Tessellation Study — Regular and Semi-Regular

Only three regular polygons tile — why? Because angles at a vertex must sum to 360°. Triangle (60°×6), square (90°×4), hexagon (120°×3) work. Pentagon (108°) leaves gaps. Indirect aim: angle geometry, aesthetic mathematics, preparation for crystallography.

• Framework: Montessori
• Unit: Sensorial
• Learning area: Geometric Cabinet
• Level: Ages 9-11

Objectives

• Tessellate the plane with triangles, squares, hexagons
• Discover why only three regular tessellations exist
• Study semi-regular (Archimedean) tessellations
• Design original tessellations

Materials

• Pattern blocks (triangle, square, hexagon, rhombus, trapezoid)
• Graph paper
• Compass and ruler
• Images of Escher tessellations

Instructions

1. Try tessellating with triangles: 6 fit around a point. 2. Squares: 4 fit. 3. Hexagons: 3 fit. 4. Try pentagons: leaves gaps — no regular tessellation. 5. Combine shapes: triangle+hexagon = semi-regular. 6. Study Escher's lizard tessellations. 7. Design an original with graph paper. 8. Photograph real-world tessellations (brick, tile).

Educator tip

Escher died in 1972 but his tessellations are immortal. A classroom wall of Escher prints inspires years of research.

Assessment

Observe correct regular tessellations, angle reasoning, and original design.

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• Fibonacci Sequence — Patterns in Nature
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• Color Wheel — Chromatic Theory
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