Chapter 1 Flashcards
Mathematics is the study of _____-
pattern and
structure
Mathematics is fundamental to the
________
physical and biological sciences, engineering
and information technology, to economics and
increasingly to the social sciences.
Mathematics is a useful way to think about ______
nature and our world
Mathematics is a tool to _____, _______ and
______ our world, _____ phenomena and
make life easier for us.
quantify
organize
control
predict
Many patterns and occurrences exists in nature,
in our world, in our life. Mathematics helps
make sense of these patterns and occurrences.
WHERE IS MATHEMATICS?
WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?
Mathematics helps ____ patterns and
regularities in our world.
organize
WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?
Mathematics helps _____ the behavior of
nature and phenomena in the world.
predict
WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?
Mathematics helps _____ nature and
occurrences in the world for our own ends.
control
WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?
Mathematics has numerous applications in the
world making it _______.
indispensable
_______ are visible regularities of
form found in the natural world and can also be seen in
the universe.
Patterns in nature
Nature patterns which are not just to be
admired, they are vital clues to the rules that govern
_______.
natural processes
Patterns can be observed even in _____ which
move in _____ across the sky each day.
stars
circles
The weather season cycle each year. All
snowflakes contains ______ which no
two are exactly the same.
sixfold symmetry
Patterns can be seen in fish patterns like
___________ These
animals and fish stripes and spots attest to
mathematical regularities in biological growth
and form.
spotted trunkfish, spotted puffer, blue spotted
stingray, spotted moral eel, coral grouper,
redlion fish, yellow boxfish and angel fish.
Zebras, tigers, cats and snakes are covered in
patterns of ______
stripes;
leopards and hyenas are
covered in pattern of _____
spots
giraffes are
covered in pattern of ______
blotches.
Natural patterns like the _________
These serves as clues to the rules that govern
the flow of water, sand and air.
intricate waves across
the oceans; sand dunes on deserts; formation
of typhoon; water drop with ripple and others.
Other patterns in nature can also be seen in the ball
of mackerel, the v-formation of geese in the sky and
the tornado formation of starlings.
okay
a sense of harmonious and
beautiful proportion of balance or an object is
invariant to any various transformations
(reflection, rotation or scaling.)
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.
Bilateral Symmetry:
Vertical Symmetry
Bilateral Symmetry:
Radial symmetry suits
organism like _______ whose adults do not move and jellyfish(dihedral-D4 symmetry).
sea anemones
A five-fold
symmetry is found in the ______, the group in
which includes starfish (dihedral-D5 symmetry), sea
urchins and sea lilies.
echinoderms
a
symmetry around a fixed point known as the center
Radial Symmetry ( or rotational symmetry ):
Radial Symmetry ( or rotational symmetry )
it can be classified as either_____
cyclic or dihedral.
a curve or geometric figure, each part
of which has the same statistical character as the
whole.
fractals
A fractal is a ______found in
nature.
never-ending pattern
The exact same shape is replicated in a
process called ______
“self similarity.”
A logarithmic spiral or growth spiral is a
self-similar spiral curve which often appears in
nature.
Spirals
Spiral was first describe by ____and
was later investigated by ________.
Rene Descartes
Jacob Bernoulli
is
a curved pattern that focuses on a center point and
a series of circular shapes that revolve around it.
spiral
Examples of spirals are_____
pine cones, pineapples,
hurricanes.
The reason for why ____ use a spiral
form is because they are constantly trying to grow
but stay secure.
plants
is a series of numbers
where a number is found by adding up the two numbers
before it.
FIBONACCI SEQUENCE
FIBONACCI SEQUENCE FORMULA
Xn = Xn−1 + Xn−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.
One of the book’s exercises which is written like
this “A man put a pair of rabbits in a place surrounded
on all sides by a wall. How many pairs of rabbits are
produced from that pair in a year, if it supposed that
every month each pair produces a new pair, which from
the second month onwards becomes productive?” This
is best understood in this diagram:
THE HABBIT RABBIT
The sequence encountered in the rabbit problem 1, 1,
2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …. is called
the ______ and its terms the_______
Fibonacci sequence
Fibonacci
numbers
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 _______ a
perfect rectangle.
golden rectangle
calla lily
1 petal,
euphorbia
2
petals,
trillium
3 petals
columbine
5
petals,
bloodroot
8 petals
black-eyed susan
13 petals
shasta daisies
21 petals
field daisies
34 petals
other types of daisies contain
55
and 89 petals.
The sunflower seed conveys the Fibonacci
sequence. The pattern of two spirals goes in opposing
directions (clockwise and counter-clockwise ). The
number of clockwise spirals and counter clockwise
spirals are consecutive Fibonacci numbers and usually
contains 34 and 55 seeds.
FIBONACCI SEQUENCE IN NATURE
Fibonacci discovery of Fibonacci sequence
happened to approach the ratio _____
asymptotically
The golden ratio was first called as the ______ in the early 1500s in Leonardo da Vinci’s
work which was explored by Luca Pacioli entitled “De
Divina Proportione” in 1509.
Divine
Proportion
Golden ratio contains the drawings
of the five platonic solids and it was probably da Vinci
who first called it _______ which is Latin for
______
“section aurea”
Golden Secion
Golden ratio can be deduced in an_____
isosceles
triangle.
GOLDEN RATIO IN NATURE
- Flower petals
- Faces
- Body parts
- Seed heads
- Fruits, Vegetables and Trees
- Shells
- Spiral Galaxies
- Hurricanes
The mouth and nose are
each positioned at golden sections of the distance
between the eyes and the bottom of the chin. Similar
proportions can been seen from the side, and even the
eye and ear itself.
Faces
The Golden Section is manifested in the structure of the
human body. The human body is based on Phi and the
number 5
- Body parts
Typically, seeds are produced at the center, and then
migrate towards the outside to fill all the space.
Seed heads
Spiraling patterns can be found on pineapples and
cauliflower. Fibonacci numbers are seen in the
branching of trees or the number of leaves on a floral
stem; numbers like 4 are not. 3’s and 5’s, however, are
abundant in nature.
Fruits, Vegetables and Trees
Snail shells and nautilus shells follow the logarithmic
spiral, as does the cochlea of the inner ear. It can also
be seen in the horns of certain goats, and the shape of
certain spider’s webs.
Shells
Spiral galaxies are the most common galaxy shape. The
Milky Way has several spiral arms, each of them a
logarithmic spiral of about 12 degrees.
Spiral Galaxies
It’s amazing how closely the powerful swirls of
hurricane match the Fibonacci sequence.
- Hurricanes
The golden ratio can be used to achieve beauty,
balance and harmony in art, architecture and design. It
can be used as a tool in art and design to achieve
balance in the composition.
GOLDEN RATIO IN ARTS
The exterior dimension of the ______ in
Athens, Greece embodies the golden ratio.
Pathernon
In “Timaeus” Plato describes five possible
regular solids that relate to the golden ratio
which is now known as ______. He also
considers the golden ratio to be the most
bringing of all mathematical relationships.
Platonic Solids
five possible regular solids
tetrahedron
hexahedron
octahedron
dodecahedron
icosahedron
____ 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”.
Euclid
Euclid proved the link of the numbers to the
construction of the _____, which is now
known as golden ratio.
pentagram
Each intersections to the
other edges of a pentagram is a _____
golden ratio
Leonardo da Vinci incorporated the golden ratio in his
own paintings such as the _______
Vitruvian Man, The
Last Supper, Monalisa and St. Jerome in the
Wilderness.
_______ was
considered the greatest living artists of his time.
Michaelangelo di Lodovico Simon
Michaelangelo di Lodovico Simon used golden ratio in his painting_____which can be seen on the
ceiling of the Sistine Chapel.
“The
Creation of Adam”
______ or more popularly
known as Raphael was also a painter and
architect from the Rennaisance.
Raffaello Sanzio da Urbino
Raffaello Sanzio’s da Urbino painting
______ the division between
the figures in the painting and their proportions
are distributed using the golden ration.
“The School of Athens,”,
The
golden triangle and pentagram can also be
found in Raphael’s painting _____
“Crucifixion”.
The golden ratio can also be found in the works
of other renowned painters such as
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”).
The ______ built 4700 BC in
Ahmes Papyrus of Egypt is with proportion
according to a “Golden Ratio”. The length of
each side of the base is 756 feet with a height of
481 feet. The ratio of the base to the height is
roughly 1.5717, which is close to the Golden
ratio.
Great Pyramid of Giza
_____ 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.
Notre Dame
The _______ 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.
Taj Mahal
The _______ in Paris,
France also exhibits the Golden ratio.
Cathedral of Our Lady of Chartres
In the______, the window
configuration reveal golden proportion.
United Nation Building
The ______ 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.
Eiffel Tower
The _____ 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.
CN Tower
BEHAVIOR OF NATURE
Symmetry
Fractals
Spirals
Trees
Meanders
Waves
Foams
Tessellations
Cracks
Stripes
Spots
Honeycombs of the bees
show specific regular repeating
_______. It uses the least
amount of wax to store the honey giving a strong structure with no gaps.
hexagons
Zebra’s coat, the
alternating pattern of _______ are due to
mathematical rules that govern
the pigmentation chemicals of
its skin.
blacks
and white
Spider _____illustrate a
beautiful pattern. The
spider creates a
structure by
performing innate
steps.
webs
The nautilus shell has
natural pattern which contains
a_____ shape called
logarithmic spiral.
spiral
Age of the trees can be
determined by applying
_____ which is a
scientific method of dating based
on the amount of rings found in
the core of a tree.
dendrochronology
- Turtles have growth
rings called _____ which
are hexagonal.
Scutes estimates the age
of the turtle.
Smallest scute is in the
center and is the oldest
one, while the largest ones
on the outside are the
newer ones.
“scutes”
Lightning during storms creates_____. Foam
bubbles formed by trapping pockets of gas in a liquid or
solid.
fractals.
____ can also be found on the barks of trees
which show some sort of weakness in the bark.
Cracks
The ____ is one of a series of regular
sinuous curves, bends, loops, turns, or windings
in the channel of the body of water.
meander
applications of mathematics
forensic science, medicine, engineering, information
technology, cryptography, archaeology, social
sciences, political science and other fields.
mathematics is applied specifically
the differential and integral calculus to clarify
the blurred image to clear image. Another
application of calculus is optimization (maximize
or minimize) surface areas, volumes, profit and
cost analysis, projectile motion, etc.
In forensic
much of a function of a protein
is determined by its shape and how the pieces
move. Many drugs are designed to change the
shape or motions of a protein by modeling
using geometry and related areas.
Mathematics is also being applied in the
development of medicine to cure diseases.
In medical field
engineers use numerical
analysis in phenomena involving heat,
electricity and magnetism, relativistic
mechanics, quantum mechanics and other
theoretical constructs.
In fluid dynamics,
modern computer
are invented through the help of mathematics.
An important area of applications of
mathematics in the development of formal
mathematical theories related to the
development of computer science. Computer science development includes logic, relations,
functions, basic set theory, counting
techniques, graph theory, combinatorics,
discrete probability, recursion, recurrence
relations and number theory, computer-
oriented numerical analysis and Operation
Research techniques.
In Information Technology
is a combination of both
mathematics and computer science and is
affiliated closely with information theory,
computer security and engineering. It is used in
applications present in technologically
advanced societies, examples include the
security of ATM cards, computer passwords and
electronic commerce.
Cryptography
archaeologists use a variety of
mathematical and statistical techniques to
present the data from archaeological surveys
and try to find patterns to shed on past human
behavior an in carbon dating artifacts.
In archaeology,
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.
In Social Sciences
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.
In Economics,
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.
In political Science,
the rhythm that we find in all
music notes is the result of innumerable
permutations and combinations. Music
theorists understand musical structure and
communicate new ways of hearing music by
applying set theory, abstract algebra, and
number theory.
In music and arts,
