The Pythagorean spiral is a geometric figure that is simple to draw but complex in the mathematical principles that it illustrates. With a little research and time, you can use the Pythagorean spiral as the basis for a math project that will demonstrate your knowledge of some geometry and math while leaving room for you to express yourself through a colourful and creative drawing.

- Skill level:
- Moderately Easy

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### Things you need

- Large sheet of paper
- Pencil
- Ruler
- Coloured pencils

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## Instructions

- 1
Visit a library. Research the Pythagorean theorem and the significance and history of the Pythagorean spiral (also sometimes known as the spiral of Theodorus). Look up the definition of irrational numbers.

- 2
Write a report in which you explain how to use the Pythagorean theorem to find the length of a triangle's third side and how this principle, combined with the Pythagorean spiral, illustrates the existence of irrational numbers.

- 3
Explain that your Pythagorean spiral will express irrational numbers through the lengths of certain line segments.

## Research and Write a Report

- 1
On a large piece of construction paper, use a pencil and ruler to lightly draw two lines connecting the page's opposite corners. Mark the page where the lines intersect to locate the paper's centre. Erase the lines but leave the centre mark.

- 2
Starting at the page's centre, draw a horizontal line 10 centimetres long.

- 3
Draw another line, 10 centimetres long, perpendicular to the first.

- 4
Connect the endpoints of the lines to form a right triangle.

- 5
Create a second right triangle using the hypotenuse of your first triangle as a base. Draw a 10-centimeter line perpendicular to the base and connect the legs to form the hypotenuse of your second triangle.

- 6
Continue to draw triangles by using the hypotenuse of the previous triangle as a base, drawing a perpendicular line ten centimetres long and connecting the legs. Fill as much of the page as is practical.

- 7
Neatly label the length of every line. The length of each of the outer lines should be 10 centimetres. The length of the first triangle's hypotenuse in centimetres is the square root of one. The hypotenuse of the second triangle will be the square root of two centimetres long. Continue to label each hypotenuse according to this pattern. Make sure you know why these lengths are correct.

- 8
Decorate the spiral by colouring each triangle a different colour. You could even disguise the finished spiral as a snail or seashell.

## Draw Your Pythagorean Spiral

- 1
Look over your report and check it for errors.

- 2
Write a final section in which you describe anything you found particularly difficult or particularly enjoyable.

- 3
Include a section about what you learnt through the course of the project.