The Tiling Collective — Printable Cards

🎨 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.

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 and draw the square on the Argand diagram.

Task 1b

Raise to the power 4. Write the equation.

Use De Moivre's theorem to raise each vertex to the power 4. What do all four results have in common? Write a single equation \(z^n=w\) that all four vertices satisfy.

💡 \(\bigl[r(\cos\theta+i\sin\theta)\bigr]^n=r^n(\cos n\theta+i\sin n\theta)\). Use \(r=1\), \(n=4\).

Phase 2

The Rotated Triangle

Task 2a

Mark all three vertices geometrically. Draw the triangle.

A second tile is an equilateral triangle on the unit circle. One vertex is at argument \(\pi/3\). Using only geometry — no algebra yet — mark all three vertices and draw the triangle. Explain your reasoning.

Task 2b

Write in polar form. Build the formula.

Write each vertex as \(\cos\theta+i\sin\theta\). Find a formula for the argument of the \(k\)-th vertex \((k=0,1,2)\).

Task 2c

Raise to the power 3. Find the equation.

Use De Moivre's theorem to raise each vertex to the power 3. What equation \(z^3=w\) do all three satisfy?

Phase 3

The Pentagon — Bigger Circle

Task 3a

Mark all five vertices. Draw the pentagon.

A regular pentagon has vertices on a circle of radius 2. One vertex is at \(2\!\left(\cos\dfrac{\pi}{5}+i\sin\dfrac{\pi}{5}\right)\). Use the geometric approach from Phase 2 to mark all five vertices. State the spacing and modulus.

Task 3b

Raise to the power 5. Find the equation.

Use De Moivre's to raise \(z_1\) to the power 5. What equation \(z^5=w\) do all five vertices satisfy? How does \(|w|\) relate to the circle radius?

Phase 4

Reverse-Engineer the Hexagon

Task 4a

Find \(n\), the radius, and the argument.

A regular hexagon has one vertex at \(-\sqrt{3}+i\). Find \(n\), the circle radius, and the argument of this vertex. Show your working.

Task 4b

Find all six vertices in exact rectangular form. Draw the hexagon.

Using the geometric formula and Task 4a, find all six vertices. Write each in exact rectangular form \(a+bi\) — you will need exact values of \(\cos\) and \(\sin\). Plot them and draw the hexagon. Remember to reduce arguments greater than \(2\pi\).

Task 4c

Find \(w\). State and verify \(z^6=w\).

Use De Moivre's to raise \(-\sqrt{3}+i\) to the power 6. Reduce the argument fully. Verify \(|w|\) and \(\arg(w)\) match Task 4a.

Task 4d — Extension

Write the general formula for all \(n\) roots of \(z^n=w\).

Express \(z_k\) in terms of \(|w|\), \(\arg(w)\), \(n\), \(k\). Label each part geometrically.