Tiling and Tessellation

Semi-regular Tessellation

Squares and Equilateral Triangles


Tessellation with Convex Hexagons


Type 1

B = D = F = 120 deg

AB = BC; CD = DE; EF = FA

worksheet for a transformation exercise based on this tessellation


Type 2

B+C+D = 360 deg



Type 3

A+B+D = 360 deg

AF = CD; BC = DE


A Special Case of Type 2

3 pairs of opposite sides equal and parallel


Tessellation with Convex Pentagons


Type 1

B+C+D =360 deg


Type 2

A+B+D = 360 deg



Type 3

A = B = D = 120 deg

CD = DE; CA + BE = AB


Type 4

A = C = 90 deg

EA = AB; BC = CD


Type 5

A = 60 deg; C = 120 deg

EA = AB; BC = CD


Type 6

A+B+D = 360 deg; A = 2C

DE = EA = AB; BC = CD


Type 7

2D+A = 360 deg; 2B+C = 360 deg

EA = AB = BC = CD


Special Case of Type 2

E = C = 90 deg

AE = DC; ED = BC


Special Case of Type 2 and 4

E = C = 90 deg

AE = ED = DC = CB

AB can be adjusted until all sides equal (Cairo Street Tiling)


Tessellation with Concave Pentagons; Rotational Symmetry


Regular Decagon (Colored and Outline)

composed of equilateral pentagons

worksheet for investigating number of pentagons in successive rings

  Regular Octagon
  Regular Nonagon




Totally Tessellated - A ThinkQuest Internet Challenge 1998 Entry
Tessellation Tutorials by Suzanne Alejandre at the Math Forum
Tessellations Theme Page at Community Learning Network

links to information about tessellations (curricular resources and instructional materials)

Tantalizing Tessellations

series of lesson plans integrating tessellations across the curriculum

Tessellations: Creative Combinations of Math, Art and Fun (at mathartfun.com)
Catalog of Isohedral Tilings by Symmetric Polygonal Tiles by Doris Schattschneider and Nikolai Dolbilin

rather technical article but full of interesting dynamic illustrations with The Geometer's Sketchpad, showing tilings of various symmetric quadrilaterals, pentagons and hexagons


Intriguing Tessellations by Marjorie Rice: Discovery and Designsof Convex Pentagon Tessellations

Pentagons that tile the Plane

it talks about all kinds of pentagons (concave and convex) and a 'recipe' for finding the tiles

The 14 Different Types of Convex Pentagons that Tile the Plane
Tessellation Resources at Geometry Centre

2 lists of online resources complied by Prof. Doris Schattschneider: references and sources for illustrations; software for creating tilings

Grotesque Geometry by Andrew Crompton

art work inspired by those of Escher and a list of artists doing similar things

Internet Lesson Plans by artist/teacher Woody Duncan
Escher Web Sketch

a Java applet for drawing repeating patterns with various symmetry; you may work online or download the applet

Penrose Tiler: a Java applet for creating and investiagting Penrose Tilings



Gardner, M. (1988), Time Travel and Other Mathematical Bewilderments, New York: W.H. Freeman and Company.
(Ch. 13, Tiling with Convex Polygons)

Nelson, D., Joseph, G.G., Williams, J. (1993), Multicultural Mathematics, Oxford: Oxford Unversity Press.
(Ch. 6, Geometry and Art).
Wells, D. (1991), The Penguin Dictionary of Curious and Interesting Geometry, Penguin Books.