The Tiling
Collective

A second tile is an equilateral triangle, also on the unit circle. One vertex sits at argument \(\dfrac{\pi}{3}\). Using only geometric reasoning — no algebra yet — mark all three vertices on your Argand diagram and draw the triangle. Be ready to explain how you decided where to place them.

Write each vertex in the form \(\cos\theta+i\sin\theta\). Then write a formula for the argument of the \(k\)-th vertex in terms of \(\dfrac{\pi}{3}\), \(\dfrac{2\pi}{3}\), and \(k\), for \(k=0,1,2\).

Use De Moivre's theorem to raise each vertex to the power 3. What single equation \(z^3=w\) do all three vertices satisfy? Why does your answer not depend on which vertex you chose?

The studio scales up. The next tile is a regular pentagon, but its vertices lie on a circle of radius 2. One vertex is at \(2\!\left(\cos\dfrac{\pi}{5}+i\sin\dfrac{\pi}{5}\right)\). Using the same geometric approach as Phase 2, mark all five vertices on your Argand diagram. State the angular spacing and the modulus of each vertex.

Use De Moivre's theorem to raise \(z_1=2\!\left(\cos\dfrac{\pi}{5}+i\sin\dfrac{\pi}{5}\right)\) to the power 5. What single equation \(z^5=w\) do all five vertices satisfy? How does the modulus of \(w\) relate to the radius of the circle?

The studio found an old tile: a regular hexagon with six vertices equally spaced on a circle. One vertex is labelled \(-\sqrt{3}+i\). From this information alone, find \(n\), the radius of the circle, and the argument of the given vertex. Show your working.

Using the geometric formula and your answer from Task 4a, find all six vertices. Write each one in exact rectangular form \(a+bi\) — you will need exact values of \(\cos\) and \(\sin\) for the arguments. Plot all six on your Argand diagram and draw the hexagon.

Use De Moivre's theorem to raise the given vertex \(-\sqrt{3}+i\) to the power 6. Find \(w\) and state the equation. Verify that \(|w|\) and \(\arg(w)\) are consistent with what you found in Task 4a.

A regular \(n\)-gon has vertices on a circle of radius \(r\), and the equation generating them is \(z^n=w\). Write the general formula for all \(n\) roots \(z_k\) in terms of \(|w|\), \(\arg(w)\), \(n\), and \(k\). Label what each part of the formula represents geometrically.