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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!

Twizzle Twists

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Make sure you've tackled Twizzle Wind Up before trying this problem.

In each of these animations the grey w twizzle is related to the blue z twizzle. As usual, you can change the value of the blue z twizzle by moving its number arrow head. This will make the value of the grey w twizzle change too.

In each of these questions your task is to discover a formula for w in terms of z .
[For example, it might be w=z2 +z+4 or or w=(z-1)(z-3) or w=z3 -8. ]

There's some useful help in the hints which will get you started, but play and think first!

Switch on tracing and turn z around its spot in a big loop. Watch how w behaves as as you do this.
Try this with smaller and smaller loops.

Q1: Full Screen Version
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Q2: Full Screen Version
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Q3: Full Screen Version
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