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Hexagon

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Hexagon

Regular hexagon
A regular hexagon
Type Regular polygon
Edges and vertices 6
Schläfli symbol {6}
Coxeter 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" and γωνία, gonía, "corner, angle") 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°.

Hexagonal structures

From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest number of hexagons. This means that honeycombs require less wax to construct and gain lots of strength under compression.

Regular hexagon

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.
The regular hexagon has a number of subsymmetries that can be seen by coloring or geometric variations

A regular hexagon has all sides of the same length, and that length is also the radius of the circumscribed circle. 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

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

A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866025404d^2.

The area can also be found by the formulas A=ap/2 and \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 \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.

Tesselations by hexagons

In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.

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.

Cyclic hexagon

The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.

If the successive sides of a 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]

If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[2]

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.

In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[3]

a+c+e=b+d+f.

Convex equilateral hexagon

A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[4]:p.184,#286.3 a principal diagonal d1 such that

\frac{d_1}{a} \leq 2

and a principal diagonal d2 such that

\frac{d_2}{a} > \sqrt{3}.

Related figures


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

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

(3D) (4D) (5D)

Cube

Octahedron

3-3 duopyramid

3-3 duoprism

5-simplex

Rectified 5-simplex

Birectified 5-simplex

Polyhedra with hexagons

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. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and .

Archimedean solids
Tetrahedral Octahedral Icosahedral

truncated tetrahedron

truncated octahedron

truncated cuboctahedron

truncated icosahedron

truncated icosidodecahedron
There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):
Tetrahedral Octahedral Icosahedral

Chamfered tetrahedron

Chamfered cube

Chamfered dodecahedron


There are also 9 Johnson solids with regular hexagons:

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 polyhedral with hexagons

Truncated triakis tetrahedron

Regular and uniform tilings with hexagons


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.

The hexagonal tiling can be distorted, like these centrosymmetric hexagons

Trihexagonal tiling

Trihexagonal tiling

Rhombitrihexagonal tiling

Truncated trihexagonal tiling

Hexagons: natural and human-made

See also

References

  1. ^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
  2. ^ Nikolaos Dergiades, "Dao's theorem on six circumcenters associated with a cyclic hexagon", Forum Geometricorum 14, 2014, 243--246. http://forumgeom.fau.edu/FG2014volume14/FG201424index.html
  3. ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [4], Accessed 2012-04-17.
  4. ^ Inequalities proposed in “Crux Mathematicorum”, [5].

External links

  • Weisstein, Eric W., "Hexagon", MathWorld.
  • 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)
  • supraHex A supra-hexagonal map for analysing high-dimensional omics data.
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