Module 4. Mathematical Patterns Flashcards
recurring geometric forms, or numbers that are sequence in such a way that they follow certain rules
Patterns
Three (3) common mathematical patterns that we often encounter daily
- number patterns
- logic patterns
- geometric patterns
set of numbers arranged in some order
sequences
each of the numbers of a sequence
term of the sequence
symbol for first term
a1
symbol for nth term
an
sequence of values follows a pattern of adding a fixed amout from one term to the next
arithmetic sequence
fixed amount of arithmetic sequence
common difference, d
how to find the common difference
d = a2 - a1
how to find the nth term in an arithmetic sequence
an = a1 + (n-1)d
Steps in finding the arithmetic sequence
- Identify a1
- Get the common differemce
- Use the formula
(4. Conclusion)
sequence of values folows a pattern of multiplying a fixed amount from one term to the next
geometric sequence
fixed amount of geometric sequence
common ratio, r
how to find the common ratio
r = an/an-1
how is common ratio found
by dividing any term by its previous term
How to find an in geometric sequence
- an = (an-1) r
- an = (a1)(r^n-1)
set of numbers developed by Leonardo Fibonacci as a means of solving practical problems
Fibonacci sequence
who developed the Fibonacci sequence
Leonardo Fibonacci
How is the Fibonacci sequence formed
starting with 1, and adding the two preceding numbers to get the next number
Real world examples with fibonacci numbers
- leaf arrangement
- petals of flowers
- bracts of pine cones
- scales of pineapples
- patterns of seashells
What do you call the ratio of two (2) successive Fibonacci numbers approach the number
Φ, Golden Ratio
Golden ratio
approx. = 1.618
Golden ratio formula
(a + b)/b
spiral draw inside fibonacci rectangles
Golden Spiral
involves flexible thinking, creativity, judgement, and logical problem solving
abstract reasoning
abstract reasoning is independent of __ and __ __
educational and cultural background
abstract reasoning questions are considered to be an accurate indicator of one’s general __ __
intellectual ability
ability to analyze information, detect patterns and relationships, and solve problems on a complex, intangible level
abstract reasoning
Factors to be able to quickly identify the pattern in abstract reasoning
- size
- location
- color or shadeness
- angles
- movement
Three (3) main transformations
- translation
- rotation
- reflection
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
another word for tesselation
tiling
when were tessellations first used
Sumerians about 4000 BC
Some of the most famous tessellations
Moorish wall tiles of Islamic architecture
first person to complete a study of tessellations
Johannes Kepler
when did Johannes Kepler become the first person to complete a study of tessellations
1591
who began the study of tessellations in mathematics
Yevgraf Fyodorov
Yevgraf Fyodorov is a Russian ___
crystallographer
Where can we often see tessellations
- nature
- walls
- floor tilings
- carpets
- wallpapers
- fabrics
- pavement
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
tessellation made up of congruent regular polygons or polygons whose sides are all the same length
regular tessellation
What are the three regular polygons tessellate in the Euclidean plane
- triangles
- squares
- hexagons
tessellation that is
- formed by regular polygons
- arrangement of polygons at every vertex point is identical
semi-regular tessellation
Two properties of semi-regular tessellation
- formed by regular polygons
- arrangement of polygons at every vertex is identical
one can explore and investigate that… …can be used as a repeating unit with which to tessellate
any triangle and any quadrilateral
How to produce some irregular shapes to tessellate
by transforming other shapes, which are known to tessellate, such as regular polygons
Methods to produce irregular shapes to tessellate
- translating the midpoint of any side of the starting shape
- rotating the midpoint of any side of the starting shape
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)
rough or fragmented geometric shapte that can be split into parts, each of which is approximately a reduced size copy of the whole
fractals
Three very simple processes that often produce fractals
- reflection
- rotation
- translation
Fourth type of symmetry fractals demonstrate
self-similar
looks essentially the same from a distance as it does closer up
self-similar
self-similar is often referred to as __ __ or __ __
- scaling symmetry
- scale invariance
examples of self-similarity that is not of fractal
- spirals
- nested dolls
natural objects exhibit __ __ but only over a limited range of scales
scaling symmetry
Besides theory, fractals were used to…
- compress data in the Encarta Encyclopedia
- create realistic landscapes in several movies
Examples of where we can find fractals in nature
- bacteria cultures
- galaxies
- our bodies
Some of wide known applications of fractals
- astrophysics
- data compression
- fractal art
- study of music
- computer graphics
- weather forecasting
- human anatomy
