The Uses of The Uses of IrrationalityIrrationalityJohn D BarrowJohn D Barrow

Paper SizesPaper Sizes

The Square Root of TwoThe Square Root of Two

1

1

2 Does Does 2 = P/Q2 = P/Q with P,Q integers ??with P,Q integers ??

AssumeAssume 2 = P/Q and P,Q integers with no common divisorP2 = 2Q2 so P2 is even and P must be even as well (because even = even x even or even x odd)So P = 2N and Q2 = ½ x 4N2 = 2N2

So Q2 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 2 cannot be written as a rational fraction, P/Q, with

P,Q integers: it is called an ‘irrational’ numberP,Q integers: it is called an ‘irrational’ number

Euclid Book 10, but known to the Pythagoreans

2 = 1.414213562.. 99/70

r

1 1 2

Height: width = r/1 Height: width = 2/r

r

2

rotate

r/1 = 2/r r2 = 2 and r = 2

Nice Irrational Aspect RatiosNice Irrational Aspect Ratios

And so on…..And so on…..If you cut format A(N) paper parallel

to its shorter side into two equal piecesof paper, these will have format A(N+1)

All sizes rounded to the nearest millimetre

International Standard Paper SizesInternational Standard Paper Sizes

TolerancesTolerances ±1.5 mm (0.06 in) for ±1.5 mm (0.06 in) for

dimensions up to 150 dimensions up to 150 mm (5.9 in) mm (5.9 in)

±2 mm (0.08 in) for ±2 mm (0.08 in) for lengths in the range lengths in the range 150 to 600 mm (5.9 to 150 to 600 mm (5.9 to 23.6 in) 23.6 in)

±3 mm (0.12 in) for ±3 mm (0.12 in) for any dimension above any dimension above 600 mm (23.6 in) 600 mm (23.6 in)

The Lichtenberg RatioThe Lichtenberg Ratio Georg Christoph Lichtenberg Georg Christoph Lichtenberg

wrote to Johann Beckmann wrote to Johann Beckmann on 25on 25thth October 1786 about October 1786 about the advantages ofthe advantages of a a 2 2 paper-size ratiopaper-size ratio

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

SizeSize Height x Width Height x Width (mm)(mm)

Height x Width Height x Width (in)(in)

4A0 2378 x 1682 mm 93.6 x 66.2 in

2A0 1682 x 1189 mm 66.2 x 46.8 in

A0 1189 x 841 mm 46.8 x 33.1 in

A1 841 x 594 mm 33.1 x 23.4 in

A2 594 x 420 mm 23.4 x 16.5 in

A3 420 x 297 mm 16.5 x 11.7 in

A4 297 x 210 mm 11.7 x 8.3 in

A5 210 x 148 mm 8.3 x 5.8 in

A6 148 x 105 mm 5.8 x 4.1 in

A7 105 x 74 mm 4.1 x. 2.9 in

A8 74 x 52 mm 2.9 x 2.0 in

A9 52 x 37 mm 2.0 x 1.5 in

A10 37 x 26 mm 1.5 x 1.0 in

A-series Paper SizesA-series Paper Sizes

B-series Paper SizesB-series Paper Sizes

Length and width of B(n) are the geometric mean size Length and width of B(n) are the geometric mean size of A(n) and A(n-1): of A(n) and A(n-1):

B(n) = B(n) = [A(n) x A(n-1)][A(n) x A(n-1)]

eg size of B1 is eg size of B1 is (A1 x A0) size(A1 x A0) size Beware Japanese standard B paper sizes!

Japanese A series has the usual 2 scaling but Japanese B seriesis defined by the arithmetic mean not the geometric mean.

This introduces other magnification scalings and is not used internationally.

SizeSize Height x Width Height x Width (mm)(mm)

Height x Width Height x Width (in)(in)

B0B0 1414 x 1000 mm1414 x 1000 mm 55.7 x 39.4 in55.7 x 39.4 in

B1B1 1000 x 707 mm1000 x 707 mm 39.4 x 27.8 in39.4 x 27.8 in

B2B2 707 x 500 mm707 x 500 mm 27.8 x 19.7 in27.8 x 19.7 in

B3B3 500 x 353 mm500 x 353 mm 19.7 x 13.9 in19.7 x 13.9 in

B4B4 353 x 250 mm353 x 250 mm 13.9 x 9.8 in13.9 x 9.8 in

B5B5 250 x 176 mm250 x 176 mm 9.8 x 6.9 in9.8 x 6.9 in

B6B6 176 x 125 mm176 x 125 mm 6.9 x 4.9 in6.9 x 4.9 in

B7B7 125 x 88 mm125 x 88 mm 4.9 x. 3.5 in4.9 x. 3.5 in

B8B8 88 x 62 mm88 x 62 mm 3.5 x 2.4 in3.5 x 2.4 in

B9B9 62 x 44 mm62 x 44 mm 2.4 x 1.7 in2.4 x 1.7 in

B10B10 44 x 31 mm44 x 31 mm 1.7 x 1.2 in1.7 x 1.2 in

B-series Paper SizesB-series Paper Sizes

The Deep Magic of Xerox The Deep Magic of Xerox MachinesMachines

All A series paper enlargements and reductions are by factors of

2 = 1.41 = 141% for enlargementsand

1/2 = 0.71 = 71% for reductions

71 %,71 %, 84%, 84%, 119%,119%, 141% 141%1/1/2,2, 1/1/2.2. 2,2, 2 2

A3 A4, B4 A4, A4 B4, A4 A3 B5 A4, A5 A4

Photos of Xerox Machine Control PanelsPhotos of Xerox Machine Control Panels

‘…‘…looks just like his dad’looks just like his dad’

fromto 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%

A10 3200% 2263% 1600% 1131% 800% 566% 400% 283% 200% 141% 100%

Go Forth and MultiplyGo Forth and Multiply

0.71 x 0.71 = 0.504, 0.71 x 0.504 = 0.3579, 0.3579 x 0.71 = 0.25440.71 x 0.71 = 0.504, 0.71 x 0.504 = 0.3579, 0.3579 x 0.71 = 0.25441/0.71 = 1.408, 1.408/0.71 = 1.983, 1.983/0.71 = 2.793, 2.793/0.71 = 3.968

NewspapersNewspapers

Broadsheet 29½ ” x 23½” (750 x 600 mm) -- depth x width)Broadsheet 29½ ” x 23½” (750 x 600 mm) -- depth x width)

Tabloid (or ‘Compact’) 17” x 11” (430 x 280 mm)Tabloid (or ‘Compact’) 17” x 11” (430 x 280 mm)

Berliner 18.5” x 12.5” (470 x 315 mm)Berliner 18.5” x 12.5” (470 x 315 mm)

C4 envelopeA4 letter fitseasily insideunfolded

C5 envelopeA4 letter fitseasily insideFolded in half

C6 envelope

SizeSize Height x Width Height x Width (mm)(mm)

Height x Width Height x Width (in)(in)

C0C0 1297 x 917 mm1297 x 917 mm 51.5 x 36.1 in51.5 x 36.1 in

C1C1 917 x 648 mm917 x 648 mm 36.1 x 25.5 in36.1 x 25.5 in

C2C2 648 x 458 mm648 x 458 mm 25.5 x 18.0 in25.5 x 18.0 in

C3C3 458 x 324 mm458 x 324 mm 18.0 x 12.8 in18.0 x 12.8 in

C4C4 324 x 229 mm324 x 229 mm 12.8 x 9.0 in 12.8 x 9.0 in

C5C5 229 x 162 mm229 x 162 mm 9.0 x 6.4 in9.0 x 6.4 in

C6C6 162 x 114 mm162 x 114 mm 6.4 x 4.5 in6.4 x 4.5 in

C7C7 114 x 81 mm114 x 81 mm 4.5 x. 3.2 in4.5 x. 3.2 in

C8C8 81 x 57 mm81 x 57 mm 3.2 x 2.2 in3.2 x 2.2 in

C9C9 57 x 40 mm57 x 40 mm 2.2 x 1.6 in2.2 x 1.6 in

C10C10 40 x 28 mm40 x 28 mm 1.6 x 1.1 in1.6 x 1.1 in

C-series Paper SizesC-series Paper SizesC(n) = [A(n)xB(n)]

A4: 297 x 210 mm

A0, A1 A0, A1 technical drawings, posters technical drawings, posters

A1, A2 A1, A2 flip charts flip charts

A2, A3 A2, A3 drawings, diagrams, large tables drawings, diagrams, large tables

A4 A4 letters, magazines, forms, catalogues, laser printer and letters, magazines, forms, catalogues, laser printer and copying machine output copying machine output

A5 A5 note pads note pads

A6 A6 European Toilet paper(!), postcards European Toilet paper(!), postcards

B5, A5, B5, A5, B6, A6 B6, A6

books books

C4, C5, C6 C4, C5, C6 envelopes for A4 letters: unfolded (C4), folded once (C5), envelopes for A4 letters: unfolded (C4), folded once (C5), folded twice (C6) folded twice (C6)

B4, A3 B4, A3 newspapers, supported by most copying machines in newspapers, supported by most copying machines in addition to A4 addition to A4

B8, A8B8, A8 playing cards playing cards

UsesUses

FormFormatat

WidthWidth (metres)

HeightHeight (metres)

A(n) 2−1/4−n/2 21/4−n/2

B(n) 2−n/2 21/2−n/2

C(n) 2−1/8−n/2 23/8−n/2

Some Handy Formulae for Paper TigersSome Handy Formulae for Paper Tigers

Quantum Gravitational Paper!Quantum Gravitational Paper!A233 has an area 2-233 m2 (10-35 m)2 Gh/c3 = 1 Planck area unit

S = kB (surface area)/(Planck area)

Bekenstein-Hawking Entropy

Breakdown of classical and quantum picture of space!

Areas and Paper Areas and Paper WeightsWeights

A0 has area 1 sq mA0 has area 1 sq m A4 has area 1/2A4 has area 1/244 = 1/16 sq m = 1/16 sq m Common paper quality is 5 gm Common paper quality is 5 gm per page for A4per page for A4 C4 envelope weighs less than 20 gmC4 envelope weighs less than 20 gm You can put 16 A4 pages in the envelope You can put 16 A4 pages in the envelope

before it weighs before it weighs (16 x 5) + 20 = 100gm(16 x 5) + 20 = 100gm Good for calculating weightGood for calculating weight of stacked papersof stacked papers

Technical Drawing Pen NibsTechnical Drawing Pen NibsStandard 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 = 1.4..

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 A4You can draw on the copy with a 0.25mm pen.

Stencil templates have similar scaling5mm 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 Real Irrationality: American Paper SizesSizes

USA, Canada and Mexico are the only three major countries thatdon’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} = 1.6180339…1/ = - 1 = 0.6180339…

The Golden RatioThe Golden Ratio

/1 = 1/(-1)

Euclid’s DefinitionEuclid’s Definition

A C B

c 300 BC

1

= AC/CB = AB/AC = (= AC/CB = AB/AC = ( + 1)/ + 1)/

22 - - -1 =0 -1 =0

=

The real number that is farthest from any rational number The real number that is farthest from any rational number

    

         

(1)

    

      

Two good approximationsTwo good approximations

(5/6) and 7/5e

Accurate to 1.2 x 10-5 and 1.6 x 10-5

Rational approximations are1, 1 + 1/1, 1 + 1/(1+1), etcie 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…And Continued fractions again…

=

Medieval Vellum and Paper Medieval Vellum and Paper FoldingFoldingFoldover

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 etcNo matter how many times you fold

Flesh side of vellum will always face flesh and hair will face hair

Manuscript of Euclid’s ElementsAdelard of Bath, 4th Dec 1480

Gutenberg Bible

Medieval Book Page CanonsMedieval 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 ConstructionTschichold’s Construction

Divide into1/9ths

Type area height = page width2:3 page size ratio = text size ratio

Give 2:3:4:6 inner:top:outer:bottom margin ratiosPage to text area ratio = (3/2)2 = 9/4

1/9th of page width2/9th

1/9th page ht

2/9th of page ht

circle