Fractal workshop

Aim

To describe a fractal and find some of them in nature. For the class to create

their own fractal, Sierpinski’s triangle, and then work together to create larger

versions of this fractal.

Workshop Topics

Look at some fractals in nature and describe them

Learn a little about some of the mathematicians that have investigated fractals

Create a version of Sierpinski’s triangle, you can also join these triangles

together to make a giant triangle.

Look for the number patterns created if you count the number of triangles in

each level.

There is an extension exploring an interesting property of fractals where the

area shrinks (or is limited) while the perimeter tends towards infinity.

Skills & learning

Mathematical language: equilateral triangle, symmetry, spiral, repeating, apex,

cone, fractal, parallel.

Mathematical skills: measuring, drawing lines, dividing by 2, counting, 3x table.

Other skills: team work, discussion, describing.

Knowledge: mathematicians, definition of a fractal.

Worksheet

What is a Fractal?

Fractals are sometimes described as the mathematics of chaos! They are

patterns created by continual repetition but with changing scale. The rise of

the computer made the creation of ever more complex fractals possible.

However, fractals aren’t just limited to the virtual world we also find them in

nature. As an example, look at a tree. If you break off a branch it looks like a

miniature version of the tree. You can then divide your branch further by taking

off a twig.

Describing a Fractal

Look at this picture of some

Romanesque Broccoli. Talk with your

partner or the children on your table

about it. What describing words would

you use?

Here are some other examples of

fractals, how would you describe them?

Pictures by Garry, Michael, Jitze

Couperous, DIY genius Wolfgang Beyer, Brilliant, Postdif.

Mathematicians

Niels Fabian Helge von Koch 1870 – 1924

The Swedish mathematician Helge von Koch wrote about a fractal, the Koch

snowflake, in 1904. It starts with an equilateral triangle and then each side is

divided into 4 line segments. This can then be repeated.

Warclaw Sierpinski 1882 – 1969

After his Doctorate Sierpinski worked at the University of Lwow. In the years

before WW1 he wrote books on the theory of number. During the Polish-Soviet

war 1919 – 21 he also helped with cryptography breaking Russian codes. From

1920 until he retired in 1969 one of the topics Sierpinski worked on while a

professor at Warsaw University was fractals and the Sierpinski curve, this is a

recursive sequence (recursive means repeating). One form of this creates an

arrowhead. Similar to this is the Sierpinski triangle.

Karl Menger 1902 – 1985

The Menger sponge is a fractal with an infinite number of cavities first

described by Austrian mathematician Karl Menger in 1926. To create a Menger

sponge start with a cube, then divide this into 27 smaller cubes (a bit like a

Rubik’s cube). Next remove the cube in the centre and the cubes (6) that share

faces with it. This leaves you with 20 cubes. You can now repeat this process

with your 20 smaller cubes and continue repeating (in theory an infinite number

of times).

Benoit Mandlebroit 1924 – 2010

In 1975 Mandlebroit first used the term

fractal to describe an intricate set of

curves generated by a computer.

Computers were able to complete complex

calculations quickly and several similar

copies can be created at different scales.

Earlier mathematicians had explored

fractals but it was the advent of the

computer that allowed an explosion of these intricate patterns to be created

and explored. Fractals are sometimes referred to as the mathematics of chaos.

Picture by Binette228

Other fractal images by

Creating our Sierpinski Triangle

We start by drawing an equilateral triangle. It is helpful to have a template so

that all the starting triangles are the same size. This will help later when we

want to combine triangles.

Find the centre of the base and draw a faint vertical line. To do this measure

the length of the base and divide by 2.

Now divide the vertical line in half and draw a line

parallel to the base.

You can use this to draw a

downward pointing triangle. I

found it interesting to count

the triangles as I go to see

what number pattern was

created. You will need a table

a bit like this.

step Total ▲triangles Total ▼triangles Total

1

2

3

4

5

Note: the ∆ triangles can be counted in different ways.

You have created 3 smaller upward pointing triangles

inside the large one. Repeat the above for each of these

smaller triangles. Remember to keep track of the

number of triangles at each step.

You can colour your triangles, what colour scheme are you going to use?

Can you cut out and put 3 completed triangles together?

What about 9 triangles, 27?

What about the triangle count?

Hopefully you got something like this.

step Total ▲triangles Total ▼triangles Total ▲+▼

1 1 0 1

2 3 1 4

3 9 4 13

4 27 13 40

5 81 22 103

Or if you counted the triangles inside triangles it might be

step Total ▲triangles Total ▼triangles Total ▲+▼

1 1 0 1

2 4 1 5

3 13 4 17

4 40 13 53

5 121 22 143

Do you notice anything about these number patterns?

You should see some 3x patterns, either directly or by taking the difference

between adjacent numbers.

Some of you might notice the pattern

1, 3, 3x3, 3x3x3, 3x3x3x3… expressed as powers of 3: 30, 31, 32, 33, 34…

Other things to try

You can use Minecraft™ to build a Menger sponge.

Search for images of fractals on the internet, what do you think about them?

Can you create your own fractal art?

Template for Sierpinski Triangle

Extension

If we compare the perimeter and surface area of a fractal we get some

interesting results. To investigate this we are going to compare a square and a

Menger carpet. A Menger carpet is similar to the Menger sponge but only in two

dimensions.

It is easy to get confused between perimeter and area. Perimeter is the

distance around the edges. Area is the space that a shape takes up.

Consider this square

Because this is a square all the sides are the same

length, here labelled ‘d’.

The perimeter is d+d+d+d or 4xd

The area is dxd

Let us now see how these change when we increase

the length of d from 1 cm to 6 cm in 1 cm steps. A

table would be a good way to collect this data.

d perimeter area

1cm

2cm

3cm

4cm

5cm

6cm

Be careful with the units perimeter

is measured in centimetres and area

in centimetres squared.

Here are the answers

d perimeter area

1cm 4cm 1cm2

2cm 8cm 4cm2

3cm 12cm 9cm2

4cm 16cm 16cm2

5cm 20cm 25cm2

6cm 24cm 36cm2

d

d

Now let us look at the first 3 levels of the Menger carpet with a side length of

3cm.

For level 1 we have a square, perimeter

will be 4 x 3cm = 12cm and area 3 cm x

3 cm = 9 cm2

Level 2, the green square is a hole so we

need to add its perimeter, so the

outside perimeter is 4 x 3 cm and the

inside (hole) is 4 x 1 cm. Total

perimeter = 12 cm + 4 cm = 16 cm.

The area however is 3 cm x 3 cm minus

the hole 1 cm x 1 cm. Total 9 cm2 – 1 cm2

= 8 cm2

Level 3, things are getting increasingly

complex and care needs to be taken.

First calculate the perimeter using the

same principles as above but we now

have 9 holes, one size 1 cm x 1 cm and 8 1/3 cm x 1/3 cm. this means the

perimeter is 12 cm + 4 cm + 102/3 cm

(can you see how I worked that out?) =

262/3 cm.

The area is 9 cm2 minus the 9 holes. So 9 – 1 – (8/9) = 71/9 cm2 lets put all

these in a table for comparison.

Level perimeter area

1 12 cm 9 cm2

2 16 cm 8 cm2

3 262/3 cm 71/9 cm2

Compare this with the results for the plain square.

For the square the perimeter and area both increase, with the Menger

carpet the perimeter increases and the area decreases, weird! In fact if

we were to keep increasing the number of holes as we approached an

infinite number of levels we would have an infinite perimeter and finitely

small (approaching zero) area. Along with the amazing patterns this is one

of the properties that makes fractals of interest to mathematicians.