Regular hexagon
A regular hexagon
Type	Regular polygon
Edges and vertices	6
Schläfli symbol	{6} t{3}
Coxeter–Dynkin diagram
Symmetry group	D6, order 2×6
Internal angle (degrees)	120°
Dual polygon	self
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°.

Regular hexagon[link]

A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15.

A regular hexagon has all sides of the same length, and all internal angles are 120 degrees. 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 is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.

An alternative formula for area is A = 1.5dt where the length d is the distance between the parallel sides (also referred to as the flat-to-flat distance), or the height of the hexagon when it sits on one side as base, or the diameter of the inscribed circle.

Another alternative formula for the area if only the flat-to-flat distance, d, is known, is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866025404d^2.

The area can also be found by the formulas Failed to parse (Missing texvc executable; please see math/README to configure.): A=ap/2

and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle A\ =\ {2}a^2\sqrt{3}\ \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 is 6t, its maximal diameter 2t, and its minimal diameter Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle d\ =\ t\sqrt{3} .

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[link]

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.^[1]

Hexagon inscribed in a conic section[link]

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[link]

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.

In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,^[2]

Failed to parse (Missing texvc executable; please see math/README to configure.): a+c+e=b+d+f.

Related figures[link]

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.

The hexagram can be created as a stellation process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices.

A concave hexagon

A self-intersecting hexagon (star polygon)

A (nonplanar) skew regular hexagon, within the edges of a cube

Petrie polygons[link]

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

(3D)		(5D)
Cube	Octahedron	5-simplex	Rectified 5-simplex	Birectified 5-simplex

Polyhedra with hexagons[link]

There is no platonic solid made of only regular hexagons, because the hexagons tessellate, 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.

Archimedean solids
truncated tetrahedron	truncated octahedron	truncated icosahedron	truncated cuboctahedron	truncated icosidodecahedron

There are also 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

Prismoids
Hexagonal prism	Hexagonal antiprism	Hexagonal pyramid

Other symmetric polyhedra
Truncated triakis tetrahedron	Truncated rhombic dodecahedron	Truncated rhombic triacontahedron

Regular and uniform tilings with hexagons[link]

The hexagon can form a regular tessellate the plane with a Schläfli symbol {6,3}, having 3 hexagons around every vertex.	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.	A third tessellation of the plane can be formed with three colored hexagons around every vertex.
Trihexagonal tiling	Trihexagonal tiling
Rhombitrihexagonal tiling	Truncated trihexagonal tiling

Hexagons: natural and human-made[link]

• 

  Assembled E-ELT mirror segments

• 

  A beehive honeycomb

• 

  The scutes of a turtle's carapace

• 

  North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini [1] [2] [3]

• 

  Micrograph of a snowflake

• 

  Benzene, the simplest aromatic compound with hexagonal shape.

• 

  Crystal structure of a molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834-1839.

• 

  Naturally formed basalt columns from Giant's Causeway in Ireland; large masses must cool slowly to form a polygonal fracture pattern

• 

  An aerial view of Fort Jefferson in Dry Tortugas National Park

• 

  The James Webb Space Telescope mirror is composed of 18 hexagonal segments.

• 

  Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France aka the "metropole" as opposed to the overseas territories such as Guadeloupe, Martinique or French Guiana.

• 

  Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals

• 

  Hexagonal barn

• 

  The Hexagon, a hexagonal theatre in Reading, Berkshire

See also[link]

• 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[link]

External links[link]

• Weisstein, Eric W., "Hexagon" from MathWorld.
• Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
• Cymatics – Hexagonal shapes occurring within water sound images^{[dead link]}
• 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)

Terry Cavanagh
29th Mayor of Edmonton
In office November 1975 – October 19, 1977
Preceded by	William Hawrelak
Succeeded by	Cec Purves
In office 1988 – October 16, 1989
Preceded by	Laurence Decore
Succeeded by	Jan Reimer
Alderman/Councillor on the Edmonton City Council (Ward 6)
In office October 22, 1992 – October 15, 2007 Serving with Sheila McKay (1992-1995), Dick Mather (1995-1997), and Dave Thiele (1998-2007)
Preceded by	Ken Kozak, Sheila McKay
Succeeded by	Amarjeet Sohi, Dave Thiele
In office October 17, 1983 – 1988 Serving with Bettie Hewes (1983-1984) and Ken Kozak (1984-1988)
Preceded by	Bettie Hewes, Ed Leger
Succeeded by	Ken Kozak, Sheila McKay
Alderman on the Edmonton City Council (Ward 4)
In office October 13, 1971 – November 1975 Serving with Una Evans and Buck Olsen (1971-1974), and Bettie Hewes and Buck Olsen (1974-1975)
Preceded by	New district
Succeeded by	William Chmiliar, Bettie Hewes, Buck Olsen
Personal details
Born	(1926-07-19) July 19, 1926 (age 85) Edmonton, Alberta
Political party	Independent
Spouse(s)	June Gould
Children	Three
Profession	Businessman
Signature

Terence James "Terry" Cavanagh (born July 19, 1926) is a Canadian politician, municipal councillor in Edmonton, Alberta, who served as mayor. He was Edmonton's first native-born mayor.

Early life[link]

Cavanagh was born in Edmonton July 19, 1926, to recent Scottish immigrants. He attended high school in Edmonton before moving to Galt, Ontario to play hockey for the Galt Red Wings of the Ontario Hockey Association, where he was a teammate of Gordie Howe. After stints with the Dallas Texans of the United States Hockey League, the Valleyfield Braves of the Quebec Senior Hockey League, and the Los Angeles Ramblers and the Trail Smoke Eaters of the Western International Hockey League, he retired from hockey and found employment in the purchasing department of Consolidated Mining and Smelting Co. in Trail, British Columbia. He married June Gould on April 12, 1948; the couple would have three children, and June would herself serve as an Edmonton alderman.

In 1957, he returned to Edmonton to work as the purchasing manager for Premier Steel Mills Ltd. He stayed on when the company was taken over by Stelco Steel in 1962, and remained in the position until 1977.

Political career[link]

Early career and hiatus[link]

Cavanagh entered politics during the 1968 Edmonton election, when he ran for Edmonton City Council as an aldermanic candidate. He was defeated, finishing fourteenth of thirty-two candidates in an election in which the top twelve were elected. He was more successful during the 1971 election; Edmonton had adopted a ward system (in which three aldermen were elected in each of four wards) to replace the previous at-large method of election, and Cavanagh placed second of seventeen candidates in Ward 4. He was re-elected in the 1974 election, when he placed first of nine candidates.

On November 7, 1975, mayor William Hawrelak died in office. After nine ballots, Council chose Cavanagh over Laurence Decore to serve out his term, but when ran to retain the position in the 1977 election, he finished third, nearly ten thousand votes behind victor Cec Purves and under a thousand behind Decore (he did finish well ahead of former mayor Ivor Dent, however). He stayed out of politics for six years following this defeat, serving as chair of the Government of Alberta's Alberta Rent Decontrol Appeal Board (tasked with the elimination of provincial rent controls) from 1977 until 1980 and as a communications advisor for the Alberta Energy Company Ltd. from 1980 until 1984. He also rejected overtures from Neil Stanley Crawford to run for the Legislative Assembly of Alberta as a Progressive Conservative, commenting later that he had little interest in sitting as a backbencher after serving as mayor of Edmonton, and that Premier Peter Lougheed had not been prepared to promise him a cabinet position.

Return to council[link]

He returned to Council after the 1983 election, when he finished second of eight candidates in Ward 6 (Edmonton's ward system had been revised such that there were two aldermen elected from each of six wards). He was re-elected in the 1986 election, placing first of nine candidates.

In 1988, Decore (who had been elected mayor in 1983) resigned to take a position as leader of the Alberta Liberal Party. Again Council chose Cavanagh to serve as interim mayor, and again he was defeated soundly when he sought re-election - this time, he finished with fewer than half the votes of Jan Reimer during the 1989 municipal election. He would win his old aldermanic seat back in the 1992 election, and would be re-elected in the 1995, 1998, 2001, and 2004 elections.

In May 2007, Cavanagh announced that he would not seek election in the 2007 municipal election.

Political views, initiatives, and reputation[link]

Terry Cavanagh has excellent ties with Edmonton's immigrant and especially Chinese communities. Members of the Chinese community threw him a dinner for his eightieth birthday, and he claims that he can say "vote for me" in "15 or 17 languages". He has also been an advocate for the development of Edmonton's North Saskatchewan River valley as a park area, despite the fact that he is not generally known as an environmentalist. Despite his experience in removing rent controls during the 1970s, he urged Alberta Premier Ed Stelmach to consider reintroducing them during Alberta's recent economic boom and accompanying tight housing market.

During the later years of his political involvement, some observers accused him of taking a less active role on city council than in years past.

Personal life and extra-political involvement[link]

Cavanagh is a communications instructor, and has taught courses on the subject at Victoria Composite High School, the Northern Alberta Institute of Technology (where he has also taught a purchasing management course), and the University of Alberta's Faculty of Extension. He has also taught a government relations course at Grant MacEwan College.

References[link]

• Terry Cavanagh profile Edmonton City Council (retrieved August 14, 2007)
• Edmonton Public Library biography of Terry Cavanagh
• "Council's 'statesman' says 27 years is enough", Edmonton Journal, May 16, 2007
• Terry Cavanagh's career stats at The Internet Hockey Database
• "Budgeting Time", Real Estate Weekly, November 23, 2006
• "Keep rent-control plan alive, say councillors", Edmonton Journal, May 6, 2007
• "Edmonton then, Edmonton now", Edmonton Journal, June 23, 2007 (reproduced on connect2edmonton.ca)

v t e Mayors of Edmonton McCauley Wilson Gallagher McDougall Edmiston MacKenzie Short MacKenzie May Griesbach McDougall Lee Armstrong Short McNamara Henry Evans Clarke Duggan Blatchford Bury Douglas Knott Clarke Fry Ainlay Parsons Hawrelak Mitchell Roper Hawrelak Dantzer Dent Hawrelak Cavanagh Purves Decore Cavanagh Reimer Smith Mandel The Edmonton portal

Ty Burrell
Ty Burrell, 2009
Born	(1967-08-22) August 22, 1967 (age 44) Grants Pass, Oregon, U.S.
Occupation	Actor, comedian
Years active	2000—present
Spouse	Holly
Children	2

Tyler Gerald "Ty" Burrell (born August 22, 1967) is an American actor and comedian. He is best known for his role in the ABC comedy Modern Family as Phil Dunphy, for which he won the Primetime Emmy Award for Outstanding Supporting Actor in a Comedy Series in 2011. He was nominated for the same award in 2010. He also received nominations for the Screen Actors Guild Award for Outstanding Performance by a Male Actor in a Comedy Series for his role, sharing the Screen Actors Guild Award for Outstanding Performance by an Ensemble in a Comedy Series with his co-stars.

Early life[link]

Burrell was born in Grants Pass in Southern Oregon,^[1] and grew up in the small Southern Oregon town of Applegate, near the California border, where his family ran a business.^[2][3] He attended Hidden Valley High School in Grants Pass, where he played football and was a lineman for the Hidden Valley Mustangs. After graduating from high school, he enrolled at Southern Oregon University in Ashland,^[3] and graduated with a bachelor's degree in theatre arts in 1993.^[2] Burrell was the commencement speaker at the school in 2008.^[4] While in college he worked as a bartender at the Oregon Shakespeare Festival.^[2] Burrell continued his education at the Penn State University, where he earned an MFA.^[2] 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.^[5]

Acting career[link]

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. He was nominated for the Emmy Award for Outstanding Supporting Actor in a Comedy Series for his role in 2010 (he submitted the episode Up All Night), but lost to his co-star Eric Stonestreet. In 2011, he was nominated for an Emmy ^[6] and won for his performance in the episode Good Cop Bad Dog.

Personal life[link]

Ty and his wife Holly married in 2000. They resided in New York City until moving to Salt Lake City^[7] in 2008.^[2] They still own a two-bedroom apartment in the Astoria, Queens neighborhood which they rent out.^[8] 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.^[9] In March 2012, Burrell and his wife adopted a second daughter.^[10] Burrell is an avid Oregon Ducks, Portland Trail Blazers and New York Mets fan.^[11]

Filmography[link]

References[link]

External links[link]

• Ty Burrell at the Internet Movie Database
• Ty Burrell at AllRovi
• Ty Burrell at Emmys.com