(1.1) Patterns and Numbers in Nature and the World Flashcards
T or F:
Patterns are regular, recurring, and repeating forms or designs that are commonly observed in natural objects
TRUE
commonly denoted Fn form a sequence, such that each number is the sum of the two preceding ones
Fibonacci sequence
T or F:
Many patterns and occurrences exist in nature, in our world, in our life. Mathematics help makes sense of patterns and occurrences.
True
T or F:
Mathematics is a tool to quantify, organize, and control our world, predict phenomena, and make life easier for us.
True
T or F:
Mathematics is just for the books, confined in the classroom.
False
T or F:
Mathematics has numerous applications in the world making it indispensable.
True
T or F:
Mathematics helps the predict the behavior of nature and phenomena in the world.
True
T or F:
Everyone uses mathematics but different people use different mathematics at different times, for different purposes, using different tools, with different attitudes
True
golden ratio =
= (1+√5)/2 or 1.6180339
golden ratio is also known as
divine ratio or proportion
to recognize, classify, and exploit these patterns, the human mind and culture have developed a formal system of thought called
mathematics
- German astronomer
- author of Six-Cornered Snowflakes
- argued that snowflakes must be made by packing tiny identical units together
Johannes Kepler
- 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
six-fold symmetry of snowflakes
T or F:
the regular nightly motion of stars is a clue to the fact the Earth rotates
True
T or F:
waves and dunes are clues to the rules that govern the flow of water, sand, and air
True
T or F:
the tiger’s stripes and hyena’s spots attest to mathematical regularities in biological growth and form
True
T or F:
patterns not only provide clues to the rules that govern natural phenomena but also they possess beauty
True
Patterns are
a. numerical
b. found in geometric patterns
c. found on land (wave patterns)
d. found in stripes and spots in the animal kingdom
T or F:
in 28 days, the phases of the moon make a complete cycle from new moon to full moon and back again
True
T or F:
Approximately, there are 365 days in a year
True
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.
Sand dunes
5 Types of Dunes
- Transverse dunes
- Barchanoid ridges
- Shield-shaped barchan
- Parabolic dunes
- Star-shaped dunes
Types of Symmetries
bilateral, n-fold, rotational
Examples of Symmetries
methane (tetrahedron)
benzene (six-fold)
buckminsterfullerene (icosahedron)
variant: truncated (cutted-off corners)
found on the proportions on human body i.e. heart
assymmetry
Examples of Patterns of Movement
walks on 2 feet, 4 feet, etc, insect scuttling, bird flight patterns, wave-like moments of snakes and fish
2 New Kinds of Pattern
a. Fractals
b. Chaos
these are geometric shapes that repeat their structure on ever-finer scales.
fractals
exhibit similar patterns at increasing small scales called self similarity, aka expanding or unfolding symmetry
fractals
types of fractals
- Natural Fractals-Branching
- Natural Fractals-Spirals
- Geometric Fractals
- Algebraic Fractals (Mandelbrot Set)
Examples of Natural Fractals-Branching
neurons
lungs
Lichtenberg “lightning”
oak tree
river network in China
Examples of Natural Fractals-Spirals
ammonite (shell)
hurricane
spiral galaxy
agave cactus (spiral)
turbulent motion of fluids
fiddlehead fern
Examples of Geometric Fractals
Sierpinski Triangle
Fractal tree (generated using sierpinski triangle)
Koch Curve or Koch Snowflake
these fractals are created by repeatedly calculating a simple equation over and over.
Algebraic fractals
Examples of Algebraic Fractals
Mandelbrot Set (discovered in 1980 shortly after the invention of the personal computer)
Mandelbrot Set:
Znew = Z old ^2 + C
____ 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
chaos
T or F:
the development of new mathematical theories helps us unravel the secrets of the more elusive of nature’s patterns
True
What are some uses of understanding nature’s patterns?
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
