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Major Facilities for
Mathematical Thinking
and Understanding.
(1) Language.
We would like to expound some
examples on how language can
help us in mathematical
thinking.
Our discussion will eventually
lead to organized descriptions
of some plants and trees.
Gao Xingjian
Nobel Prize in
Literature
for 2000
“Language is the ultimate
crystallization of human
civilization. It is intricate, incisive
and difficult to grasp ....”
“ It is a writer’s insights in
grasping truth that
determine the quality of a
work…”
Languages are wonderful
creations of human being.
Our linguistic facility is an
important tool for thinking.
For examples:
‘Ex’ equals minus ‘bee’ plus or
minus the square root of ‘bee’
squared minus four ‘ay’ ‘see’ all
over two ‘ay’
Aleph Bet/Ve Gim
t
el
‫א‬
‫ב‬
‫ג‬
Lame
d
Mem
‫ל‬
‫מ‬
‫נ‬
‫ם‬
‫ן‬
Dalet
He
‫ד‬
‫ה‬
Vav Zayi K
n
‫ו‬
‫ז‬
Nun Samek Ayi Pe/F Tsad K
h
n
e
i
‫ס‬
‫ע‬
‫פ‬
‫צ‬
‫ף‬
‫ץ‬
 b  b  4ac
x
.
2a
2
‘Ay’ square plus ‘bee’ square
equals ‘see’ square.
a b c
2
2
2
-Pythagoras’ theorem.
Besides language,
mathematicians also derive
symbols. This may dramatically
simplify the thinking process.
For examples,
A B
x A
f (x)
100
n
n 1

: A is a subset of B
: x is an element in A
: a function of x
2
: summation
: angle
PQ
: statements P and Q
P  Q : statements P or Q
negation of the
~ P : statement
P
The use of language is also
reflected in computer
programming.
After one month, the pair of
rabbits is mature and mated. One
more month later, a pair of rabbits
are born.
Let’s use the letters ‘b’ to denoted
a pair of new born rabbits, and ‘a’ a
mature pair. The situation can be
represented by the rewriting rules
a
 ab
b
 a.
Starting with ‘b’ we proceed
b
 a 
 ab 
 aba

 abaab

 abaababa

 abaababaabaab

 abaababaabaababaababa
:
Let
Cn be the number of letters
in the n-th step. E.g.
Co  0,
C1  1,
C2  1,
C3  2,
C4  3,
C5  5,
C6  8,
C7  13,
C8  21,
...
We conjecture that
Cn  Cn 1  Cn  2 .
(Think about a proof in the
tutorial.)
The Fibonacci numbers are
0, 1, 1, 2, 3, 5, 8, 13, 21, ...
(add the last two numbers to get
the next number).
There are 13 notes in the span of any note
through its octave, with 8 white keys and 5
black keys..
The first few Fibonacci
numbers appear in this plant.
1, 1, 2, 3, 5, 8, 13, 21, ...
The sequence of ratios
1 2 3 5 8 13
1, , , , ,
,
,...
2 3 5 8 13 21
‘converges’ to the golden ratio
(number) 0·6180339887...
Many books on oil painting and photo taking in your
local library will point out that it is better to position
objects not in the centre of the picture but to one
side or "about one-third" of the way across.
1 x
x 1 x

1
x
x
1
 x  x 1  0
2
1  5
2
x
 x  0.618  .
2
3
Position the objects about 1/3 of the way across.
The position of baby Christ is described
by horizontal and vertical lines using the
Golden ratio number.
The exquisite Parthenon temple
in Athens, which has
height : breadth : ~ 0.618
(recall the golden section
number 0·6180339887…).
Pantheon in Rome (~120 AD).
Classicism – clarity, symmetry and
proportion.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
Woodcut from the Divina Proportione by
Luca Pacioli (1509) depicting the golden
proportion as it applies to the human face.
http://www.beautyanalysis.com
(Extra slide for interest)
http://zhurnal.lib.ru/m/muratow_s_w/violin_design.shtml
(Extra slide for interest)
Curtate cycloid
A violin scroll usually approximates a
Logarithmic Spiral.
(Extra slide for interest)
(Extra slide for interest)
(Extra slide for interest)
(Extra slide for interest)
tan 31.6 ~ 0.615