MMW Chapter1
MMW Chapter1
MMW Chapter1
MATHEMATICS IN THE
MODERN WORLD
Introduction
Mathematics is a useful way to think about nature and our world. The nature of
mathematics underscores the exploration of patterns. Mathematics exists everywhere and it is
applied in the most useful phenomenon. Even by just looking at the ordinary part of the house,
the room and the street, mathematics is there.
Mathematics is an integral part of daily life; formal and informal. It is used in technology,
business, medicine, natural and data sciences, machine learning and construction. It helps
organize patterns and regularities in the world, predict the behavior of nature and phenomenon in
the world, control nature and occurrences in our world for our end. It has numerous applications
in the world making it indispensable.
Points to Ponder
4. Zebras, tigers, cats and snakes are covered in patterns of stripes; leopards and hyenas are
covered in pattern of spots and giraffes are covered in pattern of blotches.
5. Natural patterns like the intricate waves across the oceans; sand dunes on deserts;
formation of typhoon; water drop with ripple and others. These serves as clues to the
rules that govern the flow of water, sand and air.
a. Bilateral Symmetry: a symmetry in which the left and right sides of the organism can
be divided into approximately mirror image of each other along the midline. Symmetry
exists in living things such as in insects, animals, plants, flowers and others. Animals
have mainly bilateral or vertical symmetry, even leaves of plants and some flowers such
as orchids.
2. FRACTALS – a curve or geometric figure, each part of which has the same statistical
character as the whole. A fractal is a never-ending pattern found in nature. The exact
same shape is replicated in a process called “self similarity.” The pattern repeats itself
over and over again at different scales. For example, a tree grows by repetitive branching.
This same kind of branching can be seen in lightning bolts and the veins in your body.
Examine a single fern or an aerial view of an entire river system and you’ll see fractal
patterns.
3. SPIRALS - A logarithmic spiral or growth spiral is a self-similar spiral curve which often
appears in nature. It was first describe by Rene Descartes and was later investigated by Jacob
Bernoulli. A spiral is a curved pattern that focuses on a center point and a series of circular
shapes that revolve around it. Examples of spirals are pine cones, pineapples, hurricanes. The
FIBONACCI SEQUENCE
The Fibonacci sequence is a series of numbers where a number is found by adding up the
two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and
so forth. Written as a rule, the expression is
X n=X n−1 + X n−2
Named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano, Fibonacci
numbers were first introduced in his Liber Abbaci (Book of Calculation) in 1202. The son of a
Pisan merchant, Fibonacci traveled widely and traded extensively. Mathematics was incredibly
important to those in the trading industry, and his passion for numbers was cultivated in his
youth.
GOLDEN RECTANGLE
Leonardo of Pisa also known as Fibonacci discovered a sequence of numbers that created
an interesting numbers that created an interesting pattern the sequence 1, 1, 2, 3, 5, 8, 13, 21,
34… each number is obtained by adding the last two numbers of the sequence forms what is
known as golden rectangle a perfect rectangle. A golden rectangle can be broken down into
squares the size of the next Fibonacci number down and below. If we were to take a golden
rectangle, break it down to smaller squares based from Fibonacci sequence and divide each with
an arc, the pattern begin to take shapes, we begin with Fibonacci spiral in which we can see in
nature.
The Fibonacci sequence can also be seen in the way tree branches form or split. A main
trunk will grow until it produces a branch, which creates two growth points. Then, one of the
new stems branches into two, while the other one lies dormant. This pattern of branching is
repeated for each of the new stems. A good example is the sneezewort. Root systems and even
algae exhibit this pattern.
a+b a
φ= = =1.618033987 ….
a b
B a a/b
2 3 1.5
3 5 1.666666666…
5 8 1.6
8 13 1.625
13 21 1.615384615...
21 34 1.61905
34 55 1.61765
GOLDEN TRIANGLE
Golden ratio can be deduced in an isosceles triangle. If we take the isosceles triangle that
has the two base angles of 72 degrees and we bisect one of the base angles, we should see that
we get another golden triangle that is similar to the golden rectangle. If we apply the same
manner as the golden rectangle, we should get a set of whirling triangles. With these whirling
triangles, we are able to draw a logarithmic spiral that will converge at the intersection of the two
lines. The spiral converges at the intersection of the two lines and this ratio of the lengths of
these two lines is in the Golden Ratio.
1. Flower petals
2. In “Timaeus” Plato describes five possible regular solids that relate to the golden ratio
which is now known as Platonic Solids. He also considers the golden ratio to be the most
bringing of all mathematical relationships.
3. Euclid was the first to give definition of the golden ratio as “a dividing line in the
extreme and mean ratio” in his book the “Elements”. He proved the link of the numbers
to the construction of the pentagram, which is now known as golden ratio. Each
intersections to the other edges of a pentagram is a golden ratio. Also the ratio of the
length of the shorter segment to the segment bounded by the two intersecting lines is a
golden ratio.
5. Michaelangelo di Lodovico Simon was considered the greatest living artists of his time.
He used golden ratio in his painting “The Creation of Adam” which can be seen on the
ceiling of the Sistine Chapel. His painting used the golden ratio showing how God’s
finger and Adam’s finger meet precisely at the golden ratio point of the weight and the
height of the area that contains them.
7. The golden ratio can also be found in the works of other renowned painters such as
a.) Sandro Botticelli (Birth of Venus);
b.) George-Pierre Surat (“Bathers at Assinieres”, “Bridge of Courbevoie” and “A
Sunday on La Grande Jette”), and
c.) Salvador Dali (“The Sacrament of the Last Supper”).
2. Notre Dame is a Gothic Cathedral in Paris, which was built in between 1163 and 1250. It
appears to have a golden ratio in a number of its key proportions of designs.
3. The Taj Mahal in India used the golden ratio in its construction and was completed in
1648. The order and proportion of the arches of the Taj Mahal on the main structure keep
reducing proportionately following the golden ratio.
4. The Cathedral of Our Lady of Chartres in Paris, France also exhibits the Golden ratio.
5. In the United Nation Building, the window configuration reveal golden proportion.
6. The Eiffel Tower in Paris, France, erected in 1889 is an iron lattice. The base is broader
while it narrows down the top, perfectly following the golden ratio.
7. The CN Tower in Toronto, the tallest tower and freestanding structure in the world,
contains the golden ratio in its design. The ratio of observation deck at 342 meters to the
total height of 553.33 is 0.618 or phi, the reciprocal of phi.
BEHAVIOR OF NATURE
Behavior of nature can be observed around us.
Natural regularities of nature:
Symmetry Fractals Spirals
Trees Meanders Waves
Foams Tessellations Cracks
Golden Ratio can be found in the beauty of nature, the growth patterns of many plants,
insects, and the universe.
You may watch this video to understand why do the cells of a honeycomb have a hexagon al form.
2. Zebra’s coat, the alternating pattern of blacks and white are due to
mathematical rules that govern the pigmentation chemicals of its
skin.
4. The nautilus shell has natural pattern which contains a spiral shape called logarithmic
spiral.
8. Cracks can also be found on the barks of trees which show some sort of weakness in the
bark.
The meander is one of a series of regular sinuous curves, bends, loops, turns, or windings
in the channel of the body of water.
7. In Social Sciences such as economics, sociology, psychology and linguistics all now
make extensive use of mathematical models, using the tools of calculus, probability,
game theory, and network theory.
8. In Economics, mathematics such as matrices, probability and statistics are used. The
models may be stochastic or deterministic, linear or non-linear, static or dynamic,
continuous or discrete and all types of algebraic, differential, difference and integral
equations arise for the solution of these models.
9. In political Science, political analysts study past election results to see changes in
voting patterns and the influence of various factors on voting behavior or switching of
votes among political parties and mathematical models for Conflict Resolution using
Game Theory and Statistics.
10. In music and arts, the rhythm that we find in all music notes is the result of
innumerable permutations and combinations. Music theorists understand musical
V. Assignment
1. Collect pictures showing the application of mathematics in different work stations in your
community. Place them in your portfolio and present them as photo album, photo exhibit,
portfolio, etc. with written reports.
HISTORICAL NOTES
Fibonacci