Chapter 1 Flashcards

1
Q

Mathematics is the study of _____-

A

pattern and
structure

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2
Q

Mathematics is fundamental to the
________

A

physical and biological sciences, engineering
and information technology, to economics and
increasingly to the social sciences.

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3
Q

Mathematics is a useful way to think about ______

A

nature and our world

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4
Q

Mathematics is a tool to _____, _______ and
______ our world, _____ phenomena and
make life easier for us.

A

quantify
organize
control
predict

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5
Q

Many patterns and occurrences exists in nature,
in our world, in our life. Mathematics helps
make sense of these patterns and occurrences.

A

WHERE IS MATHEMATICS?

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6
Q

WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?

Mathematics helps ____ patterns and
regularities in our world.

A

organize

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7
Q

WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?

Mathematics helps _____ the behavior of
nature and phenomena in the world.

A

predict

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8
Q

WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?

Mathematics helps _____ nature and
occurrences in the world for our own ends.

A

control

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9
Q

WHAT ROLE DOES MATHEMATICS PLAY IN OUR
WORLD?

Mathematics has numerous applications in the
world making it _______.

A

indispensable

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10
Q

_______ are visible regularities of
form found in the natural world and can also be seen in
the universe.

A

Patterns in nature

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11
Q

Nature patterns which are not just to be
admired, they are vital clues to the rules that govern
_______.

A

natural processes

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12
Q

Patterns can be observed even in _____ which
move in _____ across the sky each day.

A

stars
circles

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13
Q

The weather season cycle each year. All
snowflakes contains ______ which no
two are exactly the same.

A

sixfold symmetry

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14
Q

Patterns can be seen in fish patterns like
___________ These
animals and fish stripes and spots attest to
mathematical regularities in biological growth
and form.

A

spotted trunkfish, spotted puffer, blue spotted
stingray, spotted moral eel, coral grouper,
redlion fish, yellow boxfish and angel fish.

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15
Q

Zebras, tigers, cats and snakes are covered in
patterns of ______

A

stripes;

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16
Q

leopards and hyenas are
covered in pattern of _____

A

spots

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17
Q

giraffes are
covered in pattern of ______

A

blotches.

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18
Q

Natural patterns like the _________
These serves as clues to the rules that govern
the flow of water, sand and air.

A

intricate waves across
the oceans; sand dunes on deserts; formation
of typhoon; water drop with ripple and others.

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19
Q

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.

A

okay

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20
Q

a sense of harmonious and
beautiful proportion of balance or an object is
invariant to any various transformations
(reflection, rotation or scaling.)

A

SYMMETRY

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21
Q
A
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22
Q

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.

A

Bilateral Symmetry:

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22
Q

Vertical Symmetry

A

Bilateral Symmetry:

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23
Q

Radial symmetry suits
organism like _______ whose adults do not move and jellyfish(dihedral-D4 symmetry).

A

sea anemones

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24
A five-fold symmetry is found in the ______, the group in which includes starfish (dihedral-D5 symmetry), sea urchins and sea lilies.
echinoderms
24
a symmetry around a fixed point known as the center
Radial Symmetry ( or rotational symmetry ):
24
Radial Symmetry ( or rotational symmetry ) it can be classified as either_____
cyclic or dihedral.
24
a curve or geometric figure, each part of which has the same statistical character as the whole.
fractals
25
A fractal is a ______found in nature.
never-ending pattern
26
The exact same shape is replicated in a process called ______
“self similarity.”
27
A logarithmic spiral or growth spiral is a self-similar spiral curve which often appears in nature.
Spirals
28
Spiral was first describe by ____and was later investigated by ________.
Rene Descartes Jacob Bernoulli
29
is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it.
spiral
30
Examples of spirals are_____
pine cones, pineapples, hurricanes.
31
The reason for why ____ use a spiral form is because they are constantly trying to grow but stay secure.
plants
32
is a series of numbers where a number is found by adding up the two numbers before it.
FIBONACCI SEQUENCE
33
FIBONACCI SEQUENCE FORMULA
Xn = Xn−1 + Xn−2
34
Named after Fibonacci, also known as _____
Leonardo of Pisa or Leonardo Pisano,
35
Fibonacci numbers were first introduced in his _____
Liber Abbaci (Book of Calculation) in 1202.
36
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
37
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
38
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
39
calla lily
1 petal,
40
euphorbia
2 petals,
41
trillium
3 petals
42
columbine
5 petals,
43
bloodroot
8 petals
44
black-eyed susan
13 petals
45
shasta daisies
21 petals
46
field daisies
34 petals
47
other types of daisies contain
55 and 89 petals.
48
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
49
Fibonacci discovery of Fibonacci sequence happened to approach the ratio _____
asymptotically
50
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
51
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
52
Golden ratio can be deduced in an_____
isosceles triangle.
53
GOLDEN RATIO IN NATURE
1. Flower petals 2. Faces 3. Body parts 4. Seed heads 5. Fruits, Vegetables and Trees 6. Shells 7. Spiral Galaxies 8. Hurricanes
54
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
55
The Golden Section is manifested in the structure of the human body. The human body is based on Phi and the number 5
3. Body parts
56
Typically, seeds are produced at the center, and then migrate towards the outside to fill all the space.
Seed heads
57
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
58
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
59
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
60
It’s amazing how closely the powerful swirls of hurricane match the Fibonacci sequence.
8. Hurricanes
61
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
62
The exterior dimension of the ______ in Athens, Greece embodies the golden ratio.
Pathernon
63
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
64
five possible regular solids
tetrahedron hexahedron octahedron dodecahedron icosahedron
65
____ 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
66
Euclid proved the link of the numbers to the construction of the _____, which is now known as golden ratio.
pentagram
67
Each intersections to the other edges of a pentagram is a _____
golden ratio
68
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.
69
_______ was considered the greatest living artists of his time.
Michaelangelo di Lodovico Simon
70
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”
71
______ or more popularly known as Raphael was also a painter and architect from the Rennaisance.
Raffaello Sanzio da Urbino
72
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,”,
73
The golden triangle and pentagram can also be found in Raphael’s painting _____
“Crucifixion”.
74
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”).
75
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
76
_____ 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
77
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
78
The _______ in Paris, France also exhibits the Golden ratio.
Cathedral of Our Lady of Chartres
79
In the______, the window configuration reveal golden proportion.
United Nation Building
80
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
81
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
82
BEHAVIOR OF NATURE
Symmetry Fractals Spirals Trees Meanders Waves Foams Tessellations Cracks Stripes Spots
83
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
84
Zebra’s coat, the alternating pattern of _______ are due to mathematical rules that govern the pigmentation chemicals of its skin.
blacks and white
85
Spider _____illustrate a beautiful pattern. The spider creates a structure by performing innate steps.
webs
86
The nautilus shell has natural pattern which contains a_____ shape called logarithmic spiral.
spiral
87
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
88
6. 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”
89
Lightning during storms creates_____. Foam bubbles formed by trapping pockets of gas in a liquid or solid.
fractals.
90
____ can also be found on the barks of trees which show some sort of weakness in the bark.
Cracks
91
The ____ is one of a series of regular sinuous curves, bends, loops, turns, or windings in the channel of the body of water.
meander
92
applications of mathematics
forensic science, medicine, engineering, information technology, cryptography, archaeology, social sciences, political science and other fields.
93
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
94
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
95
engineers use numerical analysis in phenomena involving heat, electricity and magnetism, relativistic mechanics, quantum mechanics and other theoretical constructs.
In fluid dynamics,
96
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
97
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
98
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,
99
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
100
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,
101
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,
102
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,