The four-vertex theorem of

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 ...

states that the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...

along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function 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 ...

. This theorem has many generalizations, including a version for

space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s where a vertex is defined as a point of vanishing

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

Definition and examples

The

at any point of a

smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

in the plane can be defined as the

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

of the radius of an

osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...

at that point, or as the

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

of the

second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

of a

parametric representation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...

of the curve, parameterized consistently with the length along the curve. For the vertices of a curve to be well-defined, the curvature itself should vary continuously, as happens for curves of

smoothness In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

C^2

. A vertex is then a

local maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

local minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

of curvature. If the curvature is constant over an arc of the curve, all points of that arc are considered to be vertices. The four-vertex theorem states that a smooth

closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

always has at least four vertices. An

ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse i ...

has exactly four vertices: two local maxima of curvature where it is crossed by the major axis of the ellipse, and two local minima of curvature where it is crossed by the minor axis. In a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

, every point is both a local maximum and a local minimum of curvature, so there are infinitely many vertices. Every

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width ...

has at least six vertices. Although many curves of constant width, such as the

Reuleaux triangle A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the ...

, are non-smooth or have circular arcs on their boundaries, there exist smooth curves of constant width that have exactly six vertices.

History

The four-vertex theorem was first proved for

convex curve In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, ...

s (i.e. curves with strictly positive curvature) in 1909 by

Syamadas Mukhopadhyaya Syamadas Mukhopadhyaya (22 June 1866 – 8 May 1937) was an Indian mathematician who introduced the four-vertex theorem and Mukhopadhyaya's theorem in plane geometry. Biography Syamadas Mukhopadhyaya was born at Haripal, Hooghly district, ...

. His proof utilizes the fact that a point on the curve is an extremum of the curvature function

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...

the

at that point has fourth-order

contact Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * ...

with the curve; in general the osculating circle has only third-order contact with the curve. The four-vertex theorem was proved for more general curves by

Adolf Kneser Adolf Kneser (19 March 1862 – 24 January 1930) was a German mathematician. He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany (now Wrocław, Poland). He is the father of the mathematician Hellmuth Kneser and the gra ...

in 1912 using a projective argument.

Proof

For many years the proof of the four-vertex theorem remained difficult, but a simple and conceptual proof was given by , based on the idea of the minimum enclosing circle. This is a circle that contains the given curve and has the smallest possible radius. If the curve includes an arc of the circle, it has infinitely many vertices. Otherwise, the curve and circle must be

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

at at least two points, because a circle that touched the curve at fewer points could be reduced in size while still enclosing it. At each tangency, the curvature of the curve is greater than that of the circle, for otherwise the curve would continue from the tangency outside the circle rather than inside. However, between each pair of tangencies, the curvature must decrease to less than that of the circle, for instance at a point obtained by translating the circle until it no longer contains any part of the curve between the two points of tangency and considering the last point of contact between the translated circle and the curve. Therefore, there is a local minimum of curvature between each pair of tangencies, giving two of the four vertices. There must be a local maximum of curvature between each pair of local minima (not necessarily at the points of tangency), giving the other two vertices.

Converse

The converse to the four-vertex theorem states that any

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for strictly positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres. The full converse to the four-vertex theorem was proved by shortly before his death in January 1998, and published posthumously. Dahlberg's proof uses a

winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tur ...

argument which is in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra.

Application to mechanics

One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points. A discrete version of this is that there cannot be a monostatic polygon. However, in three dimensions there do exist monostatic polyhedra, and there also exists a convex, homogeneous object with exactly 2 balance points (one stable, and the other unstable), the

Gömböc The Gömböc ( ) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. The ...

Discrete variations

There are several discrete versions of the four-vertex theorem, both for convex and non-convex polygons. Here are some of them: * ( Bilinski) The sequence of angles of a convex

equilateral polygon In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon ...

with at least four vertices has at least four extrema. * The sequence of side lengths of a convex

equiangular polygon In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons whic ...

with at least four sides has at least four extrema. * (Musin) A

circumscribed In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...

around three consecutive vertices of a polygon with at least four vertices is called ''extremal'' if it contains all remaining vertices of the polygon, or has none of them in its interior. Such a convex polygon is ''generic'' if it has no four vertices on the same circle. Then every generic convex polygon with at least four vertices has at least four extremal circles. * ( Legendre–

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

) Two convex ''n''-gons with equal corresponding side length have either zero or at least 4 sign changes in the cyclic sequence of the corresponding angle differences. * ( A.D. Alexandrov) Two convex ''n''-gons with parallel

corresponding sides In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, takin ...

and equal area have either zero or at least 4 sign changes in the cyclic sequence of the corresponding side lengths differences. Some of these variations are stronger than the other, and all of them imply the (usual) four-vertex theorem by a limit argument.

Generalizations to space curve

The

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...

from the once-punctured sphere to the plane preserves critical points of

geodesic curvature In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's ...

. Thus simple closed spherical curves have four vertices. Furthermore, on the sphere vertices of a curve correspond to points where its

vanishes. So for space curves a vertex is defined as a point of vanishing torsion. Every simple closed space curve which lies on the boundary of a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

has four vertices. This theorem can be generalized to all curves which bound a locally convex disk.

References

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Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...

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