Metatron’s Cube (From Wikipedia)

The Fruit of Life (a component of the Flower of Life) has thirteen circles. If each circle's center is considered a "node", and each node is connected to each other node with a single line, a total of seventy-eight lines are created. Within this cube, many other shapes can be found, including two-dimensionally flattened versions of the five Platonic solids. The true Metatron's Cube will include all five Platonic solids in such a way that the solids, existing in volumetric 3D space, have had their z-coordinates set to zero but their x- and y-coordinates retained, such that they are orthogonally flattened.

In early kabbalist scriptures, Metatron supposedly forms the cube from his soul. This cube can later be seen in Christian art, where it appears on his chest or floating behind him. Metatron's Cube is also considered a holy glyph, and was often drawn around an object or person to ward off demons and satanic powers. This idea is also present in alchemy, in which the cube was favoured as a containment circle or creation circle.

The simplest means of constructing Metatron's Cube is to begin with a cube flattened along a space diagonal, such that it becomes a 2D figure, equivalent to a regular hexagon divided via its own diagonals into six equilateral triangles. The vertices of this 2D figure are then connected with additional lines. Several steps later, the full Metatron's Cube figure is formed.[23] This method requires dividing vertices according to the golden ratio. There is also a method of construction from the Flower of Life.[24] The cube resembles the fourth dimensional analog of the cube, or the Tesseract.

(From http://www.sangraal.com/library/dedicated.htm) Metatron's Cube ultimately represents Magic, Alchemy and containment. Among its attributes are the three directions of up and down, side to side, and front to back, with the concept of a cube within a cube.

At the center of the world, which is also man, there is a synthesis, an equilibrium of the six directions that is of the three spatial sections and a neutral center.

It is the ultimate Magical container. It is The Box and the Incubator in which breeds the of the seed of First Light, the concept by which we believe in death and resurrection.

Metatron and Eve's Grid A question and answer dialogue describing aspects of the Flower of Life, Metatron, and more detail about the Tesseract.

Sacred Geometry and the Box. The Tesseract Drawing comes from a book published in 1928 entitled "Theosophy and the Fourth Dimension" by Alexander Horne.

Incubation is the luxury of knowing that we will repeat our lives, lessons, and achievements whenever we pass on to the other world. It is a luxury that is cherish. It was afforded us by the belief in death and resurrection.

The Ancient Egyptian belief that we have the ability to become a star in the Universe is a bit different. Expanding on the belief that we can Ascend to another consciousness, Alexander Horne offered the drawing to the left in 1928, suggesting that we can become fourth dimensional. His instruction includes a process by which we are able to view fourth dimensional consciousness by going into and perceiving a section of the cube that we otherwise would miss. MORE

(from http://www.ccds.charlotte.nc.us/mcgrail/stu0102/HP16/index.html)

One of the key concepts of sacred geometry is the flower of life. Within this figure, through various expansions, contractions, and connections, one can find a blueprint for concepts espoused by Platonic philosophy and even ancient Judaism to be the basis for the universe and life itself (Frissell, Something 197). When considering the flower of life shape, it is important to realize that this is actually a two-dimensional representation of a three-dimensional form. Therefore, these two figures actually represent the same shape:

(Alloca 1) In the figure on the left, only nineteen of the spheres are visible, but by representing the shape two-dimensionally, one can see that in actuality, there are a total of twenty-seven spheres. Through expanding the flower of life shape to contain 125 spheres, one creates a new shape, from which the “fruit of life” can be extracted (from this vantage point, one sees only thirteen spheres in the fruit of life, but in actuality, there are seventeen, with two both in front and behind of the middle sphere). By connecting the middle point of each sphere to the middle point of all others, Metatron’s Cube can be found (Allocca 1).

(Melchizidek 7)

(Three-Dimensional View of Metatron's Cube) Although the shape of Metatron’s Cube has it’s own significance in relation to sacred geometry and other forms of philosophy, it is more widely recognized as the basis for deriving the platonic solids, as seen here:

(Click to Enlarge) (Melchizidek 8) The platonic solids are a group of five three-dimensional solid shapes, each containing all congruent angles and sides. Also, if circumscribed with a sphere, all vertexes would touch the edge of the sphere. It was Euclid who would later prove in his book, entitled The Elements, that these are the only five shapes which actually fit this criteria (Weisstein 1).

(Frissell, Something 200) In his book, Timaeus, written in approximately 350 BC, Plato first described these solids, linking them to different elements of reality. The tetrahedron, containing four sides, and actually found twice within Metatron’s Cube (the star tetrahedron is a combination of two tetrahedrons), is used to represent fire. The cube, containing six sides, and also found twice within Metatron’s Cube, represents the earth. The octahedron, containing eight sides, represents the air. The icosahedron, containing twenty sides, represents the water. Finally, the dodecahedron, containing twelve sides, is used to represent the cosmos (Weisstein 1). The concept of the entire universe being made up of four basic elements (earth, fire, water, and air) dates back over a hundred years earlier than Plato’s Timaeus, with the work of the Greek philosopher Empedocles, who lived from approximately 493 to 433 BC. He theorized that all matter is made merely of varying combinations and proportions of these elements. It is possible, however, that this concept also existed long before Empedocles ever wrote about it. He was a disciple of Pythagoras, who had also been greatly influenced by the ancient Egyptians, and Empedocles could have, in theory, gotten his information handed down from either of these sources.

However, it was Plato, who was born six years after Empedocles’ death, who would apply a logical formula to assigning the elements to the Platonic Solids: “Let us assign the cube to earth, for it is the most immobile of the four bodies and most retentive of shape…the least mobile of the remaining figures (icosahedron) to water, the most mobile (tetrahedron) to fire, the intermediate (octahedron) to air” However, this still leaves the dodecahedron, which, according to Plato, “the god used for embroidering the constellations on the whole heaven" (Calter 2-6). In this way, the Platonic Solids can be used to represent the entire universe. Other shapes within sacred geometry continue to expand upon these principles and provide valuable insight into the nature of all things, from mankind to the cosmos. The most difficult part of this project was narrowing such a broad original topic like sacred geometry into a more specific subject such as Metatron’s Cube and the Platonic Solids. However, I am glad that I was able to do a large amount of general research before specifying because this way of doing things allowed me to increase my own knowledge on the subject of sacred geometry. I would definitely like to continue research on the subject in many other areas. Specifically, in my research I found some information linking parts of sacred geometry to ancient Jewish beliefs, and I would like to find out more about this connection. Although it may have seemed a daunting task at first, to gain a complete understanding of sacred geometry, the realization that I still have so much to learn is comforting in that this interesting topic will be able to stay with me for a long time.

by IDTimeTravellerfrom 3rdDimensionOnLine Website

Metatron's Cube The image you see above is the correct version of "Metatron’s Cube", or the fruit of life with the 5 platonicsolids. I made it with a 3D program and what you see are the actual edges of the platonic solids.

The dodecahedron and icosahedron actually have their"inner" corner points touching the star tetrahedron thatfits in a circle that is = 1/Ø x the circle connecting thecorners of the outer cube.

Where Ø ("phi") = 1.618 033 988 749 594 ...

The way to figure out drawing this is by beginning withthe normal flower of life "grid" (of 5 non-intersectinginner circles, or in this case: 7 intersecting full circles -left image):

Then draw a circle that is equal to 1/Ø x the outer circle.

For this, put the point of a compass on the middle of thevertical radius (in the centre of the second "flower" fromthe top along the vertical axis) and draw a circle fromwhere the horizontal axis (through the middle of thecircle) cuts the outer circle (see below);The circle you can draw when you put one point of thecompass in the centre of the grid, and draw a circle fromwhere the other circle cuts the vertical axis (through themiddle of the circle), is a circle Ø times smaller than the

outer circle:

After this, draw the appropriate startetrahedron(s) and you have found all the points necessary to draw theother platonic solids.

Here are all the platonic solids the way they fit in Metatron’s cube:

Tetrahedron Star tetrahedron

Cube Octahedron

Dodecahedron Icosahedron

All platonic solidsNot all solids can be (fully) seen, as some lines are hidden behind others.

Here are all the solids of Metatron’s cube. This is what yousee in Metatron’s cube from the top, but here it is seen from

the (right) side.

Classic representation of dodecahdron in tetrahedron

This is a correct version of a dodecahdron in astartetrahedron. It’s a classic representation that is

commonly done in a way that is incorrect.

The corners of the dodecahedron are found by drawing linesfrom the corners of the big startetrahedron to those of a

startetrahedron that is Ø times smaller. .

from Sangraal Website

The TesseractDrawing comes from a book

published in 1928 titled"Theosophy and the Fourth

Dimension" by Alexander Horne.

Drawing by Drunvalo Melchizedek

Perhaps you can see the rectangle with the triad on top and bottom. It is part of the cube nested within acube. Metatron’s Cube is the incubator of 3D. It represents a box within a box or a cube nested within acube. The Tesseract represents a process in which one might peer into the fourthdimension. The drawings of the Dodecahedron - Eve’s Grid and the Tesseractare basically the same geometry except that the Tesseract is tilted so youmight better define the six sections.

Outlined in red is one of the six sections. The idea is to go into the outlinedarea and see it as a cube. Once you have visualized being inside the cube, itrepresents the cube of 4D. Click on the Tesseract to get a larger image forvisualization.

It often takes a student several tries to actually get inside the 4D cube. Thelarger image works much better than the small one. If you have difficultyafter a couple of days, perhaps stop for a few days and try again. It took meseveral attempts to see it the first time.

Seeing the cube within a cube dimensionally leads one to consider the natureof the Holographic model. It has the nature of being exactly the samestructure if you were to tear off a piece from the edge. The integrity of thegeometry would be maintained infinitely. The image below could be that of the corner you just ripped off theedge.

Above, we see that the Tree of Life iscontained in Metatron’s Cube/Grid. The

view above comes from the Seed ofLife, therefore metaphorically, the Treeis contained within the seed, and is afractal property from the beginning.

Drawing by Drunvalo Melchizedek

When considering the Flower of life Geometries, keep in mind that curved lines represent femalecharacteristics and straight lines represent male characteristics. The drawing depicts the holographic natureof the male and female character of the grid work and a deeper, more "within" nature, of Metatron’s Cube.

Merkaba Field

Return

Example overlapping round circle figures

square circle grid

1+

4

9

Centered square lattice forms

5

13

triangular circle grid

1+

3

4

7

19

Overlapping circles gridAn overlapping circles grid is a geometric pattern ofrepeating, overlapping circles of equal radii in two-dimensional space. Commonly, designs are based on circlescentered on triangles (with the simple, two circle formnamed vesica piscis) or on the square lattice pattern ofpoints.

Patterns of seven overlapping circles appear in historicalartefacts from the 7th century BC onwards; they become afrequently used ornament in the Roman Empire period, andsurvive into medieval artistic traditions both in Islamic art(girih decorations) and in Gothic art. The name "Flower ofLife" is given to the overlapping circles pattern in New Agepublications.

Of special interest is the six petal rosette derived from the"seven overlapping circles" pattern, also known as "Sun ofthe Alps" from its frequent use in alpine folk art in the 17thand 18th century.

Triangular grid of overlapping circlesCultural significance

Near EastEuropeModern usageGallery

ConstructionProgressions

Other variationsRelated concepts

Square grid of overlapping circlesSee alsoReferencesExternal links

The triangular lattice form, with circle radii equal to theirseparation is called a seven overlapping circles grid.[1] Itcontains 6 circles intersecting at a point, with a 7th circlecentered on that intersection.

Overlapping circles with similar geometrical constructionshave been used infrequently in various of the decorative arts since ancient times. The pattern has found a wide range of usagein popular culture, in fashion, jewelry, tattoos and decorative products.

Contents

Triangular grid of overlapping circles

Cultural significance

This pattern can beextended indefinitely, seenhere with hexagonal ringsof 1, 7, 19, 37, 61, 91circles...

The oldest known occurrence of the "overlapping circles" pattern is dated to the 7th or 6th centuryBCE, found on the threshold of the palace of Assyrian king Aššur-bāni-apli in Dur Šarrukin (nowin the Louvre).[2]

The design becomes more widespread in the early centuries of the Common Era. One earlyexample are five patterns of 19 overlapping circles drawn on the granite columns at the Temple ofOsiris in Abydos, Egypt,[3] and a further five on column opposite the building. They are drawn inred ochre and some are very faint and difficult to distinguish.[4] The patterns are graffiti, and notfound in natively Egyptian ornaments. They are mostly dated to the early centuries of theChristian Era[5] although medieval or even modern (early 20th century) origin cannot be ruled outwith certainty, as the drawings are not mentioned in the extensive listings of graffiti at the templecompiled by Margaret Murray in 1904.[6]

Similar patterns were sometimes used in England as apotropaic marks to keep witches fromentering buildings.[7] Consecration crosses indicating points in churches anointed with holy water during a churchesdedication also take the form of overlapping circles.

In Islamic art, the pattern is one of several arrangements of circles (others beingused for fourfold or fivefold designs) used to construct grids for Islamicgeometric patterns. It is used to design patterns with 6- and 12-pointed stars aswell as hexagons in the style called girih. The resulting patterns howevercharacteristically conceal the construction grid, presenting instead a design ofinterlaced strapwork.[8]

Patterns of seven overlapping circles are found on a Cypro-Archaic I cup of the8th-7th century BC in Cyprus and Roman mosaics, for example at Herod'spalace in the 1st century BC. They are also found in the Hindu temple atPrambanan in Java. The design is found on one of the silver plaques of the Late Roman hoard of Kaiseraugst (discovered1961).[9] 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).[10] Leonardo da Vinciexplicitly discussed the mathematical proportions of the design.[11]

The name "Flower of Life" is modern, associated with the New Age movement, and commonlyattributed specifically to Drunvalo Melchizedek in his book The Ancient Secret of the Flower ofLife (1999).[12][13][14]

The pattern and modern name have propagated into wide range of usage in popular culture, infashion, jewelry, tattoos and decorative products. The pattern in quilting has been calleddiamond wedding ring or triangle wedding ring to contrast it from the square pattern. Besidesan occasional use in fashion,[15] it is also used in the decorative arts. For example, the albumSempiternal (2013) by Bring Me the Horizon uses the 61 overlapping circles grid as the mainfeature of its album cover,[16] whereas the album A Head Full of Dreams (2015) by Coldplayfeatures the 19 overlapping circles grid as the central part of its album cover. Teaser postersillustrating the cover art to A Head Full of Dreams were widely displayed on the LondonUnderground 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 Padaniannationalism in northern Italy since the 1990s.[18] It resembles a pattern often found in that areaon buildings.[19]

Near East

A girih pattern that canbe drawn with compass

and straight edgeWindow cage atTopkapı Palace,

using pattern

Europe

Modern usage

19-circle with arcs Pendant, silver, ⌀ 27 mm (commercial product,2013)

Gallery

1, 7, and 19-circle hexagonal variant

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

1-circle withcompleted arcs

7-circle: Mosaicfloor from abathhouse inHerod's palace,1st century BCE

19-circle symbolwith completedarcs andbounded by alarger circle

19-circle: Twosymbols drawnin red ochreTemple of Osirisat Abydos,Egypt

19-circle: Awindow at thesouthern apsisof the church ofPreveliMonastery(Moni Preveli),Crete.

19-circle: FromEphesus,Turkey

Similar patterns

In the examples below the pattern does not have a hexagonal outline.

Drawing byLeonardo daVinci (CodexAtlanticus, fol.307v)

Cup withmythologicalscenes, asphinx friezeand therepresentationof a kingvanquishing hisenemies.Cypro-Archaic I(8th–7thcenturies BC).From Idalion,Cyprus.

Ball held by themale ImperialGuardian Lionat the Gate ofSupremeHarmony,Forbidden City,Beijing, China,showing thegeometricalpattern on itssurface.

Floor decorationfrom thenorthern Iraqpalace of KingAshurbanipal,visible in theMuseum ofLouvre, dated645BC.

"Sun of theAlps" emblemused by theLega Nord

Martha Bartfeld, author of geometric art tutorial books, described her independent discovery of the design in 1968. Heroriginal definition said, "This design consists of circles having a 1-[inch] radius, with each point of intersection serving as anew center. The design can be expanded ad infinitum depending upon the number of times the odd-numbered points aremarked off."

The pattern figure can be drawn by pen and compass, by creating multiple series of interlinking circles of the same diametertouching the previous circle's center. The second circle is centered at any point on the first circle. All following circles arecentered on the intersection of two other circles.

Construction

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. Thethird 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 numberof 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 thediagonal axis. There are more spheres than circles because some are overlapping in 2 dimensions.

Rosette figures including partial circles

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)

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)

+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 radii. Richard Kershnershowed in 1939 that no arrangement of circles can cover the plane more efficiently than this hexagonal latticearrangement.[20]

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

Progressions

Other variations

Square lattice form

19 circle example

Two offset copies ofthe minimal coveringcircle pattern (left)make a rhombictiling pattern, likethis red, blueversion.

Three offset copiesof the minimalcovering circlepattern (left mostimage) make the 7-circle pattern, likethis red, green, blueversion.

Example on AyyubidRaqqa warestoneware glazedjar. Syria, 12th/13thcentury

Black and whitemosaic pavement atHadrian's Villa,Tivoli, 2nd centuryAD

The center lens of the 2-circle figure is called a Vesica piscis, from Euclid. Two circles are also called Villarceau circles as aplane 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 aunicursal 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 overlappingcircles.

Polyhedra in stereographic projection

octahedron

Cuboctahedron

Icosidodecahedron

The 7-circle pattern has also been called an Islamic seven-circles pattern for its use in Islamic art.

The square lattice form can be seen with circles that line uphorizontally and vertically, while intersecting on their diagonals. Thepattern appears slightly different when rotated on its diagonal, alsocalled a centered square lattice form because it can be seen as twosquare lattices with each centered on the gaps of the other.

It is called a Kawung motif in Indonesian batik, and is found on thewalls of the 8th century Hindu temple Prambanan in Java.

Related concepts

Square grid of overlapping circles

The circle radius is squareroot of 2 times theirseparation.

A quilt design called adouble wedding ringpattern.

Centered square lattice form

It can be seen as two half-offset square grids oftangent circles.

Egyptian design, fromOwen Jones's TheGrammar of Ornament(1856)

It is called an Apsamikkum from ancient Mesopotamianmathematics.[21]

The square grid canbe seen in a face-centered cubiclattice, with 12spheres in contactaround every sphere

The related fiveoverlapping circlesgrid is constructedby from two sets ofoverlapping circleshalf-offset.[22]

Kawung or "CoffeeBean" Batik sarong(detail), Java,Indonesia

Uniform tiling symmetry mutations - pattern mutations in 3D spaceKnot theory

1. Islamic Art and Geometric Design: Activities for Learning (http://www.metmuseum.org/learn/for-educators/publications-for-educators/islamic-art-and-geometric-design)

2. Louvre Inv.-Nr. AO 19915 (http://cartelfr.louvre.fr/cartelfr/visite?srv=car_not_frame&idNotice=24382). GeorgesPerrot, Charles Chipiez, A History of Art in Chaldæa and Assyria, vol. 1, London 1884, S. 240, (gutenberg.org)(http://www.gutenberg.org/files/28072/28072-h/28072-h.htm)

3. Weisstein, Eric W. "Flower of life" (http://mathworld.wolfram.com/FlowerofLife.html). MathWorld.4. Stewart, Malcolm (2008). "The "Flower of Life" and the Osirion – Facts are more interesting than Fantasy" (htt

p://www.davidfurlong.co.uk/egypttour_stewart.html). Egyptian Tour (David Furlong). Retrieved November 8,2015.

5. Furlong, David. "The Osirion and the Flower of Life" (http://www.davidfurlong.co.uk/egypttour_osirion.html).Retrieved November 8, 2015. Furlong states that these engravings can date no earlier than 535 BCE andprobably 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 thecircles being drawn and therefore likely to have been well after the end of the Ptolemaic dynasty.

6. Murray, Margaret Alice (1904). The Osireion at Abydos London (http://www.etana.org/node/666). p. 35.Retrieved November 4, 2015.

See also

References

Weisstein, Eric W. "Circle-circle intersection" (http://mathworld.wolfram.com/Circle-CircleIntersection.html).MathWorld.The flower of life (https://www.themystica.com/flower-of-life/) article from The Mystica

Retrieved from "https://en.wikipedia.org/w/index.php?title=Overlapping_circles_grid&oldid=932683388"

This page was last edited on 27 December 2019, at 16:10 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to theTerms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

7. Kennedy, Maev (October 31, 2016). "Witches' marks: public asked to seek ancient scratchings in buildings" (https://www.theguardian.com/culture/2016/oct/31/witches-marks-historic-england-evil-spirits). The Guardian.Retrieved October 31, 2016.

8. Broug, Eric (2008). Islamic Geometric Patterns. Thames and Hudson. pp. 22–23 and passim. ISBN 978-0-500-28721-7.

9. Hans Ulrich Instinsky: Der spätrömische Silberschatzfund von Kaiseraugst. Mainz 1971, plaque 85.10. The Cosmati Pavements in Westminster Abbey. (http://www.pleiade.org/col_geal/aux_armes-symbolism-notes.

html#note_5) Abgerufen am 14. September 2013.11. Codex Atlanticus, foll. 307r–309v, 459r (dated between 1478 and 1519).12. Bartfeld, Martha (2005). How to Create Sacred Geometry Mandalas. Santa Fe, NM: Mandalart Creations.

p. 35. ISBN 9780966228526. OCLC 70293628 (https://www.worldcat.org/oclc/70293628).13. Wolfram, Stephen (May 14, 2002). A New Kind of Science (https://archive.org/details/newkindofscience00wolf/

page/43). Wolfram Media, Inc. pp. 43 and 873–874 (https://archive.org/details/newkindofscience00wolf/page/43). ISBN 1-57955-008-8.

14. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition (https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA1079). CRC Press. p. 1079. ISBN 1420035223.

15. E.g. Zaman, Sana (May 14, 2013). "Zaeem Jamal Launches New Collection on Board a Private Yacht in DubaiMarina" (http://hauteliving.com/2013/05/zaeem-jamal-launches-new-collection-on-board-a-private-yacht-in-dubai-marina/358184/). Haute Living. Retrieved November 9, 2015.

16. Cooper, Ed (February 25, 2013). "Bring Me The Horizon: This album needs to be the one that lasts forever" (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/). The Independent. Archived from the original (http://blogs.independent.co.uk/2013/02/22/bring-me-the-horizon-this-album-needs-to-be-the-one-that-lasts-forever/) onOctober 23, 2015. Retrieved November 8, 2015.

17. Denham, Jess (November 6, 2015). "Coldplay new album: Beyonce and Noel Gallagher to feature on A HeadFull of Dreams" (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). The Independent. RetrievedNovember 8, 2015.

18. "Il significato del simbolo del Sole delle Alpi" (https://web.archive.org/web/20140112135050/http://www.prov-varese.leganord.org/doc/solealpi.htm) (in Italian). Lega Nord. Archived from the original (http://www.prov-varese.leganord.org/doc/solealpi.htm) on January 12, 2014. Retrieved December 1, 2014.

19. Ivano Dorboló (June 6, 2010). "The church of S.Egidio and the Sun of the Alps symbol" (http://www.matajur.it/Sito%20English/21-S.Egidio/S.Egidio.htm). Storia di Confine – Valli di Natisone. Retrieved November 9, 2015.

20. Sphere Packings, Lattices and Groups, John Conway, Neil J. A. Sloane, Chapter 2, section 1.1, Coveringspace with overlapping circle. pp. 31-32. Figure 2.1 Covering the plane with circles (b) The more efficient orthinner covering in a hexagonal lattice. [1] (https://books.google.com/books?id=upYwZ6cQumoC&pg=PA31&dq=overlapping+circles&hl=en&sa=X&ved=0ahUKEwjP8N_0_OLKAhWK4iYKHeK4DeoQ6AEIUjAH#v=onepage&q=overlapping%20circles&f=false)

21. Mesopotamian Mathematics 2100-1600 BC: Technical Constants in Bureaucracy and Education (OxfordEditions of Cuneiform Texts), Eleanor Robson, Clarendon Press, 1999, ISBN 978-0198152460 [2] (https://books.google.com/books?id=5gdVQRDYjYsC&pg=PA53&dq=Mesopotamian+Mathematics+2100-1600+BC+Apsamikkum&hl=en&sa=X&ved=0ahUKEwjg4K2gtu3JAhWM5CYKHS_5BhAQ6AEIJjAA#v=onepage&q=Mesopotamian%20Mathematics%202100-1600%20BC%20Apsamikkum&f=false) at books.google.com

22. Creating Square Grids from Circles (http://www.zso7.home.pl/art/?now=103)

External links