geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 ...

) with the property that the reflection 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 modu ...

to each other. There are eight possible edge tessellations in

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, 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 ge ...

. 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 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 Sir David Brewster Knight of the Royal Guelphic Order, KH President of the Royal Society of Edinburgh, PRSE Fellow of the Royal Society of London, FRS Fellow of the Society of Antiquaries of Scotland, FSA Scot Fellow of the Scottish Society of ...

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 symmetrical pattern when viewed fro ...

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