Coordinates	35°27′″N139°38′″N
name	Regular hexagon
type	''general type of this shape''
edges	6
schläfli	{6} t{3}
coxeter
symmetry	Dihedral (D6)
area	$A\; \backslash frac\{3\; \backslash sqrt\{3\}\}$

{2}t^2
$\backslash simeq\; 2.598076211\; t^2.$ (with ''t'' = edge length) | angle = 120° | dual = | properties = convex, cyclic, equilateral, isogonal, isotoxal }}

In geometry, a hexagon (from Greek ἕξ ''hex'', 'six') is a polygon with six edges and six vertices. A regular hexagon has Schläfli symbol {6}. The total of the internal angles of any hexagon is 720 degrees.

Regular hexagon

A regular hexagon has all sides of the same length, and all internal angles are 120°. A regular hexagon has 6 rotational symmetries ''(rotational symmetry of order six)'' and 6 reflection symmetries ''(six lines of symmetry)'', making up the dihedral group D₆. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.

Like squares and equilateral triangles, regular hexagons fit together without any gaps to ''tile the plane'' (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.

The area of a regular hexagon of side length $t\backslash ,\backslash !$ is given by $A\; =\; \backslash frac\{3\; \backslash sqrt\{3\}\}\{2\}t^2\; \backslash simeq\; 2.598076211\; t^2.$

An alternative formula for area is $\backslash \; A\; =\; 1.5dt,$ where the length ''d'' is the distance between the parallel sides, or the height of the hexagon when it sits on one side as base, or the diameter of the inscribed circle.

The area can also be found by the formulas $A=ap/2$ and $A\; =\; \{2\}a^2\backslash sqrt\{3\}\; \backslash simeq\; 3.464102\; a^2.$, where ''a'' is the apothem and ''p'' is the perimeter.

The perimeter of a regular hexagon of side length $t\backslash ,\backslash !$ is $6t\backslash ,\backslash !$, its maximal diameter $2t\backslash ,\backslash !$, and its minimal diameter $d\; =\; t\backslash sqrt\{3\}\backslash ,\backslash !$.

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C, then PE + PF = PA + PB + PC + PD.

Cyclic hexagon

A cyclic hexagon is any hexagon inscribed in a circle. If the successive sides of the cyclic hexagon are ''a, b, c, d, e, f'', then the three main diagonals intersect in a single point if and only if ''ace=bdf''.

Hexagon inscribed in a conic section

Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.

Hexagon tangential to a conic section

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.

Related figures

{| class=wikitable width=640 |- valign=top |160px
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. This form only has D₃ symmetry. In this figure, the remaining edges of the original triangle are drawn blue, and new edges from the truncation are red. |160px
The hexagram can be created as a stellation process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices. |160px
A concave hexagon |160px
A self-intersecting hexagon (star polygon) | |}

Petrie polygons

The regular hexagon is the Petrie polygon for these regular and uniform polytopes, shown in these skew orthogonal projections:

{| class=wikitable width=360 |- align=center !colspan=2|(3D) !colspan=3|(5D) |- align=center | | | | | |}

Polyhedra with hexagons

There is no platonic solid made of regular hexagons, because the hexagons tesselate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron.

{| class=wikitable width=500 !colspan=5|Archimedean solids |- valign=top | | | | | |}

And 9 Johnson solids:

triangular cupola, elongated triangular cupola, gyroelongated triangular cupola, augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism, triaugmented hexagonal prism, augmented truncated tetrahedron, triangular hebesphenorotunda

{| class=wikitable width=300 !colspan=2|Prismoids |- valign=top | | | |}

{| class=wikitable width=500 !colspan=4|Other symmetric polyhedra |- valign=top | | | | |}

Regular and uniform tilings with hexagons

{| class=wikitable width=480 |- valign=top |160px
The hexagon can form a regular tessellate the plane with a Schläfli symbol {6,3}, having 3 hexagons around every vertex. |160px
A second hexagonal tessellation of the plane can be formed as a truncated triangular tiling or rhombille tiling, with one of three hexagons colored differently. |160px
A third tessellation of the plane can be formed with three colored hexagons around every vertex. |- valign=top | | |- valign=top | | |}

Hexagons: natural and human-made

See also

Hexagram: 6-sided star within a regular hexagon

Unicursal hexagram: single path, 6-sided star, within a hexagon

Hexagonal tiling: a regular tiling of hexagons in a plane

Hexagonal number

Hexagonal crystal system

Icositetrachoron: a four-dimensional figure which, like the hexagon, has orthoplex facets and is self-dual

References

External links

Definition and properties of a hexagon With interactive animation and construction with compass and straightedge.

Cymatics - Hexagonal shapes occurring within water sound images

Cassini Images Bizarre Hexagon on Saturn

Saturn's Strange Hexagon

A hexagonal feature around Saturn's North Pole

"Bizarre Hexagon Spotted on Saturn" - from Space.com (27 March 2007)

Category:Polygons Category:Geometry

af:Seshoek ar:سداسي أضلاع ast:Hexágonu az:Düzgün altıbucaqlı bg:Шестоъгълник ca:Hexàgon cs:Šestiúhelník da:Sekskant de:Sechseck es:Hexágono eo:Seslatero eu:Hexagono fa:شش‌ضلعی fr:Hexagone gv:Shey-lhiattaghan gl:Hexágono ko:육각형 io:Hexagono id:Segienam it:Esagono he:משושה jv:Segienem ht:Egzagòn lv:Sešstūris lb:Sechseck lt:Šešiakampis hu:Hatszög ms:Heksagon mn:Зургаан өнцөгт nl:Zeshoek ja:六角形 no:Heksagon nn:Heksagon nrm:Siêx-carres ps:شپږڅنډی pl:Sześciokąt pt:Hexágono ro:Hexagon ru:Шестиугольник scn:Esàgunu simple:Hexagon sk:Šesťuholník sl:Šestkotnik sr:Шестоугао su:Sisi genep fi:Kuusikulmio sv:Hexagon ta:அறுகோணம் te:షడ్భుజి th:รูปหกเหลี่ยม tr:Altıgen uk:Шестикутник zh:六边形

Coordinates	35°27′″N139°38′″N
birth date	August 22, 1967
birth place	Grants Pass, Oregon, U.S.
occupation	Actor, comedian
yearsactive	2000—present }}

Tyler Gerald "Ty" Burrell (born August 22, 1967) is an American actor and comedian. He currently stars in the ABC comedy ''Modern Family'' as Phil Dunphy. He has been nominated for two Emmy Awards and a Screen Actors Guild Award for his role. He also shared the Screen Actors Guild Award for Outstanding Performance by an Ensemble in a Comedy Series with his co-stars.

Early life

Burrell was born in Grants Pass in Southern Oregon, and grew up in the Southern Oregon town of Applegate near the California border. He attended Hidden Valley High School where he played football and was a lineman for the Hidden Valley Mustangs. After graduating from Hidden Valley High School in Grants Pass, he enrolled at Southern Oregon University in Ashland and graduated with a bachelor's degree in theatre arts in 1993. Burrell was the commencement speaker at the school in 2008. While in college he worked as a bartender at the Oregon Shakespeare Festival. Burrell continued his education at the Pennsylvania State University, where he earned an MFA. In 1999, Burrell worked as a festival actor at the Utah Shakespeare Festival. Burrell has also stated that for a period of time in graduate school, he lived out of his van to save money.

Acting career

Some of his earlier work includes appearances in the movies ''Black Hawk Down'', the 2004 remake of ''Dawn of the Dead'', and appearances in several stage performances (like 2000's Broadway production of ''Macbeth'', and the off-Broadway plays ''Corners'', ''The Blue Demon'', ''Burn This'', and ''Show People''). He was a co-writer and actor in the original production of the offbeat comedy ''The Red Herring O' Happiness'' directed by Russell Dyball. Burrell's stage work also includes writing and working in the off-Broadway play ''Babble'' with his brother, Duncan. He has also made an appearance as a probation officer in ''Law & Order: Special Victims Unit''.

After that, he was cast as Oliver Barnes, a shallow but well-meaning plastic surgeon, on the CBS sitcom ''Out of Practice'' (2006), also created by Christopher Lloyd. The show was canceled with episodes unaired in spring 2007. After that show's cancellation, he played Allan Arbus in the film ''Fur: An Imaginary Portrait of Diane Arbus''.

Burrell was also in the Marvel movie adaptation of the comic ''The Incredible Hulk'', in which he played Doc Samson. He has gone on record in various interviews saying he does not get his powers in this movie but hopes to do so in later movies.

Ty recently had a small role in the film ''National Treasure: Book of Secrets''. He starred in the sitcom ''Back to You'' on Fox. On the show, created by Steven Levitan and Christopher Lloyd, Burrell played a field reporter (alongside Kelsey Grammer and Patricia Heaton). The show was canceled in 2008.

He now stars in the ABC comedy ''Modern Family'' as Phil Dunphy. On July 8, 2010, Burrell was nominated for the Primetime Emmy Award for Outstanding Supporting Actor - Comedy Series; he submitted the episode, (Modern Family) (Game Changer)

Personal life

Ty and his wife Holly married in 2000. They resided in New York City until moving to Salt Lake City in 2008. The couple have since relocated to Southern California for Burrell's work on ''Modern Family''. On March 18, 2010, it was announced that Burrell and his wife had adopted a baby girl, Frances. Burrell is an avid Oregon Ducks, Portland Trail Blazers and New York Mets fan.

Filmography

! Year	! Title	! Type of Media	! Role	Notes
2000	''Law & Order''	TV	Paul Donatelli	Appeared again in 2003 as a different character
2001		Movie	Flemming
2001	''''	TV	Tom Starks	1 episode
2001		Movie	U.S. Air Force Pararescueman Timothy A. Wilkinson
2002	''Law & Order: Special Victims Unit''	TV	Alan Messinger	1 episode
2003	''Law & Order''	TV	Herman Capshaw	Second appearance
2004		Movie	Steve
2004		Movie	Enrique Colon
2005	''Down in the Valley''	Movie	Sheriff/Cowboy
2006	''Friends with Money''	Movie	Other Aaron
2006	''''	Movie	Emile
2006	''Out of Practice''	TV	Dr. Oliver Barnes	20 episodes
2006		Movie
2007	''National Treasure: Book of Secrets''	Movie	Connor
2008		Movie	Christopher
2008	''''	Movie	Doc Samson
2008	''Back to You''	TV	Gary Crezyzewski	17 episodes
2009		TV	Douglas Schiff	1 episode
2009		Movie		Release date December 25, 2009
2009–present	''Modern Family''	TV	Phil Dunphy	Nominated for 2010 Television Critics Association Award	Nominated - Primetime Emmy Award for Outstanding Supporting Actor in a Comedy Series (2010)	Pending - Primetime Emmy Award for Outstanding Supporting Actor in a Comedy Series (2011)
2010		Movie	Fred
2010	''Glenn Martin, DDS''	TV	(voice)	1 episode
2010	''The Super Hero Squad Show''	TV		2 episodes
2010		Movie	Paul McVee
2010		TV	Himself	Big Star in a Small CarLap Time 1:46.6
2011		Movie	Bob Pickler	Post-production
2011	''Goats''			In production

References

External links

Category:1967 births Category:Actors from Oregon Category:American film actors Category:Living people Category:Outstanding Performance by an Ensemble in a Comedy Series Screen Actors Guild Award winners Category:Pennsylvania State University alumni Category:People from Grants Pass, Oregon Category:Southern Oregon University alumni Category:People from Ashland, Oregon

de:Ty Burrell es:Ty Burrell fr:Ty Burrell it:Ty Burrell nl:Ty Burrell pl:Ty Burrell ru:Баррелл, Ти fi:Ty Burrell sv:Ty Burrell

Coordinates	35°27′″N139°38′″N
name	Boards of Canada
landscape	Yes
background	group_or_band
alias	Hell Interface
origin	Edinburgh, Scotland
genre	ElectronicAmbientIDMExperimentalPsychedelicDowntempoIllbient
years active	1986–present
label	Warp, Skam, Music70
website	boardsofcanada.com
current members	Mike SandisonMarcus Eoin }}

Boards of Canada (commonly abbreviated BoC) are a Scottish electronic music duo consisting of brothers Mike Sandison (born June 1, 1970) and Marcus Eoin (born July 21, 1971). They are signed with Warp Records and have released several works on that label with little advertising and few interviews, while also having an elusive and obscure back-catalogue of releases on their self-run Music70 label. They have also recorded four tracks under the alias Hell Interface.

Musical style

Boards of Canada's music is reminiscent of the warm, analogue sounds of 1970s media and contains themes of childhood, nostalgia and the natural world. Mike and Marcus have mentioned the documentary films of the National Film Board of Canada, from which the group's name is derived, as a source of inspiration.

History

Early years (1986–1995)

Growing up in a musical family, brothers Mike Sandison and Marcus Eoin began playing instruments at a young age. They experimented with recording techniques at around the age of 10, using tape machines to layer cut-up samples of found sounds over compositions of their own. In their teens they participated in a number of amateur bands; however, it was not until 1986 when Marcus was invited to Mike's band that Boards of Canada was born. By 1989, the band had been reduced to Sandison and Eoin. In the early 1990s, a number of collaborations took place and the band put on small shows among the "''Hexagon Sun''" collective.

In early 2000, the website for the band, Music70.com, removed the early discography of Boards of Canada, although some information has been preserved by fans. Early tape releases by Boards of Canada include ''Play By Numbers'', ''Acid Memories'', ''Hooper Bay'', and the earliest known release by the band is titled ''Catalog 3''. None of the material from those days is readily available, and since official Boards of Canada sources ignore the existence of this material, there seems to be little chance for this early material ever to resurface.

From ''Twoism'' onwards (1995–present)

In 1995, the band made their first Hexagon Sun studio release, the EP ''Twoism''. Like earlier Music70 releases, it was produced in a self-financed limited run and was privately distributed, primarily to friends and labels. Unlike previous releases, however, a small number of copies were also released to the public through the IDM mailing list. Though not a widespread commercial release, it was considered of sufficient quality and worth to be subsequently re-pressed in 2002. The band made another release in 1996; titled ''Boc Maxima'', it was a semi-private release that was notable for being a full-length album, and was the precursor to ''Music Has the Right to Children'', with which it shares many tracks.

Boards of Canada's first commercial release occurred after attracting the attention of Autechre's Sean Booth, of the English label Skam Records, one of many people sent a demo EP. Skam issued what was considered Boards of Canada's first "findable" work, ''Hi Scores'', in 1996. ''Music has the Right to Children'' was subsequently released in 1998. John Peel featured Boards of Canada on his BBC Radio 1 program in January of that year. The session featured two remixes from ''Music Has the Right to Children'' — "Aquarius (Version 3)" and "Olson (Version 3)" — along with the tracks "Happy Cycling" and "XYZ". Excluding "XYZ", the set was released on a Warp Records CD titled ''Peel Session TX 21 July 1998''.

Though never an actively touring band, Boards of Canada did perform a handful of shows. Early shows saw them supporting Warp labelmates Seefeel and Autechre in a handful of UK dates. They also participated in a few festivals and multi-artist bills, including two Warp parties: Warp's 10th Anniversary Party in 1999, and The Incredible Warp Lighthouse Party almost one year later. They made their most prominent showing in 2001 as one of the headliners at the Tortoise-curated All Tomorrow's Parties festival. They have not performed a live show since.

The band released a four-track EP, ''In a Beautiful Place Out in the Country'', in November 2000, their first original release in two years. The LP edition was pressed on sky blue vinyl. A full-length album, ''Geogaddi'', was then released in 2002. It was described by Sandison as "a record for some sort of trial-by-fire, a claustrophobic, twisting journey that takes you into some pretty dark experiences before you reach the open air again."

Their third album for Warp Records, ''The Campfire Headphase'', was released on 17 October 2005 (18 October in the United States). The album featured fifteen tracks, including "Peacock Tail", "Chromakey Dreamcoat," and "Dayvan Cowboy". Two versions of "Dayvan Cowboy" — the original and a remix by Odd Nosdam — are on the six-track EP, ''Trans Canada Highway'', which was released on 26 May 2006.

In late 2009 the ''Warp20 (Recreated)'' compilation featured two BoC covers, one by Bibio of their song "Kaini Industries" and one by Mira Calix of "In a Beautiful Place Out in the Country". Warp20 (Recreated) is part of the larger Warp20 boxed set, which also includes two previously unreleased Boards of Canada tracks, "Seven Forty Seven" and a 1.8 second sample of "Spiro".

Sound and methods

BoC's sound is a product of their use of analog equipment, mix of electronic and conventional instrumentation, use of distorted samples as well as live and recorded lyrics, and their layering and blending of these elements. Because they avoid a purely synthetic sound, their music is often described as having a warm, emotive quality often meant to inspire nostalgia. This is helped by the use of samples from 1970s television shows and other media prevalent in the era of the brothers' shared childhood, especially the nature-inspired documentaries produced by the National Film Board of Canada.

Brief interludes or vignettes feature prominently in BoC's music. Such songs are often weaving melodies or speech accompanied by atmospherics to capture a specific moment or mood. They often last less than two minutes, but, as BoC state, "''we write far more of [these] than the so-called 'full-on' tracks, and, in a way, they are our own favorites''". BoC have written an enormous number of such fragments as well as full-length tracks, most of which have been held back from release. It does not appear that their music is made exclusively for commercial release; rather, albums seem to be the result of selecting complementary songs from current work. For instance, ''Geogaddi'''s development allegedly involved the creation of 400 song fragments and 64 complete songs, of which 22 were selected (possibly 23, if the final track of complete silence is included). Says Marcus: "''The idea of the perfect album is this amorphous thing that we're always aiming at […] the whole point of making music is at least to aim at your own idea of perfection.''"

Interviews with the Sandison brothers provide some insight into their creative inspiration. They have cited several acts that have influenced their work, including Joni Mitchell, The Incredible String Band ("''we have all the String Band records […] our rural sensibilities are similar''"), The Beatles ("''[they] really became enthralling to us through their psychedelism''") and My Bloody Valentine ("''even if we don't sound like them, there's a connection in terms of the approach to music''").

They have also expressed a strong interest in the power of subliminal messaging and their work is full of cryptic messages, including references to numerology and cult figures such as David Koresh of the Branch Davidians, amongst other vague hints. When questioned about their aims in making such references, BoC express themselves in neutral terms ("''We're not religious at all [...] and if we're spiritual at all it's purely in the sense of caring about art and inspiring people with ideas.''") while remaining fascinated with the ability of music to influence the minds of others ("''[We] do actually believe that there are powers in music that are almost supernatural. I think you actually manipulate people with music...''").

Discography

Major releases

1995: "Twoism" - (Music70)

1996: "Hi Scores" - (Skam)

1998: "Aquarius" - (Skam)

1998: ''Music Has the Right to Children'' – (Warp/Skam) #193 UK

1999: ''Peel Session TX 21/07/1998'' (Warp)

2000: ''In a Beautiful Place Out in the Country'' – (Warp/Music70)

2002: ''Geogaddi'' – (Warp/Music70) #21 UK

2005: ''The Campfire Headphase'' – (Warp) #41 UK

2006: ''Trans Canada Highway'' (Warp)

''Old Tunes'' and other oddities

Supposed early Boards of Canada recordings have been found online, claiming to be original tracks from ''Catalog 3'', ''Acid Memories'', ''Hooper Bay'' and other pre-''Twoism'' albums. All unreleased albums are presumed to be inauthentic BoC recordings, with the exceptions of Boc Maxima and the Old Tunes tapes. While the Old Tunes tapes are believed to be authentic, mostly due to the inclusion of very similar tracks found elsewhere on official releases, there has never been any official recognition of their authenticity.

Media usage

Many Boards of Canada tracks are used in the BBC's motoring show, Top Gear, such as ''1969'' from the album ''Geogaddi'', and ''Pete Standing Alone'' from the predecessor album ''Music Has the Right to Children''.

The song "Dayvan Cowboy" from ''The Campfire Headphase'' was used in an episode of ''CSI: Miami''.

The song "Rue The Whirl" was used in an episode of Spaced.

The song "Amo Bishop Roden" is used in Season 3, episode 5 of Skins.

The songs "Turquoise Hexagon Sun", "Aquarius", and "Sixtyniner" were used in the dark humour animated series ''Monkey Dust'', which aired on BBC Three from 2003-2005.

The songs "Gyroscope" and "You Could Feel the Sky" were used in the Norwegian Black Metal documentary ''Until The Light Takes Us''.

David Firth often uses Boards of Canada songs in his animations, notably "Beware the Friendly Stranger," a track from Geogaddi which was the backing to Firth's cartoon series Salad Fingers.

Adult Swim on Cartoon Network sometimes uses samples from Campfire Headphase for its Bumps or messages to its audience.

See also

Warp Records

Music70

Hexagon Sun Collective

References

External links

Official sites

Boards of Canada official website

Boards of Canada at MySpace

at Warp Records

Music70

Hexagon Sun

Boards of Canada official YouTube channel

Discographies

Boards of Canada at xltronic

Unofficial/fan sites

Boards of Canada wiki

Boards of Canada forum at Twoism.org

Boards of Canada forum at We Are The Music Makers

Pre-2004 boardsofcanada.com Mirror

* Category:Scottish musical groups Category:Scottish electronic musicians Category:Ambient music groups Category:British electronic music groups Category:Intelligent dance music musicians Category:Electronic music duos Category:Sibling musical duos Category:Warp Records artists Category:1970 births Category:Living people Category:Trip hop musicians

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