Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". [1]
It is one of a series of four woodcuts by Escher depicting ideas from hyperbolic geometry. Dutch physicist and mathematician Bruno Ernst called it "the best of the four". [2]
Escher became interested in tessellations of the plane after a 1936 visit to the Alhambra in Granada, Spain, [3] [4] and from the time of his 1937 artwork Metamorphosis I he had begun incorporating tessellated human and animal figures into his artworks. [4]
In a 1958 letter from Escher to H. S. M. Coxeter, Escher wrote that he was inspired to make his Circle Limit series by a figure in Coxeter's article "Crystal Symmetry and its Generalizations". [2] [3] Coxeter's figure depicts a tessellation of the hyperbolic plane by right triangles with angles of 30°, 45°, and 90°; triangles with these angles are possible in hyperbolic geometry but not in Euclidean geometry. This tessellation may be interpreted as depicting the lines of reflection and fundamental domains of the (6,4,2) triangle group. [5] An elementary analysis of Coxeter's figure, as Escher might have understood it, is given by Casselman (2010). [6]
Escher seems to have believed that the white curves of his woodcut, which bisect the fish, represent hyperbolic lines in the Poincaré disk model of the hyperbolic plane, in which the whole hyperbolic plane is modeled as a disk in the Euclidean plane, and hyperbolic lines are modeled as circular arcs perpendicular to the disk boundary. Indeed, Escher wrote that the fish move "perpendicularly to the boundary". [1] However, as Coxeter demonstrated, there is no hyperbolic arrangement of lines whose faces are alternately squares and equilateral triangles, as the figure depicts. Rather, the white curves are hypercycles that meet the boundary circle at angles of cos−121/4 − 2−1/4/2, approximately 80°. [2]
The symmetry axes of the triangles and squares that lie between the white lines are true hyperbolic lines. The squares and triangles of the woodcut closely resemble the alternated octagonal tiling of the hyperbolic plane, which also features squares and triangles meeting in the same incidence pattern. However, the precise geometry of these shapes is not the same. In the alternated octagonal tiling, the sides of the squares and triangles are hyperbolically straight line segments, which do not link up in smooth curves; instead they form polygonal chains with corners. In Escher's woodcut, the sides of the squares and triangles are formed by arcs of hypercycles, which are not straight in hyperbolic geometry, but which connect smoothly to each other without corners.
The points at the centers of the squares, where four fish meet at their fins, form the vertices of an order-8 triangular tiling, while the points where three fish fins meet and the points where three white lines cross together form the vertices of its dual, the octagonal tiling. [2] Similar tessellations by lines of fish may be constructed for other hyperbolic tilings formed by polygons other than triangles and squares, or with more than three white curves at each crossing. [7]
Euclidean coordinates of circles containing the three most prominent white curves in the woodcut may be obtained by calculations in the field of rational numbers extended by the square roots of two and three. [8]
Viewed as a pattern, ignoring the colors of the fish, in the hyperbolic plane, the woodcut has three-fold and four-fold rotational symmetry at the centers of its triangles and squares, respectively, order-three dihedral symmetry (the symmetry of an equilateral triangle) at the points where the white curves cross. In John Conway's orbifold notation, this set of symmetries is denoted 433. Each fish provides a fundamental region for this symmetry group. Contrary to appearances, the fish do not have bilateral symmetry: the white curves of the drawing are not axes of reflection symmetry. [9] [10] For example, the angle at the back of the right fin is 90° (where four fins meet), but at the back of the much smaller left fin it is 120° (where three fins meet).
The fish in Circle Limit III are depicted in four colors, allowing each string of fish to have a single color and each two adjacent fish to have different colors. Together with the black ink used to outline the fish, the overall woodcut has five colors. It is printed from five wood blocks, each of which provides one of the colors within a quarter of the disk, for a total of 20 impressions. The diameter of the outer circle, as printed, is 41.5 cm (16+3⁄8 in). [11]
As well as being included in the collection of the Escher Museum in The Hague, copies of Circle Limit III are included in the collections of the National Gallery of Art [12] and the National Gallery of Canada. [13]
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
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 of geometries.
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
In hyperbolic geometry, a horocycle, sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics ( normals) through a point on a horocycle are limiting parallel, and all converge asymptotically to a single ideal point called the centre of the horocycle. In some models of hyperbolic geometry it looks like the two "ends" of a horocycle get closer and closer to each other and closer to its centre, this is not true; the two "ends" of a horocycle get further and further away from each other and stay at an infinite distance off its centre. A horosphere is the 3-dimensional version of a horocycle.
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.
In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.
In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}.
In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.
In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,7,3}.
In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,8,3}.
In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,∞,3}.
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