(1.1) Patterns and Numbers in Nature and the World

Question 1

T or F:
Patterns are regular, recurring, and repeating forms or designs that are commonly observed in natural objects

Answer 1

TRUE

Question 2

commonly denoted Fn form a sequence, such that each number is the sum of the two preceding ones

Answer 2

Fibonacci sequence

Question 3

T or F:
Many patterns and occurrences exist in nature, in our world, in our life. Mathematics help makes sense of patterns and occurrences.

Answer 3

True

Question 4

T or F:
Mathematics is a tool to quantify, organize, and control our world, predict phenomena, and make life easier for us.

Answer 4

True

Question 5

T or F:
Mathematics is just for the books, confined in the classroom.

Answer 5

False

Question 6

T or F:
Mathematics has numerous applications in the world making it indispensable.

Answer 6

True

Question 7

T or F:
Mathematics helps the predict the behavior of nature and phenomena in the world.

Answer 7

True

Question 8

T or F:
Everyone uses mathematics but different people use different mathematics at different times, for different purposes, using different tools, with different attitudes

Answer 8

True

Question 9

golden ratio =

Answer 9

= (1+√5)/2 or 1.6180339

Question 10

golden ratio is also known as

Answer 10

divine ratio or proportion

Question 11

to recognize, classify, and exploit these patterns, the human mind and culture have developed a formal system of thought called

Answer 11

mathematics

Question 12

• German astronomer
• author of Six-Cornered Snowflakes
• argued that snowflakes must be made by packing tiny identical units together

Answer 12

Johannes Kepler

Question 13

• a natural consequence of natural packing
• if you pack a large no. of identical coins by placing them as closely as possible, you get a honeycomb arrangement in which every coin except those at the edges is surrounded by 6 others, arranged in a perfect hexagon

Answer 13

six-fold symmetry of snowflakes

Question 14

T or F:
the regular nightly motion of stars is a clue to the fact the Earth rotates

Answer 14

True

Question 15

T or F:
waves and dunes are clues to the rules that govern the flow of water, sand, and air

Answer 15

True

Question 16

T or F:
the tiger’s stripes and hyena’s spots attest to mathematical regularities in biological growth and form

Answer 16

True

Question 17

T or F:
patterns not only provide clues to the rules that govern natural phenomena but also they possess beauty

Answer 17

True

Question 18

Patterns are

Answer 18

a. numerical
b. found in geometric patterns
c. found on land (wave patterns)
d. found in stripes and spots in the animal kingdom

Question 19

T or F:
in 28 days, the phases of the moon make a complete cycle from new moon to full moon and back again

Answer 19

True

Question 20

T or F:
Approximately, there are 365 days in a year

Answer 20

True

Question 21

The most visible mathematical landscape on Earth are found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. ____ _____ form when the wind blows steadily in a fixed direction.

Answer 21

Sand dunes

Question 22

5 Types of Dunes

Answer 22

1. Transverse dunes
2. Barchanoid ridges
3. Shield-shaped barchan
4. Parabolic dunes
5. Star-shaped dunes

Question 23

Types of Symmetries

Answer 23

bilateral, n-fold, rotational

Question 24

Examples of Symmetries

Answer 24

methane (tetrahedron)
benzene (six-fold)
buckminsterfullerene (icosahedron)
variant: truncated (cutted-off corners)

Question 25

found on the proportions on human body i.e. heart

Answer 25

assymmetry

Question 26

Examples of Patterns of Movement

Answer 26

walks on 2 feet, 4 feet, etc, insect scuttling, bird flight patterns, wave-like moments of snakes and fish

Question 27

2 New Kinds of Pattern

Answer 27

a. Fractals b. Chaos

Question 28

these are geometric shapes that repeat their structure on ever-finer scales.

Answer 28

fractals

Question 29

exhibit similar patterns at increasing small scales called self similarity, aka expanding or unfolding symmetry

Answer 29

fractals

Question 30

types of fractals

Answer 30

1. Natural Fractals-Branching 2. Natural Fractals-Spirals 3. Geometric Fractals 4. Algebraic Fractals (Mandelbrot Set)

Question 31

Examples of Natural Fractals-Branching

Answer 31

neurons lungs Lichtenberg "lightning" oak tree river network in China

Question 32

Examples of Natural Fractals-Spirals

Answer 32

ammonite (shell) hurricane spiral galaxy agave cactus (spiral) turbulent motion of fluids fiddlehead fern

Question 33

Examples of Geometric Fractals

Answer 33

Sierpinski Triangle Fractal tree (generated using sierpinski triangle) Koch Curve or Koch Snowflake

Question 34

these fractals are created by repeatedly calculating a simple equation over and over.

Answer 34

Algebraic fractals

Question 35

Examples of Algebraic Fractals

Answer 35

Mandelbrot Set (discovered in 1980 shortly after the invention of the personal computer)

Question 36

Mandelbrot Set:

Answer 36

Znew = Z old ^2 + C

Question 37

____ a kind of apparent randomness whose origins are entirely deterministic, that tis, the future behavior of a system is fully determined by their initial conditions, with no random elements involved

Answer 37

chaos

Question 38

T or F: the development of new mathematical theories helps us unravel the secrets of the more elusive of nature's patterns

Answer 38

True

Question 39

What are some uses of understanding nature's patterns?

Answer 39

a. to steer artificial satellites to new destinations with far less fuel b. to help avoid wear on the wheels of locomotives and other vehicles that run on a railroad c. to improve the effectiveness of heart pacemakers d. to manage forests and fisheries e. to make more efficient dishwashers f. gives us the opportunity to deepen our views abt the universe