- •Unit 1: Fractals
- •Identify the stressed sound in each word as in the example.
- •Put the following words into the corresponding column according to the sound of the adjective or regular past ‘ed’.
- •Organize the information from the text in the form of a mind map or a flow chart. For example, types properties
- •Fractals
- •Work in pairs and see if you can remember the following words and phrases. Take turns to ask each other. Use the Mini-dictionary (unit 1) if necessary.
- •In the text, find one example of the passive infinitive, five examples of verbs in the passive voice.
- •Explain the meaning of the words and phrases below.
- •Find the words in the text which match the following definitions.
- •Match the words and phrases (1-10) from the text with their definitions (a-j).
- •Fill the gaps in the sentences with the correct words from the text. The first letter has been given.
- •Fill the gaps with the words given below.
- •Find and cross out extra words in the text. There are 10 unnecessary words.
- •Match the words in the left-hand column with the words in the right-hand column to make phrases from the text. Use each word only once. Translate the collocations into Ukrainian.
- •Put the phrases and words in the correct order to make meaningful sentences. The first word of each sentence starts with the capital letter.
- •Fill in the gaps with a suitable preposition.
- •Work in pairs. Look through the applications if fractal analysis below and put them into the corresponding row of the chart.
- •Put the words in brackets into the correct form.
- •One of the underlined parts of a sentence contains a mistake. Find it and correct.
- •Rewrite the sentences using the passive voice where possible. Omit the agent where it can be omitted.
- •. Put the verbs in brackets into the correct passive form.
- •Fill in ‘by’ or ‘with’.
- •Help your friend to translate the following sentences into English.
- •Discuss the following questions in small groups or pairs.
- •Prepare a short two-minute presentation for your classmates on one of the topics below. You may use the information from the unit or do some research on the Internet.
- •Translate the following sentences into Ukrainian for your friend’s poster.
- •Using the information from this unit and further reading section write your own short wiki entry about fractals. (up to 200 words).
- •Do the quiz to find out how much you found out about fractals. (Problem-Solving, unit 1)
- •Fractals
- •Fractal patterns spotted in the quantum realm
- •Fractals in the biological sciences
- •The common prepositions in english.
- •Passive voice
Translate the following sentences into Ukrainian for your friend’s poster.
Extending beyond the typical perception of mathematics as a body of complicated, boring formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers.
Turbulence shapes both the clouds in the sky and the clouds in space, giving them an irregular but repetitive pattern that would be impossible to describe without the help of fractal geometry.
On a smaller scale, some rivers display visual fractals created by a combination of gravity and irregular rock surfaces.
Built by tectonic forces, sculpted by water and wind that act in a similar fashion over many scales, mountains' fractality comes as little surprise.
Many image compression schemes use fractal algorithms to compress computer graphics files to less than a quarter of their original size.
Before Mandelbrot, mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically.
Fractals are often the visible remains of chaotic systems—systems that obey internal rules of organization but are so sensitive to slight changes that their long-term behavior is difficult to predict.
The Mandelbrot set is a dynamic calculation based on the iteration of complex numbers with zero as the starting point.
Using the information from this unit and further reading section write your own short wiki entry about fractals. (up to 200 words).
PROBLEM-SOLVING
Do the quiz to find out how much you found out about fractals. (Problem-Solving, unit 1)
"Heaven and Hell" M.C. Escher
PROBLEM-SOLVING
UNIT 1
Fractals
When was the term "fractal" first used?
1871
1975
1992
When a fractal produces the same shape at smaller and smaller scales, it demonstrates what?
Complexity
Self-similarity
Redundancy
Which fractal helped to calculate the length of the coastline of England?
Menger Sponge
Koch Snowflake
London Sausage
The Mandelbrot Set sums up the work of which mathematician?
Lewis Richardson
Georg Cantor
Gaston
According to fractal geometry, the coastline of the United Kingdom is how long?
7,723 miles (12,429 kilometers)
infinite
6,944 miles (11,175 kilometers)
If you wanted to simulate a mountain range using fractal geometry, what basic Euclidean shape would you want to use?
Square
Mountain shape
Triangle
In the fractal equation z = z2 + c, the variable c stands for what?
A real number
A complex number
The speed of light
Mandelbrot coined the word "fractal" to explain what?
That fractal shapes can have non-integer dimensions
The fractured nature of fractal geometry
The fractious and chaotic aspect of fractals
Fractal dimension is a measure of what?
How many iterations it takes to create a shape
How far your mind has to bend to understand these things
How rough the surface of a shape is
The Hausdorff-Besicovitch Dimension of a fractal is always:
Hard to say after a glass of wine
Less than its topological dimension
Greater than its topological dimension
What kind of fractal has been effectively used in telecommunications devices?
Julia Set
Dragon Curve
Menger Sponge
What's the highest dimension a fractal can have?
One
Two
Three
What meteorologist first discovered the Butterfly Effect in 1960?
David Letterman
Carl-Gustaf Arvid Rossby
Edward Lorenz
Why didn't Gaston Julia pursue his work with the Julia Set further?
He couldn't get funding.
He died before he could complete his work.
He didn't have a supercomputer.
Helge von Koche is best known for creating the Koch Snowflake, a fractal that illustrates what paradox?
An infinite perimeter with a dimension of zero
An infinite space surrounded by a finite perimeter
An infinite perimeter surrounding a finite space
The Lorentz Butterfly is an example of what type of fractal?
Julia Set
Mandelbrot Set
Strange Attractor
FURTHER READING
UNIT 1