geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the two ears theorem states that every simple polygon with more than three vertices has at least two ears, vertices that can be removed from the polygon without introducing any crossings. The two ears theorem is equivalent to the existence of

polygon triangulation In computational geometry, polygon triangulation is the partition of a polygonal area ( simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may ...

s. It is frequently attributed to Gary H. Meisters, but was proved earlier by

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. H ...

Statement of the theorem

An ear of a polygon is defined as a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

such that the line segment between the two neighbors of lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon has at least two ears.

Ears from triangulations

An ear and its two neighbors form a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

within the polygon that is not crossed by any other part of the polygon. Removing a triangle of this type produces a polygon with fewer sides, and repeatedly removing ears allows any simple polygon to be

triangulated In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

. Conversely, if a polygon is triangulated, the weak dual of the triangulation (a graph with one vertex per triangle and one edge per pair of adjacent triangles) will be a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

and each leaf of the tree will form an ear. Since every tree with more than one vertex has at least two leaves, every triangulated polygon with more than one triangle has at least two ears. Thus, the two ears theorem is equivalent to the fact that every simple polygon has a triangulation.

Related types of vertex

An ear is called ''exposed'' when it forms a vertex of the convex hull of the polygon. However, it is possible for a polygon to have no exposed ears. Ears are a special case of a ''principal vertex'', a vertex such that the line segment connecting the vertex's neighbors does not cross the polygon or touch any other vertex of it. A principal vertex for which this line segment lies outside the polygon is called a ''mouth''. Analogously to the two ears theorem, every non-convex simple polygon has at least one mouth. Polygons with the minimum number of principal vertices of both types, two ears and a mouth, are called anthropomorphic polygons.

History and proof

The two ears theorem is often attributed to a 1975 paper by Gary H. Meisters, from which the "ear" terminology originated. However, the theorem was proved earlier by

(circa 1899) as part of a proof of the

Jordan curve theorem In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterio ...

. To prove the theorem, Dehn observes that every polygon has at least three convex vertices. If one of these vertices, , is not an ear, then it can be connected by a diagonal to another vertex inside the triangle formed by and its two neighbors; can be chosen to be the vertex within this triangle that is farthest from line . This diagonal decomposes the polygon into two smaller polygons, and repeated decomposition by ears and diagonals eventually produces a triangulation of the whole polygon, from which an ear can be found as a leaf of the dual tree..

References

External links

Meisters' Two Ears Theorem
Cut-the-Knot
The Two-Ears Theorem

Godfried Toussaint Godfried Theodore Patrick Toussaint (1944 – July 2019) was a Canadian computer scientist, a professor of computer science, and the head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirate ...

*{{mathworld, id=Two-EarsTheorem, title=Two-Ears Theorem, mode=cs2 Theorems about polygons