The Tiling Collective

Stand Up & Math · Complex numbers · \(n\)th roots · HL 1.12–1.14
🎨 The Scenario

Background — read before you begin

A design studio creates geometric tile patterns. Every pattern is built from a regular polygon whose vertices are equally spaced around a circle, centred at the origin. Each vertex is a complex number in polar form \(r(\cos\theta+i\sin\theta)\). Your job: find those vertices, discover the rule behind them, and reverse-engineer a mystery tile. All work goes on the whiteboard first.

1
Phase 1
The Square Tile
Task 1a
Find all four vertices. Draw the square.
The studio's first tile is a square. Its four vertices sit equally spaced on a circle of radius 1. One vertex is at \((1,0)\). Find all four vertices as complex numbers in the form \(\cos\theta+i\sin\theta\).
Task 1b
Raise each vertex to the power 4. Write the equation.
Use De Moivre's theorem on each vertex. What do all four results have in common? Write a single equation \(z^n=w\) that all four vertices satisfy.
💡 \(\bigl[r(\cos\theta+i\sin\theta)\bigr]^n=r^n(\cos n\theta+i\sin n\theta)\)
2
Phase 2
The Rotated Triangle
Task 2a
Mark all three vertices geometrically.
An equilateral triangle, unit circle. One vertex at argument \(\pi/3\). Geometric reasoning only — no algebra yet.
Task 2b
Write the vertices. Build the formula.
Write each vertex in polar form. Find the formula for the argument of the \(k\)-th vertex \((k=0,1,2)\).
Task 2c
Raise to the power 3. Find the equation.
Apply De Moivre's. What single equation \(z^3=w\) do all three satisfy? Why doesn't it depend on which vertex you chose?
3
Phase 3
The Pentagon
Task 3a
Mark all five vertices. Draw the pentagon.
Regular pentagon on a circle of radius 2. One vertex at \(2\!\left(\cos\tfrac{\pi}{5}+i\sin\tfrac{\pi}{5}\right)\). State the angular spacing and modulus of each vertex.
Task 3b
Raise to the power 5. Find the equation.
Apply De Moivre's to \(z_1=2\!\left(\cos\tfrac{\pi}{5}+i\sin\tfrac{\pi}{5}\right)\). What equation \(z^5=w\) do all five satisfy? How does \(|w|\) relate to the radius?
4
Phase 4
Reverse-Engineer the Hexagon
Task 4a
Find \(n\), the radius, and the argument.
A regular hexagon — one vertex is \(-\sqrt{3}+i\). From this alone, find \(n\), the radius of the circle, and the argument of the given vertex.
Task 4b
Find all six vertices in exact rectangular form.
Use the geometric formula to find all six vertices. Write each as \(a+bi\) using exact trig values. Plot and draw the hexagon.
Task 4c
Find \(w\). State and verify \(z^6=w\).
Raise \(-\sqrt{3}+i\) to the power 6. Find \(w\). Verify \(|w|\) and \(\arg(w)\) are consistent with Task 4a.
Task 4d — Extension
Write the general formula.
For \(z^n=w\), write the general formula for all \(n\) roots \(z_k\) in terms of \(|w|,\;\arg(w),\;n,\;k\).