THE
BEAUTY AND REALITY
OF MATHEMATICS
HEARTFELT THANKS TO THOSE
BEFORE US……..WITH US
NOW…………AND TO COME
ALL THE GIFTS HAVE MADE LIFE
BETTER
The
Greeks
The Chinese
The Indians
The Sumerians The Egyptians
The Arabs
Beauty: What do we define it to be?
The quality present in a thing or person that
gives intense pleasure or deep satisfaction to
the mind, whether arising from sensory
manifestations (as shape, colour, sound, etc.),
a meaningful design or pattern, or something
else (as a personality in which high spiritual
qualities are manifest).
Chaos
Theory
Fractals – the delight of Chaos Theory.
A fractal expression looks like Z = Fn1(Z); Z = Z*Z + Fn2(C)
Mathematics and Physical Beauty
Mathematics and Physical Beauty
Leonardo da Vinci's drawings of
the human body emphasised its
proportion. The ratio of the
following distances is the
Golden Ratio:
(foot to navel) : (navel to head)
Mathematics and Physical Beauty
 Why
do we find people to be attractive?
 Because the proportions of the length of
the nose, the position of the eyes and the
length of the chin all conform to some
aspect of the Golden Ratio.
The Golden Ratio
The Fibonacci Sequence: the first 20
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597,
2584, 4181, 6765.
Where did 1.6180339887……. come from?
Let’s look at the ratio of each number in The Fibonacci
sequence to the one before it:
The Golden Ratio
Measure the length and width of your face. Divide the
length by the width. This should give approximately
1.6, which means a beautiful person’s face is about
11/2 times longer than it is wide.
The Golden Ratio: Some Other Examples
The Golden Ratio: Some Other Examples
The Golden Ratio: Some Other Examples
In seed heads such as the sunflower
shown here and the coneflower
previously, spirals curve left and
right. The number of spirals curving
left and the number of spirals curving
right are neighbours in the Fibonacci
sequence, for example, the number
of spirals curving left is 34 and the
number of spirals curving right is 55.
Mathematics and Architecture
In ancient times architecture was a field of mathematics.
Architects were simply mathematicians that someone
would hire.
Geometry is the guiding principle between the two areas.
Mathematics, however, is indispensible to the
understanding of structural concepts and calculations.
Mathematics and Architecture
Both areas search for order
and beauty- Mathematics in
nature and architecture in
construction.
Mathematics and Architecture
The tallest building in the world:
The Burj Khalifa in Dubai.
Very tall buildings are in danger of
many things depending on where
they are. Stability against
earthquakes is important as well as
ensuring aerodynamic designing is
done perfectly to mitigate against
swaying.
Mathematics and Architecture
Some are purely
utilitarian such as
the Great Wall of
China.
Mathematics and Architecture
Some failed miserably and
provide us with mirth and
wonder.
Mathematics and Architecture
 The
Alhambra Palace in
Andalucia, Spain
 Building
Watch
started in 1238
Mathematics and Architecture
Not to be outdone,
we have our own
styles.
Topology – Some Games
Answers
Numerology
It has always been, and still is the desire to understand people and
ourselves. Numbers were used a very long time ago, in the absence of more
scientific means, to tell of one’s personality and future. In this example we
look at calculating the Soul Urge Number.
Write out your full birth name (this includes your middle name (s). Using only
the vowels in your name, assign these values:
A = 1, E = 5, I = 9, O = 6 and U = 3.
Example: Heather Ina Brown, the vowels are eae ia o
5 + 1 + 5 + 9 + 1 + 6 = 27 = 9
So this persons soul urge number is 9 and they can go read up about their
personality.
Mathematicians – believe it or not, we are human!
Galois
Nash
Noether
Newton
Germain
Gauss
Einstein
Green
Mathematicians – our minds
Mathematicians – our minds
Mathematicians – our minds
Numbers are to
mathematics what words
are to language.
To the ‘distress’ of the
general society
mathematicians have
‘dreamt’ up types of
numbers.
Mathematics in life - numbers
The first securely datable
Mathematical Table in World History,
circa 2600 BCE was developed by the
Sumerians, in ancient African Tribe.
Ye Gads!!! Algebra!!!
THE DISTRIBUTIVE PROPERTY
y
x
z
5
x+4
(x+4)5
x
(x+4)x
Area = (x+4)(5+x)
= (x+4)5 +(x+4)x
= 5x+20+x2+4x
= x2+9x+20
Perfect-Square Trinomials
a
b
a
b
a2
ab
ab
b2
You can see from the diagram that the
area of the large square is both (a+b)2
and a2 +2ab+b2 .
Completing the Square
Incomplete square
Completed square
x2
x
x
6
x2
6x
6x
6x
??
6
6x
62
RESOURCES TO TEACH
MATHEMATICS
WHY ARE WE DOING
ALL OF THIS?
WHERE DO WE GO
FROM HERE?