Mathematical Patterns Flashcards
Recurring geometric forms, or numbers that are sequence in such a way that they follow certain rules.
Mathematical patterns
Common mathematical patterns that we encounter daily
Number patterns, the logic patterns and the geometric patterns
Number patterns
Arithmetic sequence
Geometric sequence
Fibonacci sequence
Set of numbers arranged in some orders.
Sequences
Each of the numbers of a sequence is called _____________.
Term of the sequence
The first term is symbolized by
a sub 1
Second term is symbolized by
a sub 2
The nth term is symbolized by
a sub n
A sequence of values follows a pattern of adding a fixed amount (always the same) from one term to the next.
Arithmetic sequence
The fixed amount in arithmetic sequence is called __________.
Common difference
How to find the common difference?
Subtract the first term from the second term.
Formula in finding the common difference
d=a sub 2 - a sub 1
A sequence of values follows a pattern of multiplying a fixed amount from one term to the next.
Geometric sequence
The fixed amount that is used to multiply one term to the next.
Common ratio
Geometric sequence formula
a sub n = r (a sub n - 1)
Common ratio formula
r = a sub n/ a sub n - 1
How to find the common ratio
Common ratio is found by dividing any term by its previous term.
This can be used to find any term in geometric sequence.
Common ratio formula
A set of numbers developed by Leonardo Fibonacci.
Fibonacci sequence
Who developed the Fibonacci sequence?
Leonardo Fibonacci
Fibonacci sequence as a means of _________
Solving practical problems
How the Fibonacci sequence formed?
The sequence is formed by starting with 1,1 and adding the two preceding numbers to get the next number.
Nature’s numbering system
Fibonacci sequence
Where does Fibonacci sequence appear?
In leaf arrangements
Petals of flowers
Bracts of pine cones
Scales of pineapples and
Patterns of seashells
The ration of 2 successive Fibonacci numbers.
Golden ratio
The golden ratio is approximated by __________.
a+b/b =b/a
The set of squares is called the ___________.
Fibonacci rectangles
Logic Patterns
Abstract reasoning
Involves flexible thinking, creativity, judgment, and logical problem solving.
Abstract reasoning
Refers to the ability to analyze information, detect patterns and relationships, and solve problems on a complex, intangible level.
Abstract reasoning
Factors to identify the pattern in abstract reasoning
Size
Location
Color and shades
Angles
Movement
Geomatric Patterns
Tessellation
Fractals
Created when a shape or combination of shapes are repeated over and over again covering a plane without any gaps or overlaps using transformations.
Tessellation
Another word for tessellation
Tiling
First used the one shape tessellations
Sumerians at about 4000 B. C. E.
Where did the sumerians uses the tessellation?
To build wall decorations in pattern of clay tiles.
Some of the most famous tessellations
Moorish wall tiles of Islamic architecture
First person to complete a study of tessellation
Johannes Kepler
When did Johannes Kepler complete the study of tessellations
After he explored the hexagonal structure of honeycomb and snowflakes.
Who began the study of tessellation in mathematics?
Ygraf Fyodorov
Ygraf Fyodorov is a ________.
Russian crystallographer
Tessellations can be made by
a. Involving a repeated use of one polygon
b. Involving repeated use of a unit of shape of two or more different regular polygons.
c. Involving triangles or quadrilaterals.
d. Of irregular shapes obtained by transformation of other ‘more regular’ tessellations shapes.
e. Involving other irregular shapes.
A tessellation made up of congruent regular polygons or polygons whose sides are all the same length.
Regular Tessellation
It means that the polygons that you put together are all the same size and shape.
Congruent
Three regular polygons tessellate in the Euclidean plane:
Triangles
Squares or hexagons
Combinations of regular tessellations.
Semi-regular tessellation
Properties of semi-regular tessellations:
-it is formed by regular polygons.
-the arrangement of polygons at every vertex point is identical.
Eight semi-regular tessellations
-3.3.3.4.4
-3.3.4.3.4
-3.4.6.4
-3.6.3.6
-4.8.8
-4.6.12
-3.3.3.3.6
-3.12.12
Examples of tessellating quadrilaterals
-tessellating triangles
-tessellating rectangles
-tessellating a rhombus
-tessellating a trapezium and a kite
How to produce come irregular shapes to tessellate?
By transforming other shapes which are known to tessellate such as regular polygons.
Suitable shapes from which to start making some irregular tessellation
Squares
Rectangles
Equilateral triangles and
Hexagons
Methods in making some irregular tessellations
By translating (or sliding) the midpoint of any side of the starting shape making some curved lines.
By rotating the midpoint of any side of the starting shape.
World famous graphic artist that attributed to the idea of transformation of shapes to create new, irregular, tessellating shapes.
Mauritis Cornelis Escher (1898-1972)
A rough or fragmented geometric shape that can be split into two parts, each of which is approximately a reduced size copy of the whole.
Fractals
Fractals are often produce by very simple processes—
Reflection, rotation and translation
Fractals demonstrate a fourth time of symmetry. They possess ___________.
Self-symmetry
A shape is _________ when it looks essentially the same from a distance as it does closer up.
Self-similar
Appear the same under magnification.
Self-similar objects
Self-similar objects composed of smaller copies of themselves. This characteristic is often referred to as ____________ or _____________.
Scaling symmetry or scale variance
Objects like spirals ans nested dolls that are Self-similar around a single point are __________.
Not fractals
Exhibit scaling symmetry but only over a limited range of scales.
Natural objects/Fractals in Nature
Fractals from mathematical constructions
Coch curve
Sierpincki’s Triangle
Most amazing thing about fractals is the variety of their __________.
Applications
Where can we find fractals?
In almost every part of the universe, from bacteria, cultures, to galaxies, and to our bodies.
Some of the wide known applications of fractals
In astrophysics
Data compression
Fractal art
Study of music
Computer graphics
Weather forecasting
Human anatomy
