Overlapping circles grid: Difference between revisions

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.

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.<ref name="murray">{{cite book | first=Margaret Alice |last=Murray | title=The Osireion at Abydos London | date=1904 | page=35 | url=http://www.etana.org/node/666 | access-date=November 4, 2015}}</ref>

Similar patterns were sometimes used in England as [[apotropaic mark]]s to keep witches from entering buildings.<ref name=Kennedy2016>{{cite web |last1=Kennedy |first1=Maev |title=Witches' marks: public asked to seek ancient scratchings in buildings |url=https://www.theguardian.com/culture/2016/oct/31/witches-marks-historic-england-evil-spirits |publisherwork=The Guardian|access-date=October 31, 2016 |date=October 31, 2016}}</ref> [[Consecration crosses]] indicating points in churches anointed with holy water during a churches dedication also take the form of overlapping circles.

Patterns of seven overlapping circles are found on a Cypro-Archaic I cup of the 8th-7th century BC in Cyprus{{cn|date=March 2018}} and Roman mosaics, for example at [[Herod's Palace (Herodium)|Herod's palace]] in the 1st century BC.<ref>{{cite journal

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).<ref>[http://www.pleiade.org/col_geal/aux_armes-symbolism-notes.html#note_5 ''The Cosmati Pavements in Westminster Abbey.''] Abgerufen amRetrieved 14.&nbsp;September 2013.</ref>

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).<ref name="HTCSGM">{{cite book | title=How to Create Sacred Geometry Mandalas | first=Martha | last=Bartfeld | location=Santa Fe, NM | publisher=Mandalart Creations | date=2005 | isbn=9780966228526 | oclc=70293628 | page=35}}</ref><ref>{{cite book |title= A New Kind of Science |author= Wolfram, Stephen |publisher= [[Wolfram Research|Wolfram Media, Inc.]] |publication-date= May 14, 2002 |isbn= 1-57955-008-8 |pages= [https://archive.org/details/newkindofscience00wolf/page/43 43 and 873–874] |url= https://archive.org/details/newkindofscience00wolf/page/43 |url-access= registration }}</ref><ref>{{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}}</ref>

Besides an occasional use in fashion,<ref>E.g. {{cite news |url=http://hauteliving.com/2013/05/zaeem-jamal-launches-new-collection-on-board-a-private-yacht-in-dubai-marina/358184/ |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 |access-date=November 9, 2015}}</ref> it is also used in the decorative arts. For example, the album ''[[Sempiternal (album)|Sempiternal]]'' (2013) by [[Bring Me the Horizon]] uses the [[#Progressions|61 overlapping circles grid]] as the main feature of its album cover,<ref>{{cite news |url=http://blogs.independent.co.uk/2013/02/22/bring-me-the-horizon-this-album-needs-to-be-the-one-that-lasts-forever/ |title=Bring Me The Horizon: This album needs to be the one that lasts forever |last=Cooper |first=Ed |publisherwork=[[The Independent]] |date=February 25, 2013 |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 }}</ref> 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.<ref>{{cite news |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 |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 |url-access=subscription |url-status=live |title=Coldplay new album: Beyonce and Noel Gallagher to feature on A Head Full of Dreams |last=Denham |first=Jess |publisherwork=[[The Independent]] |date=November 6, 2015 |access-date=November 8, 2015}}</ref>

The "Sun of the Alps" ([[Italian language|Italian]] ''Sole delle Alpi'') symbol has been used as the emblem of [[Padanian nationalism]] in northern Italy since the 1990s.<ref>{{cite web |publisher=Lega Nord |title=Il significato del simbolo del Sole delle Alpi |url=http://www.prov-varese.leganord.org/doc/solealpi.htm |access-date=December 1, 2014 |language=it |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 }}</ref> It resembles a pattern often found in that area on buildings.<ref>{{cite web|author=Ivano Dorboló|title=The church of S.Egidio and the Sun of the Alps symbol|date=June 6, 2010|url=http://www.matajur.it/Sito%20English/21-S.Egidio/S.Egidio.htm|website=Storia di Confine – Valli di Natisone|access-date=November 9, 2015}}</ref>

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."

!1-circle<br />&nbsp;

!7-circle<br />(8-1)

!19-circle<br />(27-8)

!37-circle<br />(64-27)

!61-circle<br />(125-64)

!91-circle<br />(216-125)

!127-circle...<br />(343-216)

!1-sphere<br />(1×1×1)

!8-sphere<br />(2×2×2)

!27-sphere<br />(3×3×3)

!64-sphere<br />(4×4×4)

!125-sphere<br />(5×5×5)

!216-sphere<br />(6×6×6)

!343-sphere<br />(7×7×7)

|[[File:Flower37-Circle ofSymbol life-4levelwith Completed Arcs and Bounding Circle.png|100px]]

|[[File:Flower61-Circle ofSymbol life-5levelwith Completed Arcs and Bounding Circle.png|100px|]]

|[[File:Flower91-Circle ofSymbol life-6levelwith Completed Arcs and Bounding Circle.png|100px|]]

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 [[Disk covering problem|arrangement of circles can cover the plane]] more efficiently than this hexagonal lattice arrangement.<ref>''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&pg=PA31&dq=overlapping+circles&hlpg=en&sa=X&ved=0ahUKEwjP8N_0_OLKAhWK4iYKHeK4DeoQ6AEIUjAH#v=onepage&q=overlapping%20circles&f=falsePA31]</ref>

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

The center [[lens (geometry)|lens]] of the 2-circle figure is called a [[Vesicavesica 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 (mathematics)#Plane geometry|lune]].

The 3-circle figure resembles a depiction of [[Borromean rings#History and symbolism|borromean rings]] and is used in [[Intersection (set theory)|3-set]] theory [[Venn diagram]]s. Its interior makes a [[unicursal curve|unicursal]] path called a [[triquetra]]. The center of the 3-circle figure is called a [[reuleaux triangle]].

|[[File:Tetratetrahedron stereographic projection.png|120px]]<BRbr />[[Octahedron#Spherical_tiling|octahedron]]

|[[File:Cuboctahedron stereographic projection square.png|120px]]<BRbr />[[Cuboctahedron#Spherical_tiling|Cuboctahedron]]

|[[File:Icosidodecahedron stereographic projection pentagon.png|120px]]<BRbr />[[Icosidodecahedron#Spherical_tiling|Icosidodecahedron]]

|[[File:Square_overlapping_circle_grid4.svg|160px]]<BRbr />The circle radius is the reciprocal of the square root of 2 times their separation (distance between their centers).

|[[File:Double Wedding Ring Quilt.jpg|160px]]<br />A [[Quilt#Block designs|quilt design]] called a double wedding ring pattern.

|[[File:Square_overlapping_circle_rotated_grid4.svg|160px]]<BRbr />It can be seen as two half-offset square grids of tangent circles.

|[[Image:Wallpaper group-p4m-5.jpg|160px]]<br />[[Egypt]]ian design, from Owen Jones's ''[[The Grammar of Ornament]]'' (1856)

It is called an '''Apsamikkum''' from ancient [[Mesopotamian]] mathematics.<ref>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&pg=PA53&dq=Mesopotamian+Mathematics+2100-1600+BC+Apsamikkum&hlpg=en&sa=X&ved=0ahUKEwjg4K2gtu3JAhWM5CYKHS_5BhAQ6AEIJjAA#v=onepage&q=Mesopotamian%20Mathematics%202100-1600%20BC%20Apsamikkum&f=falsePA53] at books.google.com</ref>

{{Commons category|SunMetatron's of the AlpsCube}}