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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
Q

involves flexible thinking, creativity, judgement, and logical problem solving

A

abstract reasoning

26
Q

abstract reasoning is independent of __ and __ __

A

educational and cultural background

27
Q

abstract reasoning questions are considered to be an accurate indicator of one’s general __ __

A

intellectual ability

28
Q

ability to analyze information, detect patterns and relationships, and solve problems on a complex, intangible level

A

abstract reasoning

29
Q

Factors to be able to quickly identify the pattern in abstract reasoning

A
  • size
  • location
  • color or shadeness
  • angles
  • movement
30
Q

Three (3) main transformations

A
  1. translation
  2. rotation
  3. reflection
31
Q

created when a shape or combination of shapes are repeated over and over again covering a plane without any gaps or overlaps using tansformations

A

tessellations

32
Q

another word for tesselation

A

tiling

33
Q

when were tessellations first used

A

Sumerians about 4000 BC

34
Q

Some of the most famous tessellations

A

Moorish wall tiles of Islamic architecture

35
Q

first person to complete a study of tessellations

A

Johannes Kepler

36
Q

when did Johannes Kepler become the first person to complete a study of tessellations

A

1591

37
Q

who began the study of tessellations in mathematics

A

Yevgraf Fyodorov

38
Q

Yevgraf Fyodorov is a Russian ___

A

crystallographer

39
Q

Where can we often see tessellations

A
  • nature
  • walls
  • floor tilings
  • carpets
  • wallpapers
  • fabrics
  • pavement
40
Q

Tessellations can be made involving

A

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
Q

tessellation made up of congruent regular polygons or polygons whose sides are all the same length

A

regular tessellation

42
Q

What are the three regular polygons tessellate in the Euclidean plane

A
  1. triangles
  2. squares
  3. hexagons
43
Q

tessellation that is
- formed by regular polygons
- arrangement of polygons at every vertex point is identical

A

semi-regular tessellation

44
Q

Two properties of semi-regular tessellation

A
  1. formed by regular polygons
  2. arrangement of polygons at every vertex is identical
45
Q

one can explore and investigate that… …can be used as a repeating unit with which to tessellate

A

any triangle and any quadrilateral

46
Q

How to produce some irregular shapes to tessellate

A

by transforming other shapes, which are known to tessellate, such as regular polygons

47
Q

Methods to produce irregular shapes to tessellate

A
  1. translating the midpoint of any side of the starting shape
  2. rotating the midpoint of any side of the starting shape
48
Q

world famous graphic artis whose works can be attributed to the idea of transformation of shapes to create new, irregular, tessellating shapes

A

Maurits Cornelis Escher (1898-1972)

49
Q

rough or fragmented geometric shapte that can be split into parts, each of which is approximately a reduced size copy of the whole

A

fractals

50
Q

Three very simple processes that often produce fractals

A
  1. reflection
  2. rotation
  3. translation
51
Q

Fourth type of symmetry fractals demonstrate

A

self-similar

52
Q

looks essentially the same from a distance as it does closer up

A

self-similar

53
Q

self-similar is often referred to as __ __ or __ __

A
  • scaling symmetry
  • scale invariance
54
Q

examples of self-similarity that is not of fractal

A
  • spirals
  • nested dolls
55
Q

natural objects exhibit __ __ but only over a limited range of scales

A

scaling symmetry

56
Q

Besides theory, fractals were used to…

A
  1. compress data in the Encarta Encyclopedia
  2. create realistic landscapes in several movies
57
Q

Examples of where we can find fractals in nature

A
  1. bacteria cultures
  2. galaxies
  3. our bodies
58
Q

Some of wide known applications of fractals

A
  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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