Mathematics in the Modern World - Lesson 1.pdf

- Lesson 1 in the course Mathematics in the Modern World (GNED 03)
- Types of Patterns
- Fibonacci sequence and the Golden Ratio

1. MATHEMATICS IN OUR WORLD
ELAINE C. RICOHERMOSO_LPT
Instructor 1

2. MATHEMATICS
♥Is the study of the relationships among numbers, quantities, and
shapes.
♥Includes arithmetic, algebra, trigonometry, geometry, statistics,
and calculus.
♥Nurtures human characteristics like power of creativity,
reasoning, critical thinking, spatial thinking and others.
♥Helps organize patterns and regularities in the world.
♥Predict the behavior of nature and phenomena in the world.

3. PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Patterns in nature are visible regularities found in
the natural world. These patterns persist in
different contexts and can be modelled
mathematically. Natural patterns may consist of
spirals, symmetries, mosaics, stripes, spots, etc.

4. SYMMETRY

5. SPIRAL

6. RADIAL

7. TESSELLATIONS
The tiling of a plane using one or more geometric shapes, called tiles, with no
overlaps and no gaps.

8. IN 19
TH
CENTURY
♀Belgian physicist Joseph Plateau examined
soap films, leading him to formulate the
concept of minimal surface.
♀German biologist and artist Ernst Haeckel
painted hundreds of marine organisms to
emphasize their symmetry.
♀Scottish biologist D’arcyThompson pioneered
the study of growth patterns in both plants and
animals, showing that simple equations could
explain spiral growth.

9. ♀British mathematician Alan Turing predicted mechanism of
morphogenesis which give rise to patterns of spots and
stripes.
♀Hungarian biologist AristidLindenmayerand french
americanmathematician Benoit Maldenbrotshowed how
the mathematics of fractals could create plant growth
patterns.
IN 20
TH
CENTURY

10. ♀W. Gary smith adopts eight patterns in his landscape work, namely:
scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial
and dendritic.
IN 20
TH
CENTURY

11. NUMBERS ARE EVERYWHERE IN NATURE.
MATHEMATICIANS NOTICED THAT NUMBERS APPEAR IN
MANY DIFFERENT PATTERNS IN NATURE:
•BIRD’S TWO WINGS
•CLOVER’S THREE LEAFLETS
•DEER’S FOUR HOOVES
•BUTTERCUP’S FIVE PETALS
•INSECT’S SIX LEGS
•RAINBOW’S SEVEN COLORS
•OCTOPUS’ EIGHT ARMS

12. THE FIBONACCI SEQUENCE
♀LEONARDO PISANO BIGOLLO LIVED BETWEEN 1170 AND 1250
IN ITALY. His nickname, “fibonacci” roughly means “Son of
Bonacci”
He also helped spread Hindu Arabic numerals through
Europe in place of Roman Numerals.
Fibonacci day –November 23
The fibonaccisequence goes like this:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
987 1597 2584 4181 6765 10946 17711 28657
46368 75025 121393 196418 317811, …….

14. The ratio of any two successive fibonacci numbers is very close to the Golden
Ratio, referred to and represented as phi (Φ),(φ)which is approximately equal
to 1.618034…
A B B/A = ??????
2 3 1.5
3 5 1.6666666667
5 8 1.6
8 13 1.625
… ... ...
144 233 1.6180555556
233 377 1.6180257511
… … …
75025 121393 1.6180339887
121393 196418 1.6180339888
196418 317811 1.6180339887
… … …

15. In geometry, a golden spiral is a
logarithmic spiral whose growth factor is
phi (uppercase Φ, lowercase φ or ϕ),the
golden ratio. That is, a golden spiral gets
wider (or further from its origin) by a factor
of Φ for every quarter turn it makes.

16. TheGolden Ratiois a mathematicalratio. It
is commonly found in nature, and when
used in a design, it fosters organic and
natural-looking compositions that are
aesthetically pleasing to the eye.

17. FIBONACCI SPIRAL

18. FIBONACCI PETALS

19. PATTERNS AND REGULARITIES IN THE
WORLD AS ORGANIZED BY
MATHEMATICS
Patterns, relationship and functions constitute a unifying
theme of mathematics. So many of the beautiful
phenomena observed in nature can be described in
mathematical term.
Scientific and mathematical principles undergird these
spectacular patterns: rainbows, water waves, cloud
formation, tree branching patterns and etc.

20. SPECTACULAR PATTERNS

21. Patterns that build into a simple repetitive
shapes that are reduced in size every time
they are repeated.
FRACTAL PATTERNS

22. FRACTAL PATTERNS

23. The concept of symmetry fascinates
philosophers, astronomers, mathematicians,
artists, architects and physicists.
The motion of pendulum, the reflection in a
plane mirror, the motion of a falling object and
the action-reaction pair of forces are all guided
and organized by mathematics. They exhibit
regularities and symmetry in motion and
behavior according to mathematical laws.

24. REGULARITIES IN THE WORLD
ORGANIZED BY MATHEMATICS
Themathematicsofpendulumisquitecomplicatedbut
harmonic.Itsperiodorthetimeittakestoswingbacktoits
originalpositionisrelatedtoitslength,buttherelationshipis
notlinear.Thepatternsandregularitiesintheswingingmotion
ofapendulumcanbeexplainedbymathematics.

25. REGULARITIES IN THE WORLD
ORGANIZED BY MATHEMATICS
An image formed by an object in a plane mirror can
be explained mathematically by the law of reflection.

26. REGULARITIES IN THE WORLD
ORGANIZED BY MATHEMATICS
A free-falling object is an object that is falling under the sole
influence of gravity. Any object that is moving and being
acted upon the force of gravity is said to be in a state of free
fall. Its motion obeys the equations of uniformly accelerated
vertical motion.

27. REGULARITIES IN THE WORLD
ORGANIZED BY MATHEMATICS
In every interaction, there is a pair of forces acting on the two
interacting objects. The amount of force on the first object equals the
size of the force on the second object. The direction of the force on the
first object is opposite to the direction of the force on the second
object. Forces always come in pairs –equal and opposite action-
reaction pairs.

28. APPLICATIONS OF MATHEMATICS
IN THE WORLD
Mathematics is a universal language in different places, in
different times, in different settings and different
circumstances. The physical world seems to consist of
countable things and any infinity encountered is a result of
extending a counting process. Farming and gardening also
provide rich mathematical opportunities. Within the broad
concept of farming, there are two very important elements:
time and money. At the root of both of these is mathematics.
The art of applying mathematics to complex real-world
problems is called engineering mathematics.

29. APPLICATIONS OF MATHEMATICS
IN THE WORLD
Though some of the more abstract mathematical concepts
seldom come into play, the essential skills developed in basic
math lessons resonate throughout a student’s lifetime and
often resurface to help solve various problems in real life
situations in the workplace and in the world.

30. THANK YOU FOR LISTENING! 