The Uses of Irrationality John D Barrow

Paper Sizes

The Square Root of Two 2 1 Does 2 = P/Q with P, Q integers ? ? 2 = 1. 414213562. . 99/70 1 Assume 2 = P/Q and P, Q integers with no common divisor Assume P 2 = 2 Q 2 so P 2 is even and P must be even as well (because even = even x even or even x odd) So P = 2 N and Q 2 = ½ x 4 N 2 = 2 N 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

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

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 n n n ± 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) ± 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 ‘Love is blind but marriage restores its sight’ 1742 -99

A-series Paper Sizes Size Height x Width (mm) Height x Width (in) 4 A 0 2378 x 1682 mm 93. 6 x 66. 2 in 2 A 0 1682 x 1189 mm 66. 2 x 46. 8 in A 0 1189 x 841 mm 46. 8 x 33. 1 in A 1 841 x 594 mm 33. 1 x 23. 4 in A 2 594 x 420 mm 23. 4 x 16. 5 in A 3 420 x 297 mm 16. 5 x 11. 7 in A 4 297 x 210 mm 11. 7 x 8. 3 in A 5 210 x 148 mm 8. 3 x 5. 8 in A 6 148 x 105 mm 5. 8 x 4. 1 in A 7 105 x 74 mm 4. 1 x. 2. 9 in A 8 74 x 52 mm 2. 9 x 2. 0 in A 9 52 x 37 mm 2. 0 x 1. 5 in A 10 37 x 26 mm 1. 5 x 1. 0 in

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 B 1 is (A 1 x A 0) 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.

B-series Paper Sizes Siz e Height x Width (mm) Height x Width (in) B 0 1414 x 1000 mm 55. 7 x 39. 4 in B 1 1000 x 707 mm 39. 4 x 27. 8 in B 2 707 x 500 mm 27. 8 x 19. 7 in B 3 500 x 353 mm 19. 7 x 13. 9 in B 4 353 x 250 mm 13. 9 x 9. 8 in B 5 250 x 176 mm 9. 8 x 6. 9 in B 6 176 x 125 mm 6. 9 x 4. 9 in B 7 125 x 88 mm 4. 9 x. 3. 5 in B 8 88 x 62 mm 3. 5 x 2. 4 in B 9 62 x 44 mm 2. 4 x 1. 7 in B 10 44 x 31 mm 1. 7 x 1. 2 in

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 A 3 A 4, B 4 A 4, A 4 B 4, A 4 A 3 B 5 A 4, A 5 A 4

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

Go Forth and Multiply t from o A 0 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10 A 0 100% 71% 50% 35% 25% 18% 12. 5% 8. 8% 6. 2% 4. 4% 3. 1% A 1 141% 100% 71% 50% 35% 25% 18% 12. 5% 8. 8% 6. 2% 4. 4% 200% 141% 100% 71% 50% 35% 25% 18% 12. 5% 8. 8% 6. 2% A 3 283% 200% 141% 100% 71% 50% 35% 25% 18% 12. 5% 8. 8% A 4 400% 283% 200% 141% 100% 71% 50% 35% 25% 18% 12. 5% A 5 566% 400% 283% 200% 141% 100% 71% 50% 35% 25% 18% A 6 800% 566% 400% 283% 200% 141% 100% 71% 50% 35% 25% A 7 1131% 800% 566% 400% 283% 200% 141% 100% 71% 50% 35% A 8 1600% 1131% 800% 566% 400% 283% 200% 141% 100% 71% 50% A 9 2263% 1600% 1131% 800% 566% 400% 283% 200% 141% 100% 71% A 10 3200% 2263% 1600% 1131% 800% 566% 400% 283% 200% 141% 100% A 2 0. 71 x 0. 71 = 0. 504, 0. 71 x 0. 504 = 0. 3579, 0. 3579 x 0. 71 = 0. 2544 1/0. 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)

C-series Paper Sizes C(n) = [A(n)x. B(n)] Size Height x Width (mm) Height x Width (in) C 0 1297 x 917 mm 51. 5 x 36. 1 in C 1 917 x 648 mm 36. 1 x 25. 5 in C 2 648 x 458 mm 25. 5 x 18. 0 in C 3 458 x 324 mm 18. 0 x 12. 8 in C 4 324 x 229 mm 12. 8 x 9. 0 in C 5 229 x 162 mm 9. 0 x 6. 4 in C 6 162 x 114 mm 6. 4 x 4. 5 in C 7 114 x 81 mm 4. 5 x. 3. 2 in C 8 81 x 57 mm 3. 2 x 2. 2 in C 9 57 x 40 mm 2. 2 x 1. 6 in C 10 40 x 28 mm C 4 envelope A 4 letter fits easily inside unfolded 1. 6 x 1. 1 in A 4: 297 x 210 mm C 5 envelope A 4 letter fits easily inside Folded in half C 6 envelope

Uses A 0, A 1 technical drawings, posters A 1, A 2 flip charts A 2, A 3 drawings, diagrams, large tables A 4 letters, magazines, forms, catalogues, laser printer and copying machine output A 5 note pads A 6 European Toilet paper(!), postcards B 5, A 5, B 6, A 6 books C 4, C 5, C 6 envelopes for A 4 letters: unfolded (C 4), folded once (C 5), folded twice (C 6) B 4, A 3 newspapers, supported by most copying machines in addition to A 4 B 8, A 8 playing cards

Some Handy Formulae for Paper Tigers Format A(n) B(n) C(n) Width Height (metres) 2− 1/4−n/2 2− 1/8−n/2 21/4−n/2 21/2−n/2 23/8−n/2

Quantum Gravitational Paper! A 233 has an area 2 -233 m 2 (10 -35 m)2 Gh/c 3 = 1 Planck area unit Breakdown of classical and quantum picture of space! S = k. B (surface area)/(Planck area) Bekenstein-Hawking Entropy

Areas and Paper Weights A 0 has area 1 sq m n A 4 has area 1/24 = 1/16 sq m n Common paper quality is 5 gm per page for A 4 n C 4 envelope weighs less than 20 gm n You can put 16 A 4 pages in the envelope before it weighs n (16 x 5) + 20 = 100 gm n Good for calculating weight of stacked papers n

Technical Drawing Pen Nibs Standard sizes: 2. 00 mm, 1. 40 mm, 1. 00 mm, 0. 70 mm, 0. 50 mm 0. 35 mm, 0. 25 mm, 0. 18 mm, 0. 13 mm They all differ by a factor of approx 2 = 1. 4. . Four colour-coded standards: 0. 25 , 0. 35 , 0. 50, 0. 70 mm Draw with 0. 35 mm pen on A 3 paper and reduce to A 4 You can draw on the copy with a 0. 25 mm pen. Stencil templates have similar scaling 5 mm high letters have thickness 0. 5 mm (brown nib) in A 0 Copy to A 1 and text is 3. 5 mm 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!

The Golden Ratio /1 = 1/( -1) 2 - -1 =0 = ½ {1 + 5} = 1. 6180339… 1/ = - 1 = 0. 6180339…

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

Two good approximations (5 /6) and 7 /5 e Accurate to 1. 2 x 10 -5 and 1. 6 x 10 -5 ( 1 ) And Continued fractions again… = Rational approximations are 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, …. .

Medieval Vellum and Paper Folding quarto Fold over In half again folio Fold over 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: 2 R for page proportion 1: R (Van der Graaf)] Height of text area = page width for R=3/2

Tschichold’s Construction 1/9 th page ht circle Divide into 1/9 ths 2/9 th of page ht 2/9 th 1/9 th of page width 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