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Mathematics in the Modern World > Module 4. Mathematical Patterns > Flashcards

Module 4. Mathematical Patterns Flashcards

1
Q

recurring geometric forms, or numbers that are sequence in such a way that they follow certain rules

A

Patterns

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2
Q

Three (3) common mathematical patterns that we often encounter daily

A
  1. number patterns
  2. logic patterns
  3. geometric patterns
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3
Q

set of numbers arranged in some order

A

sequences

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4
Q

each of the numbers of a sequence

A

term of the sequence

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5
Q

symbol for first term

A

a1

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6
Q

symbol for nth term

A

an

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7
Q

sequence of values follows a pattern of adding a fixed amout from one term to the next

A

arithmetic sequence

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8
Q

fixed amount of arithmetic sequence

A

common difference, d

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9
Q

how to find the common difference

A

d = a2 - a1

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10
Q

how to find the nth term in an arithmetic sequence

A

an = a1 + (n-1)d

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11
Q

Steps in finding the arithmetic sequence

A
  1. Identify a1
  2. Get the common differemce
  3. Use the formula
    (4. Conclusion)
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12
Q

sequence of values folows a pattern of multiplying a fixed amount from one term to the next

A

geometric sequence

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13
Q

fixed amount of geometric sequence

A

common ratio, r

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14
Q

how to find the common ratio

A

r = an/an-1

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15
Q

how is common ratio found

A

by dividing any term by its previous term

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16
Q

How to find an in geometric sequence

A
  1. an = (an-1) r
  2. an = (a1)(r^n-1)
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17
Q

set of numbers developed by Leonardo Fibonacci as a means of solving practical problems

A

Fibonacci sequence

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18
Q

who developed the Fibonacci sequence

A

Leonardo Fibonacci

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19
Q

How is the Fibonacci sequence formed

A

starting with 1, and adding the two preceding numbers to get the next number

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20
Q

Real world examples with fibonacci numbers

A
  1. leaf arrangement
  2. petals of flowers
  3. bracts of pine cones
  4. scales of pineapples
  5. patterns of seashells
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21
Q

What do you call the ratio of two (2) successive Fibonacci numbers approach the number

A

Φ, Golden Ratio

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22
Q

Golden ratio

A

approx. = 1.618

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23
Q

Golden ratio formula

A

(a + b)/b

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24
Q

spiral draw inside fibonacci rectangles

A

Golden Spiral

25
involves flexible thinking, creativity, judgement, and logical problem solving
abstract reasoning
26
abstract reasoning is independent of __ and __ __
educational and cultural background
27
abstract reasoning questions are considered to be an accurate indicator of one's general __ __
intellectual ability
28
ability to analyze information, detect patterns and relationships, and solve problems on a complex, intangible level
abstract reasoning
29
Factors to be able to quickly identify the pattern in abstract reasoning
- size - location - color or shadeness - angles - movement
30
Three (3) main transformations
1. translation 2. rotation 3. reflection
31
created when a shape or combination of shapes are repeated over and over again covering a plane without any gaps or overlaps using tansformations
tessellations
32
another word for tesselation
tiling
33
when were tessellations first used
Sumerians about 4000 BC
34
Some of the most famous tessellations
Moorish wall tiles of Islamic architecture
35
first person to complete a study of tessellations
Johannes Kepler
36
when did Johannes Kepler become the first person to complete a study of tessellations
1591
37
who began the study of tessellations in mathematics
Yevgraf Fyodorov
38
Yevgraf Fyodorov is a Russian ___
crystallographer
39
Where can we often see tessellations
- nature - walls - floor tilings - carpets - wallpapers - fabrics - pavement
40
Tessellations can be made involving
repeated 1. one regular polygon 2. two or more different regular polygons 3. triangles or quadrilaterals 4. irregular shapes obtained by transformation of other 'more regular' shapes 5. other irregular shapes
41
tessellation made up of congruent regular polygons or polygons whose sides are all the same length
regular tessellation
42
What are the three regular polygons tessellate in the Euclidean plane
1. triangles 2. squares 3. hexagons
43
tessellation that is - formed by regular polygons - arrangement of polygons at every vertex point is identical
semi-regular tessellation
44
Two properties of semi-regular tessellation
1. formed by regular polygons 2. arrangement of polygons at every vertex is identical
45
one can explore and investigate that... ...can be used as a repeating unit with which to tessellate
any triangle and any quadrilateral
46
How to produce some irregular shapes to tessellate
by transforming other shapes, which are known to tessellate, such as regular polygons
47
Methods to produce irregular shapes to tessellate
1. translating the midpoint of any side of the starting shape 2. rotating the midpoint of any side of the starting shape
48
world famous graphic artis whose works can be attributed to the idea of transformation of shapes to create new, irregular, tessellating shapes
Maurits Cornelis Escher (1898-1972)
49
rough or fragmented geometric shapte that can be split into parts, each of which is approximately a reduced size copy of the whole
fractals
50
Three very simple processes that often produce fractals
1. reflection 2. rotation 3. translation
51
Fourth type of symmetry fractals demonstrate
self-similar
52
looks essentially the same from a distance as it does closer up
self-similar
53
self-similar is often referred to as __ __ or __ __
- scaling symmetry - scale invariance
54
examples of self-similarity that is not of fractal
- spirals - nested dolls
55
natural objects exhibit __ __ but only over a limited range of scales
scaling symmetry
56
Besides theory, fractals were used to...
1. compress data in the Encarta Encyclopedia 2. create realistic landscapes in several movies
57
Examples of where we can find fractals in nature
1. bacteria cultures 2. galaxies 3. our bodies
58
Some of wide known applications of fractals
1. astrophysics 2. data compression 3. fractal art 4. study of music 5. computer graphics 6. weather forecasting 7. human anatomy

Mathematics in the Modern World (13 decks)

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