geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, an edge tessellation is a partition of the plane into non-overlapping polygons (a

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...

) with the property that the

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

of any of these polygons across any of its edges is another polygon in the tessellation. All of the resulting polygons must be

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

, and

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

to each other. There are eight possible edge tessellations in Euclidean geometry, but others exist in

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...

. The eight Euclidean edge tessellations are: In the first four of these, the tiles have no obtuse angles, and the degrees of the vertices are all even. Because the degrees are even, the sides of the tiles form lines through the tiling, so each of these four tessellations can alternatively be viewed as an

arrangement of lines In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestr ...

. In the second four, each tile has at least one obtuse angle at which the degree is three, and the sides of tiles that meet at that angle do not extend to lines in the same way. These tessellations were considered by 19th-century inventor David Brewster in the design of

kaleidoscope A kaleidoscope () is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a regular symmetrical pattern when v ...

s. A kaleidoscope whose mirrors are arranged in the shape of one of these tiles will produce the appearance of an edge tessellation. However, in the tessellations generated by kaleidoscopes, it does not work to have vertices of odd degree, because when the image within a single tile is asymmetric there would be no way to reflect that image consistently to all the copies of the tile around an odd-degree vertex. Therefore, Brewster considered only the edge tessellations with no obtuse angles, omitting the four that have obtuse angles and degree-three vertices.

Citations

{{reflist, refs= {{citation, first=David, last=Brewster, authorlink=David Brewster, title=A Treatise on the Kaleidoscope, title-link=iarchive:b29295440, contribution=Chapter XI: On the construction and use of polycentral kaleidoscopes, contribution-url=https://archive.org/details/b29295440/page/92, pages=92–100, year=1819, location=Edinburgh, publisher=Archibald Constable & Co. {{citation , last1 = Kirby , first1 = Matthew , last2 = Umble , first2 = Ronald , arxiv = 0908.3257 , doi = 10.4169/math.mag.84.4.283 , issue = 4 , journal = Mathematics Magazine , mr = 2843659 , pages = 283–289 , title = Edge tessellations and stamp folding puzzles , volume = 84 , year = 2011. Tessellation

See also

Citations