Overlapping circles grid

File:Flower of life 6-levels.png

The triangular lattice form, with circle radii equal to their separation is called a seven overlapping circles grid.[https://www.metmuseum.org/met-publications/islamic-art-and-geometric-design-activities-for-learning Islamic Art and Geometric Design: Activities for Learning ] 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).Louvre Inv.-Nr. [https://collections.louvre.fr/en/ark:/53355/cl010144627 AO 19915]. Georges Perrot, Charles Chipiez, A History of Art in Chaldæa and Assyria, vol. 1, London 1884, S. 240, [https://www.gutenberg.org/files/28072/28072-h/28072-h.htm (gutenberg.org)]

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,{{MathWorld|FlowerofLife|Flower of life}} and a further five on column opposite the building. They are drawn in red ochre and some are very faint and difficult to distinguish.{{cite web |last=Stewart |first=Malcolm |title=The "Flower of Life" and the Osirion – Facts are more interesting than Fantasy |work=Egyptian Tour (David Furlong) |date=2008 |url=https://www.davidfurlong.co.uk/egypttour_stewart.html |access-date=November 8, 2015}}

The patterns are graffiti, and not found in natively Egyptian ornaments. They are mostly dated to the early centuries of the Christian Era{{cite web |last=Furlong |first=David |title=The Osirion and the Flower of Life |url=https://www.davidfurlong.co.uk/egypttour_osirion.html |access-date=November 8, 2015}} Furlong states that these engravings can date no earlier than 535 BCE and probably date to the 2nd and 4th century CE. His research is based on photographic evidence of Greek text, yet to be fully deciphered. The text is seen alongside the designs and the position close to the top of columns, which are greater than 4 meters in height.

Furlong suggests the Osirion was half-filled with sand prior to the circles being drawn and therefore likely to have been well after the end of the Ptolemaic dynasty. 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.{{cite book |last=Murray |first=Margaret Alice |title=The Osireion at Abydos London |date=1904 |page=35 |url=https://etana.org/node/666 |access-date=November 4, 2015}}

Similar patterns were sometimes used in England as apotropaic marks to keep witches from entering buildings.{{cite web |last1=Kennedy |first1=Maev |title=Witches' marks: public asked to seek ancient scratchings in buildings |work=The Guardian |date=October 31, 2016 |url=https://www.theguardian.com/culture/2016/oct/31/witches-marks-historic-england-evil-spirits |access-date=October 31, 2016}} Consecration crosses indicating points in churches anointed with holy water during a church's dedication also take the form of overlapping circles.

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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.{{cite book |author=Broug, Eric |title=Islamic Geometric Patterns |publisher=Thames and Hudson |year=2008 |isbn=978-0-500-28721-7 |pages=22–23 and passim}}

Patterns of seven overlapping circles are found on Roman mosaics, for example at Herod's palace in the 1st century BC.{{cite journal

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|last2=Jin |first2=Bih-Yaw

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|journal=The Mathematical Intelligencer

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|title=Fullerenes, Polyhedra, and Chinese Guardian Lions

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The design is found on one of the silver plaques of the Late Roman hoard of Kaiseraugst (discovered 1961).Hans Ulrich Instinsky: Der spätrömische Silberschatzfund von Kaiseraugst. Mainz 1971, plaque 85.

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).[http://www.pleiade.org/col_geal/aux_armes-symbolism-notes.html#note_5 The Cosmati Pavements in Westminster Abbey.] Retrieved 14 September 2013.

Leonardo da Vinci explicitly discussed the mathematical proportions of the design.Codex Atlanticus, foll. 307r–309v, 459r (dated between 1478 and 1519).

File:Flower of Life pendant (2).jpg
Pendant, silver, ⌀ 27 mm
(commercial product, 2013)]]

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).{{cite book |title=How to Create Sacred Geometry Mandalas |last=Bartfeld |first=Martha |location=Santa Fe, NM |publisher=Mandalart Creations |date=2005 |isbn=9780966228526 |oclc=70293628 |page=35}}{{cite book |page=1079 |title=CRC Concise Encyclopedia of Mathematics, Second Edition |author=Weisstein, Eric W. |date=12 December 2002 |publisher=CRC Press |publication-date=2002 |isbn=1420035223 |url=https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA1079}}

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,E.g. {{cite news |title=Zaeem Jamal Launches New Collection on Board a Private Yacht in Dubai Marina |last=Zaman |first=Sana |publisher=Haute Living |date=May 14, 2013 |url=https://hauteliving.com/2013/05/zaeem-jamal-launches-new-collection-on-board-a-private-yacht-in-dubai-marina/358184/ |access-date=November 9, 2015}} 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,{{cite news |title=Bring Me The Horizon: This album needs to be the one that lasts forever |last=Cooper |first=Ed |work=The Independent |date=February 25, 2013 |url=http://blogs.independent.co.uk/2013/02/22/bring-me-the-horizon-this-album-needs-to-be-the-one-that-lasts-forever/ |access-date=November 8, 2015 |url-status=dead |archive-url=https://web.archive.org/web/20151023041236/http://blogs.independent.co.uk/2013/02/22/bring-me-the-horizon-this-album-needs-to-be-the-one-that-lasts-forever/ |archive-date=October 23, 2015}} 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.{{cite news |title=Coldplay new album: Beyonce and Noel Gallagher to feature on A Head Full of Dreams |last=Denham |first=Jess |work=The Independent |date=November 6, 2015 |url=https://www.independent.co.uk/arts-entertainment/music/news/coldplay-new-album-beyonce-and-noel-gallagher-to-feature-on-a-head-full-of-dreams-a6723696.html |url-access=subscription |access-date=November 8, 2015 |url-status=live |archive-url=https://ghostarchive.org/archive/20220524/https://www.independent.co.uk/arts-entertainment/music/news/coldplay-new-album-beyonce-and-noel-gallagher-to-feature-on-a-head-full-of-dreams-a6723696.html |archive-date=2022-05-24}}

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.{{cite web |publisher=Lega Nord |title=Il significato del simbolo del Sole delle Alpi |language=it |url=http://www.prov-varese.leganord.org/doc/solealpi.htm |access-date=December 1, 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140112135050/http://www.prov-varese.leganord.org/doc/solealpi.htm |archive-date=January 12, 2014}} It resembles a pattern often found in that area on buildings.{{cite web |author=Ivano Dorboló |title=The church of S.Egidio and the Sun of the Alps symbol |date=June 6, 2010 |website=Storia di Confine – Valli di Natisone |url=http://www.matajur.it/Sito%20English/21-S.Egidio/S.Egidio.htm |access-date=November 9, 2015}}

A seven-circle "Flower of Life" is also used in the coat of arms of Asgardia the space nation.

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;1, 7, and 19-circle hexagonal variant

In the examples below the pattern has a hexagonal outline, and is further circumscribed.

{{gallery |mode=nolines |width=120 |height=120

|File:Leonardo da Vinci - Codex Atlanticus folio 309v detail1.png |Leonardo da Vinci's drawing from Codex Atlanticus, between 1478 and 1519.

|File:Leonardo da Vinci – Codex Atlanticus folio 307v.jpg|Drawing by Leonardo da Vinci (Codex Atlanticus, fol. 307v)

|File:RozetaSymbol.svg|1-circle with completed arcs|class3=skin-invert-image

|File:Mosaic floor from a bathhouse in Herod's palace - Google Art Project.jpg|7-circle: Mosaic floor from a bathhouse in Herod's palace, 1st century BCE

|File:Flower-of-Life-small.svg|19-circle symbol with completed arcs and bounded by a larger circle|class5=skin-invert-image

|File:Temple-of-Osiris Flower-of-Life 02.jpg|19-circle: Two symbols drawn in red ochre Temple of Osiris at Abydos, Egypt

|File:Moni Preveli Church Apsis Window South.JPG|19-circle: A window at the southern apsis of the church of Preveli Monastery (Moni Preveli), Crete.

|File:Flower of life ephesos square.jpg|19-circle: From Ephesus, Turkey

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;Similar patterns

In the examples below, the pattern does not have a hexagonal outline:

File:Cup Idalion Louvre N3454.jpg|Cup with mythological scenes, a sphinx frieze, and the representation of a king vanquishing his enemies. Cypro-Archaic I (8th–7th centuries BC). From Idalion, Cyprus.

China-beijing-forbidden-city-P1000157-detail.jpg|Ball held by the male Imperial Guardian Lion at the Gate of Supreme Harmony, Forbidden City, Beijing, China, showing the geometrical pattern on its surface.

File:Floor decoration from the palace of King Ashurbanipal.jpg|Floor decoration from the northern Iraq palace of King Ashurbanipal, visible in the Museum of Louvre, dated 645BC.

File:Coffee cup with overlapping circles grid ornament (Germany, 2022).jpg|Coffee cup (Germany, 2022)

File:Sun of the Alps.svg|"Sun of the Alps" emblem used by the Lega Nord

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.

File:Flower-construction.svg

The pattern can be extended outward 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.

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.Sphere Packings, Lattices and Groups, John Conway, Neil J. A. Sloane, Chapter 2, section 1.1, Covering space with overlapping circle. pp. 31-32. Figure 2.1 Covering the plane with circles (b) The more efficient or thinner covering in a hexagonal lattice. [https://books.google.com/books?id=upYwZ6cQumoC&dq=overlapping+circles&pg=PA31]

Two offset copies of this circle pattern makes a rhombic tiling pattern, while three copies make the original triangular pattern.

{{gallery |width=120 |height=120 |noborder=yes

|File:Flower of life 0577-19-circle.svg|19 circle example|class1=skin-invert-image

|File:Flower of life 0577-2compound.png|Two offset copies of the minimal covering circle pattern (left) make a rhombic tiling pattern, like this red, blue version.

|File:Circlemesh hexagonal tiling compound.png|Three offset copies of the minimal covering circle pattern (left most image) make the 7-circle pattern, like this red, green, blue version.

|File:Jar Met 56.185.15.jpg|Example on Ayyubid Raqqa ware stoneware glazed jar. Syria, 12th/13th century

|File:Pavement Hospitalia Villa Hadriana n8.jpg|Black and white mosaic pavement at Hadrian's Villa, Tivoli, 2nd century AD

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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.

Some spherical polyhedra with edges along great circles can be stereographically projected onto the plane as overlapping circles.

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.Mesopotamian Mathematics 2100-1600 BC: Technical Constants in Bureaucracy and Education (Oxford Editions of Cuneiform Texts), Eleanor Robson, Clarendon Press, 1999, {{ISBN|978-0198152460}} [https://books.google.com/books?id=5gdVQRDYjYsC&dq=Mesopotamian+Mathematics+2100-1600+BC+Apsamikkum&pg=PA53] at books.google.com

File:Square circle grid spheres.png|The square grid can be seen in a face-centered cubic lattice, with 12 spheres in contact around every sphere

File:Square five overlapping circle grid.svg|The related five overlapping circles grid is constructed by from two sets of overlapping circles half-offset.[http://www.zso7.home.pl/art/?now=103 Creating Square Grids from Circles]

File:Batik pattern - kawung.jpg|Kawung or "Coffee Bean" Batik sarong (detail), Java, Indonesia

• Knot theory
• Uniform tiling symmetry mutations – pattern mutations in 3D space

{{Reflist|28em}}

{{Commons category|Flower of Life}}

{{Commons category|Metatron's Cube}}

• {{MathWorld |urlname=Circle-CircleIntersection |title=Circle-circle intersection}}
• [https://www.themystica.com/flower-of-life/ The Flower of Life Secrets: Meaning, History, and how to draw it] on The Mystica

Category:Circles

Category:Patterns

Category:Religious symbols

Category:Sacred geometry