + 1/(1+1), etc ie 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, … Ie successive approximations are ratios of consecutive Fibonacci numbers! 1,1,2,3,5,8,13,….. And Continued fractions again…  =

Medieval Vellum and Paper Folding Fold over In half again Fold over folio quarto octavo If start with coloured side up: you always have Page 1 white Pages 2-3 coloured Pages 4-5 white Pages 6-7 coloured etc No matter how many times you fold Flesh side of vellum will always face flesh and hair will face hair

Manuscript of Euclid’s Elements Adelard of Bath, 4 th Dec 1480

Gutenberg Bible

Medieval Book Page Canons Margin proportions 2:3:4:6 (inner:top:outer:bottom) when the page proportion is 2:3 [more generally 1:R:2:2R for page proportion 1:R (Van der Graaf)] Height of text area = page width for R=3/2

Tschichold’s Construction Divide into 1/9 ths Type area height = page width 2:3 page size ratio = text size ratio Give 2:3:4:6 inner:top:outer:bottom margin ratios Page to text area ratio = (3/2) 2 = 9/4 1/9 th of page width 2/9 th 1/9 th page ht 2/9 th of page ht circle