Every triangle can be used to tile or tessellate a plane because the 3 angles add up to 180° and so can be arranged to form a straight line. In a similar way every quadrilateral will tessellate because the 4 angles can be grouped round a point.

The importance of recreational mathematics and the involvement of amateur mathematicians has been dramatically demonstrated recently in connection with the prob/ lem of tiling the plane with congruent pentagons. The problem is to describe completely all pentagons whose congruent images will tile the plane (without overlaps or gaps). The problem was thought to have been solved by R. B. Kershner, who announced his results in 1968 [18], [191. In July, 1975, Kershner's article was the main topic of Martin Gardner's column, "Mathematical Games" in Scientific American. Inspired by the challenge of the problem, at least two readers attempted their own tilings of pentagons and each discovered pentagons missing from Kershner's list. New interest in the problem has been aroused and both amateur and professional mathematicians are presently working on its solution.

Additional excerpts

Tessellations with Pentagons

Tiling the Plane with Congruent Pentagons

A personal account of an interdisciplinary inquiry into the study of Islamic geometric design and architectural decoration touching on the fields of History, History of Science, Scientific Theory of Symmetry and History of Art. The study stresses the necessity of the use of a common scientific language of Symmetry Notation in order to discuss and communicate in a precise manner about Islamic geometric pattern. To understand Islamic geometric design, it is necessary to move beyond the symmetry issues, to the step-by-step process of design. This is based on primary sources of scientific manuscripts of practical geometry written specifically for the Muslim artisans. The research demonstrates not only a direct meeting but a collaborative work between science and art in Islamic civilization.

In the Tower of Babel: Beyond symmetry in islamic design

Isohedral tilings of the plane by polygons

The art of tiling goes back thousands of years. For centuries builders of floors and walls have sought out arrangements that are pleasing to the eye, and many ancient tilings and mosaics survive today. More recently, similar regular arrangements have been observed in the molecular structure of crystals. Repeating patterns have also been used in modern drawings (e.g., those by Escher). Tilings frequently occur in books on recreational mathematics; indeed, amateur mathematicians have contributed much to this subject.

Tessellations with Pentagons

Citations

Tiling the Plane with Congruent Pentagons

In the Tower of Babel: Beyond symmetry in islamic design

Isohedral tilings of the plane by polygons

Tiling and Dissection

Related Papers (5)

Maximum Rectangle Inscribed in a Triangle.

Experimental Mathematics — Tesselations of Convex Polygons in a Hexagonal Lattice

Dissections of regular polygons into similar triangles

Tilings of regular pentagons with similar right triangles

Intersecting hexagons in 3-space