🎨 2D Tessellation Explorer

A tessellation is when shapes fit together perfectly to cover a flat surface with no gaps or overlaps, like tiles on a floor. Your mission is to discover which regular polygons can tessellate by themselves, and which ones need other shapes to help them create beautiful patterns.

🔍 Shape Toolkit

Load up your workspace with different shapes. Try dragging them around to see which ones fit together nicely!

🧮 Angle Detective Work

The secret to tessellations lies in angles! For shapes to fit perfectly around a point, their corner angles must add up to exactly 360°.

🔍 Regular Polygon Interior Angles:
• Triangle: 60° each corner
• Square: 90° each corner
• Pentagon: 108° each corner
• Hexagon: 120° each corner
• Octagon: 135° each corner

🧩 Tessellation Test:
• 6 triangles: 6 × 60° = 360° ✅
• 4 squares: 4 × 90° = 360° ✅
• 3 hexagons: 3 × 120° = 360° ✅
• Pentagons: 108° doesn't divide 360° evenly ❌
• Octagons: 135° doesn't divide 360° evenly ❌
🎯 What About Pentagons and Octagons?
A single pentagon or octagon can't tessellate by itself because their angles don't add up to 360°. But you can create semi-regular tessellations by combining different shapes whose angles add up to 360°.
🎨 Try These Combinations:
• Squares + Octagons: Square-Octagon-Octagon around each point (90° + 135° + 135° = 360°)
• Hexagons + Triangles: Multiple patterns work! Try it with 1 or 2 hexagons.
• Some pairs of shapes can work in different orders, but beware - to be a semi-regular tessellation, the shapes must be in the same order at every vertex. (If vertices are different, the tessellation is only demi-regular.)
• Some shapes can't form a tessellation no matter what other shapes you combine them with.
💡 Order Matters! The same shapes can often tessellate in different arrangements - experiment to find them all!

🎨 Custom Tessellation Builder

Choose up to 3 different shapes to experiment with. You'll get 20 of each shape to play with. Since most semi-regular tessellations use only 2-3 different shapes, this gives you everything you need to discover new patterns!

🏆 Tessellation Master Challenge

Now that you know the secret, can you predict which combinations will work before trying them? Remember: the angles around each point must add up to exactly 360°!

🤯 Mind-Blowing Fact: There are exactly 8 different semi-regular tessellations possible, and mathematicians have proven that no more exist! You can discover them all through experimentation, or by thinking about the angles.