Tessellations

You ask, “What is a tessellation, exactly?” A tessellation is any repeating pattern of interlocking shapes. In English, a way to tile a floor with no overlapping pieces and no gaps.

Everything has rules, even tessellations! There can be no overlapping and no gaps. (Heard that one already!) The polygons must be regular. The sides of the figure must all be the same length The interior angles of the figure must be the same measure Each vertex must look the same. What’s a vertex? Where the “corners” meet

What type of polygons will work?? Triangles? Squares? Pentagons? Hexagons? Heptagons? Octagons? Let’s find out!

Finding the Interior Angle Where” n” is the number of sides

Triangles Squares Yes, they do! Yes, they do! The interior angle of each triangle is 60o At each “vertex” the sum of the measure is 360o Yes, they do! The interior angle of each square is 90o At each “vertex” the sum of the measure is 360o

Pentagons Hexagons No, they don’t! Yes, they do! Do you see the gap? The interior angle of each pentagon is 108o At each “vertex” the sum of the measure is 324o Do you see the gap? Yes, they do! The interior angle of each hexagon is 120o At each “vertex” the sum of the measure is 360o

Heptagons Octagons What do you think? ? No, they don’t! ? The interior angle of each heptagon is 257.1428…o At each “vertex” the sum of the measure is 771.428…o Do you see the overlap? What do you think? ? The interior angle of each octagon is 135o At each “vertex” the sum of the measure is …

You’re right! Octagons won’t tessellate. In fact, any polygon with more than 6 sides won’t tessellate So, can you remember which polygons tessellate? Triangles Squares Hexagons

Fairly Simple Tessellations… G e o m e t r y Fairly Simple Tessellations…

Semi-Regular Tessellations 3, 6, 3, 6 3, 3, 3, 3, 6 These tessellations are made up of triangles and hexagons, but the configuration of the vertices are different. That’s why they are named differently! To name a tessellation, simply work your way around the vertex naming the number of sides of each polygon. Always put the smallest possible number first!

Semi-Regular Tessellation?? This tessellations is also made up of triangles and hexagons, can you see why this isn’t an “official” semi-regular tessellation? It breaks the vertex rule, can you tell where?

More Semi-Regular Tessellations… G e o m e t r y

What’s M.C. Escher? He was a Dutch graphic artist (1898-1972) who made prints involving tessellations, impossible figures or worlds, polyhedra, and unusual perspective systems.

“Convex and Concave” m e t r y

e t r y “C y c l e”

“Bats and Owls” e t r y

“Ascending & Descending”

“Bond of Union”

“Drawing Hands”

“Encounter”

“Gravitation”

“Hand with Reflecting Sphere”

“House of Stairs”

“Liberation”

“Sky and Water I, 1938” G e om e t r y

“Metamorphose II, 1940” G e om e t r y

Some more Escher tessellations…

and more…

Are you ready to tessellate?