The Uses of Irrationality
John D Barrow
Paper Sizes
The Square Root of Two
1
2
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
P2 = 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
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  r2 = 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



±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 25th 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)
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
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.
B-series Paper Sizes
Size
Height x Width
(mm)
Height x Width
(in)
B0
1414 x 1000 mm
55.7 x 39.4 in
B1
1000 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
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’
Go Forth and Multiply
t
from
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%
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%
A2
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)xB(n)]
Size
Height x Width
(mm)
Height x Width
(in)
C0
1297 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
A4: 297 x 210 mm
C4 envelope
A4 letter fits
easily inside
unfolded
C5 envelope
A4 letter fits
easily inside
Folded in half
C6 envelope
Uses
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
Some Handy Formulae for Paper Tigers
Format
A(n)
B(n)
C(n)
Width
(metres)
−1/4−n/2
2
−n/2
2
−1/8−n/2
2
Height
(metres)
1/4−n/2
2
1/2−n/2
2
3/8−n/2
2
Quantum Gravitational Paper!
A233 has an area 2-233 m2  (10-35 m)2  Gh/c3 = 1 Planck area unit
Breakdown of classical and quantum picture of space!
S = kB (surface area)/(Planck area)
Bekenstein-Hawking Entropy
Areas and Paper Weights
A0 has area 1 sq m
 A4 has area 1/24 = 1/16 sq m
 Common paper quality is 5 gm
per page for A4
 C4 envelope weighs less than 20 gm
 You can put 16 A4 pages in the envelope
before it weighs
 (16 x 5) + 20 = 100gm
 Good for calculating weight
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 = 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 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!
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
= AC/CB = AB/AC = ( + 1)/ 
2 -  -1 =0
=
The real number that is farthest from any rational number
B
Two good approximations
  (5/6) and   7/5e
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 + 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, 4th 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
1/9th page ht
circle
Divide into
1/9ths
2/9th of page ht
2/9th
1/9th 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