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An overlapping circles grid is a geometric pattern of repeating, overlapping circles of an equal radius in two-dimensional space. Commonly, designs are based on circles centered on triangles (with the simple, two circle form named vesica piscis ) or on the square lattice pattern of points.
Patterns of seven overlapping circles appear in historical artefacts from the 7th century BC onwards; they become a frequently used ornament in the Roman Empire period, and survive into medieval artistic traditions both in Islamic art (girih decorations) and in Gothic art. The name "Flower of Life" is given to the overlapping circles pattern in New Age publications.
Of special interest is the six petal rosette derived from the "seven overlapping circles" pattern, also known as "Sun of the Alps" from its frequent use in alpine folk art in the 17th and 18th century.
The triangular lattice form, with circle radii equal to their separation is called a seven overlapping circles grid. [1] It contains 6 circles intersecting at a point, with a 7th circle centered on that intersection.
Overlapping circles with similar geometrical constructions have been used infrequently in various of the decorative arts since ancient times. The pattern has found a wide range of usage in popular culture, in fashion, jewelry, tattoos and decorative products.
The oldest known occurrence of the "overlapping circles" pattern is dated to the 7th or 6th century BCE, found on the threshold of the palace of Assyrian king Aššur-bāni-apli in Dur Šarrukin (now in the Louvre). [2]
The design becomes more widespread in the early centuries of the Common Era. One early example are five patterns of 19 overlapping circles drawn on the granite columns at the Temple of Osiris in Abydos, Egypt, [3] and a further five on column opposite the building. They are drawn in red ochre and some are very faint and difficult to distinguish. [4] The patterns are graffiti, and not found in natively Egyptian ornaments. They are mostly dated to the early centuries of the Christian Era [5] although medieval or even modern (early 20th century) origin cannot be ruled out with certainty, as the drawings are not mentioned in the extensive listings of graffiti at the temple compiled by Margaret Murray in 1904. [6]
Similar patterns were sometimes used in England as apotropaic marks to keep witches from entering buildings. [7] Consecration crosses indicating points in churches anointed with holy water during a churches dedication also take the form of overlapping circles.
In Islamic art, the pattern is one of several arrangements of circles (others being used for fourfold or fivefold designs) used to construct grids for Islamic geometric patterns. It is used to design patterns with 6- and 12-pointed stars as well as hexagons in the style called girih . The resulting patterns however characteristically conceal the construction grid, presenting instead a design of interlaced strapwork. [8]
Patterns of seven overlapping circles are found on Roman mosaics, for example at Herod's palace in the 1st century BC. [9]
The design is found on one of the silver plaques of the Late Roman hoard of Kaiseraugst (discovered 1961). [10] It is later found as an ornament in Gothic architecture, and still later in European folk art of the early modern period.
High medieval examples include the Cosmati pavements in Westminster Abbey (13th century). [11] Leonardo da Vinci explicitly discussed the mathematical proportions of the design. [12]
The name "Flower of Life" is modern, associated with the New Age movement, and commonly attributed specifically to Drunvalo Melchizedek in his book The Ancient Secret of the Flower of Life (1999). [13] [14]
The pattern and modern name have propagated into wide range of usage in popular culture, in fashion, jewelry, tattoos and decorative products. The pattern in quilting has been called diamond wedding ring or triangle wedding ring to contrast it from the square pattern. Besides an occasional use in fashion, [15] it is also used in the decorative arts. For example, the album Sempiternal (2013) by Bring Me the Horizon uses the 61 overlapping circles grid as the main feature of its album cover, [16] whereas the album A Head Full of Dreams (2015) by Coldplay features the 19 overlapping circles grid as the central part of its album cover. Teaser posters illustrating the cover art to A Head Full of Dreams were widely displayed on the London Underground in the last week of October 2015. [17]
The "Sun of the Alps" (Italian Sole delle Alpi) symbol has been used as the emblem of Padanian nationalism in northern Italy since the 1990s. [18] It resembles a pattern often found in that area on buildings. [19]
A seven-circle "Flower of Life" is also used in the coat of arms of Asgardia the space nation.
In the examples below the pattern has a hexagonal outline, and is further circumscribed.
In the examples below the pattern does not have a hexagonal outline.
Martha Bartfeld, author of geometric art tutorial books, described her independent discovery of the design in 1968. Her original definition said, "This design consists of circles having a 1-[inch; 25 mm] radius, with each point of intersection serving as a new center. The design can be expanded ad infinitum depending upon the number of times the odd-numbered points are marked off."
The pattern figure can be drawn by pen and compass, by creating multiple series of interlinking circles of the same diameter touching the previous circle's center. The second circle is centered at any point on the first circle. All following circles are centered on the intersection of two other circles.
The pattern can be extended outwards in concentric hexagonal rings of circles, as shown. The first row shows rings of circles. The second row shows a three-dimensional interpretation of a set of n×n×n cube of spheres viewed from a diagonal axis. The third row shows the pattern completed with partial circle arcs within a set of completed circles.
Expanding sets have 1, 7, 19, 37, 61, 91, 127, etc. circles, and continuing ever larger hexagonal rings of circles. The number of circles is n3-(n-1)3 = 3n2-3n+1 = 3n(n-1)+1.
These overlapping circles can also be seen as a projection of an n-unit cube of spheres in 3-dimensional space, viewed on the diagonal axis. There are more spheres than circles because some are overlapping in 2 dimensions.
1-circle | 7-circle (8-1) | 19-circle (27-8) | 37-circle (64-27) | 61-circle (125-64) | 91-circle (216-125) | 127-circle... (343-216) |
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1-sphere (1×1×1) | 8-sphere (2×2×2) | 27-sphere (3×3×3) | 64-sphere (4×4×4) | 125-sphere (5×5×5) | 216-sphere (6×6×6) | 343-sphere (7×7×7) |
(image missing) | (image missing) | (image missing) | (image missing) | |||
+12 arcs | +24 arcs | +36 arcs | +48 arcs | +60 arcs | +72 arcs | +84 arcs |
Another triangular lattice form is common, with circle separation as the square root of 3 times their radius. Richard Kershner showed in 1939 that no arrangement of circles can cover the plane more efficiently than this hexagonal lattice arrangement. [20]
Two offset copies of this circle pattern makes a rhombic tiling pattern, while three copies make the original triangular pattern.
The center lens of the 2-circle figure is called a vesica piscis, from Euclid. Two circles are also called Villarceau circles as a plane intersection of a torus. The areas inside one circle and outside the other circle is called a lune.
The 3-circle figure resembles a depiction of borromean rings and is used in 3-set theory Venn diagrams. Its interior makes a unicursal path called a triquetra. The center of the 3-circle figure is called a reuleaux triangle.
Vesica piscis | Borromean rings | Venn diagram | Triquetra | Reuleaux triangle |
Some spherical polyhedra with edges along great circles can be stereographically projected onto the plane as overlapping circles.
Octahedron | Cuboctahedron | Icosidodecahedron |
The 7-circle pattern has also been called an Islamic seven-circles pattern for its use in Islamic art.
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The square lattice form can be seen with circles that line up horizontally and vertically, while intersecting on their diagonals. The pattern appears slightly different when rotated on its diagonal, also called a centered square lattice form because it can be seen as two square lattices with each centered on the gaps of the other.
It is called a Kawung motif in Indonesian batik, and is found on the walls of the 8th century Hindu temple Prambanan in Java.
It is called an Apsamikkum from ancient Mesopotamian mathematics. [21]
Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions. It is associated with the belief of a divine creator of the universal geometer. The geometry used in the design and construction of religious structures such as churches, temples, mosques, religious monuments, altars, and tabernacles has sometimes been considered sacred. The concept applies also to sacred spaces such as temenoi, sacred groves, village greens, pagodas and holy wells, Mandala Gardens and the creation of religious and spiritual art.
In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. It is available either as loose leaf paper or bound in notebooks or Graph Books.
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.
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension which spans the vector space . For any basis of , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regular tiling of a space by a primitive cell.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as p4m, Coxeter notation as [4,4], and orbifold notation as *442.
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.
Girih are decorative Islamic geometric patterns used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern.
In geometry, circle packing is the study of the arrangement of circles on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.
Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures.
The Topkapı Scroll is a Timurid dynasty patterned scroll in the collection of the Topkapı Palace museum.
In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and Middleton on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete lattice of points. This result, also known as the Petersen–Middleton theorem, is a generalization of the Nyquist–Shannon sampling theorem for sampling one-dimensional band-limited functions to higher-dimensional Euclidean spaces.
In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposite points of tangency, with their centers on logarithmic spirals of three different shapes.