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8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

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Napoleon's Theorem

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

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Twizzle Wind Up

A loopy exploration of z^2+1=0 (z squared plus one) with an eye on winding numbers. Try not to get dizzy!

Twizzles Venture Forth

Stage: 4 Challenge Level: Challenge Level:1

At this point it is worth taking some time out to get really familiar with twizzle arithmetic. Make your own drawings, or use the twizzle addition or twizzle multiplication animations to get some practice.

You should begin to see how it's quite easy to add without a picture if the twizzle is written like 2+3i or 3-4i .

When you multiply you'll find the cis notation easier - e.g. use 3cis(30) or 2cis(10) .

Check that you agree with each of these results:

\begin{eqnarray} 2 cis (30) \times 3 cis (60) = 6i \\ (1 cis (120))^3 = 1 \\ i^4 = 1 \\ (1 cis (72))^5 = 1 \\ (2 cis (60))^3 = -8 \end{eqnarray}

1cis(120) is a cube root of 1. Can you think of any others? How many are there?

i is a fourth root of 1. Can you think of any others? How many are there?

1cis(72) is a fifth root of 1. Can you think of any others? How many are there?

Are you beginning to see a pattern?

Tackle Twizzle Wind Up and Twizzle Twists and you might begin to see how general this is and why this pattern is appearing.