Duality (projective geometry)

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https://en.wikipedia.org/wiki/Duality_(projective_geometry)

In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

Principle of duality

A projective plane C may be defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. These sets can be used to define a plane dual structure.

Interchange the role of "points" and "lines" in

C = (P, L, I)

to obtain the dual structure

C^∗ = (L, P, I^∗),

where I^∗ is the converse relation of I. C^∗ is also a projective plane, called the dual plane of C.

If C and C^∗ are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) K are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.

In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement.

If a statement is true in a projective plane C, then the plane dual of that statement must be true in the dual plane C^∗. This follows since dualizing each statement in the proof "in C" gives a corresponding statement of the proof "in C^∗".

The principle of plane duality says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C.

The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the principle of space duality.

These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point").

The validity of the principle of plane duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane. The dual of a true statement in a projective plane is therefore a true statement in the dual projective plane and the implication is that for self-dual planes, the dual of a true statement in that plane is also a true statement in that plane.

Dual theorems

As the real projective plane, PG(2, R), is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:

• Desargues' theorem ⇔ Converse of Desargues' theorem
• Pascal's theorem ⇔ Brianchon's theorem
• Menelaus' theorem ⇔ Ceva's theorem

Dual configurations

Dual configurations

Not only statements, but also systems of points and lines can be dualized.

A set of m points and n lines is called an (m_c, n_d) configuration if c of the n lines pass through each point and d of the m points lie on each line. The dual of an (m_c, n_d) configuration, is an (n_d, m_c) configuration. Thus, the dual of a quadrangle, a (4₃, 6₂) configuration of four points and six lines, is a quadrilateral, a (6₂, 4₃) configuration of six points and four lines.

The set of all points on a line, called a projective range has as its dual a pencil of lines, the set of all lines on a point.

Duality as a mapping

Plane dualities

A plane duality is a map from a projective plane C = (P, L, I) to its dual plane C^∗ = (L, P, I^∗) (see § Principle of duality above) which preserves incidence. That is, a plane duality σ will map points to lines and lines to points (P^σ = L and L^σ = P) in such a way that if a point Q is on a line m (denoted by Q I m) then Q I m ⇔ m^σ I^∗Q^σ. A plane duality which is an isomorphism is called a correlation. The existence of a correlation means that the projective plane C is self-dual.

The projective plane C in this definition need not be a Desarguesian plane. However, if it is, that is, C = PG(2, K) with K a division ring (skewfield), then a duality, as defined below for general projective spaces, gives a plane duality on C that satisfies the above definition.

In general projective spaces

A duality δ of a projective space is a permutation of the subspaces of PG(n, K) (also denoted by KPⁿ) with K a field (or more generally a skewfield (division ring)) that reverses inclusion, that is:

S ⊆ T implies S^δ ⊇ T^δ for all subspaces S, T of PG(n, K).

Consequently, a duality interchanges objects of dimension r with objects of dimension n − 1 − r ( = codimension r + 1). That is, in a projective space of dimension n, the points (dimension 0) correspond to hyperplanes (codimension 1), the lines joining two points (dimension 1) correspond to the intersection of two hyperplanes (codimension 2), and so on.

Classification of dualities

In this section the traditional geometric terminology of "right (left) vector spaces over skewfields" is used instead of the algebraic terms "R-modules over a division ring".

The dual V^∗ of a finite-dimensional (right) vector space V over a skewfield K can be regarded as a (right) vector space of the same dimension over the opposite skewfield K^o. There is thus an inclusion-reversing bijection between the projective spaces PG(n, K) and PG(n, K^o). If K and K^o are isomorphic then there exists a duality on PG(n, K). Conversely, if PG(n, K) admits a duality for n > 1, then K and K^o are isomorphic.

Let π be a duality of PG(n, K) for n > 1. If π is composed with the natural isomorphism between PG(n, K) and PG(n, K^o), the composition θ is an incidence preserving bijection between PG(n, K) and PG(n, K^o). By the Fundamental theorem of projective geometry θ is induced by a semilinear map T: V → V^∗ with associated isomorphism σ: K → K^o, which can be viewed as an antiautomorphism of K. In the classical literature, π would be called a reciprocity in general, and if σ = id it would be called a correlation (and K would necessarily be a field). Some authors suppress the role of the natural isomorphism and call θ a duality. When this is done, a duality may be thought of as a collineation between a pair of specially related projective spaces and called a reciprocity. If this collineation is a projectivity then it is called a correlation.

Let T_w = T(w) denote the linear functional of V^∗ associated with the vector w in V. Define the form φ: V × V → K by:

${\displaystyle \phi (v,w)=T_w(v).}$

φ is a nondegenerate sesquilinear form with companion antiautomorphism σ.

Any duality of PG(n, K) for n > 1 is induced by a nondegenerate sesquilinear form on the underlying vector space (with a companion antiautomorphism) and conversely.

Homogeneous coordinate formulation

Homogeneous coordinates may be used to give an algebraic description of dualities. To simplify this discussion we shall assume that K is a field, but everything can be done in the same way when K is a skewfield as long as attention is paid to the fact that multiplication need not be a commutative operation.

The points of PG(n, K) can be taken to be the nonzero vectors in the (n + 1)-dimensional vector space over K, where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of n-dimensional projective space are the 1-dimensional vector subspaces, which may be visualized as the lines through the origin in Kⁿ⁺¹. Also the n- (vector) dimensional subspaces of Kⁿ⁺¹ represent the (n − 1)- (geometric) dimensional hyperplanes of projective n-space over K, i.e., PG(n, K).

A nonzero vector u = (u₀, u₁, ..., u_n) in Kⁿ⁺¹ also determines an (n − 1) - geometric dimensional subspace (hyperplane) H_u, by

H_u = {(x₀, x₁, ..., x_n) : u₀x₀ + ... + u_nx_n = 0}.

When a vector u is used to define a hyperplane in this way it shall be denoted by u_H, while if it is designating a point we will use u_P. They are referred to as point coordinates or hyperplane coordinates respectively (in the important two-dimensional case, hyperplane coordinates are called line coordinates). Some authors distinguish how a vector is to be interpreted by writing hyperplane coordinates as horizontal (row) vectors while point coordinates are written as vertical (column) vectors. Thus, if u is a column vector we would have u_P = u while u_H = u^T. In terms of the usual dot product, H_u = {x_P : u_H ⋅ x_P = 0}. Since K is a field, the dot product is symmetrical, meaning u_H ⋅ x_P = u₀x₀ + u₁x₁ + ... + u_nx_n = x₀u₀ + x₁u₁ + ... + x_nu_n = x_H ⋅ u_P.

A fundamental example

A simple reciprocity (actually a correlation) can be given by u_P ↔ u_H between points and hyperplanes. This extends to a reciprocity between the line generated by two points and the intersection of two such hyperplanes, and so forth.

Specifically, in the projective plane, PG(2, K), with K a field, we have the correlation given by: points in homogeneous coordinates (a, b, c) ↔ lines with equations ax + by + cz = 0. In a projective space, PG(3, K), a correlation is given by: points in homogeneous coordinates (a, b, c, d) ↔ planes with equations ax + by + cz + dw = 0. This correlation would also map a line determined by two points (a₁, b₁, c₁, d₁) and (a₂, b₂, c₂, d₂) to the line which is the intersection of the two planes with equations a₁x + b₁y + c₁z + d₁w = 0 and a₂x + b₂y + c₂z + d₂w = 0.

The associated sesquilinear form for this correlation is:

φ(u, x) = u_H ⋅ x_P = u₀x₀ + u₁x₁ + ... + u_nx_n,

where the companion antiautomorphism σ = id. This is therefore a bilinear form (note that K must be a field). This can be written in matrix form (with respect to the standard basis) as:

φ(u, x) = u_H G x_P,

where G is the (n + 1) × (n + 1) identity matrix, using the convention that u_H is a row vector and x_P is a column vector.

The correlation is given by:

${\displaystyle \pi ({\mathbf{x}}_P)=(G{\mathbf{x}}_P{)}^{\mathsf{T}}=({\mathbf{x}}_P{)}^{\mathsf{T}}={\mathbf{x}}_H.}$

Geometric interpretation in the real projective plane

This correlation in the case of PG(2, R) can be described geometrically using the model of the real projective plane which is a "unit sphere with antipodes identified", or equivalently, the model of lines and planes through the origin of the vector space R³. Associate to any line through the origin the unique plane through the origin which is perpendicular (orthogonal) to the line. When, in the model, these lines are considered to be the points and the planes the lines of the projective plane PG(2, R), this association becomes a correlation (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centered at the origin. The lines meet the sphere in antipodal points which must then be identified to obtain a point of the projective plane, and the planes meet the sphere in great circles which are thus the lines of the projective plane.

That this association "preserves" incidence is most easily seen from the lines and planes model. A point incident with a line in the projective plane corresponds to a line through the origin lying in a plane through the origin in the model. Applying the association, the plane becomes a line through the origin perpendicular to the plane it is associated with. This image line is perpendicular to every line of the plane which passes through the origin, in particular the original line (point of the projective plane). All lines that are perpendicular to the original line at the origin lie in the unique plane which is orthogonal to the original line, that is, the image plane under the association. Thus, the image line lies in the image plane and the association preserves incidence.

Matrix form

As in the above example, matrices can be used to represent dualities. Let π be a duality of PG(n, K) for n > 1 and let φ be the associated sesquilinear form (with companion antiautomorphism σ) on the underlying (n + 1)-dimensional vector space V. Given a basis { e_i } of V, we may represent this form by:

${\displaystyle \phi (\mathbf{u},\mathbf{x})={\mathbf{u}}^{\mathsf{T}}G({\mathbf{x}}^{\sigma }),}$

where G is a nonsingular (n + 1) × (n + 1) matrix over K and the vectors are written as column vectors. The notation x^σ means that the antiautomorphism σ is applied to each coordinate of the vector x.

Now define the duality in terms of point coordinates by:

${\displaystyle \pi (\mathbf{x})=(G({\mathbf{x}}^{\sigma }){)}^{\mathsf{T}}.}$

Polarity

A duality that is an involution (has order two) is called a polarity. It is necessary to distinguish between polarities of general projective spaces and those that arise from the slightly more general definition of plane duality. It is also possible to give more precise statements in the case of a finite geometry, so we shall emphasize the results in finite projective planes.

Polarities of general projective spaces

If π is a duality of PG(n, K), with K a skewfield, then a common notation is defined by π(S) = S^⊥ for a subspace S of PG(n, K). Hence, a polarity is a duality for which S^⊥⊥ = S for every subspace S of PG(n, K). It is also common to bypass mentioning the dual space and write, in terms of the associated sesquilinear form:

${\displaystyle S^{\mathrm{\perp }}=\{\mathbf{u}\text{ in }V:\phi (\mathbf{u},\mathbf{x})=0\text{ for all }\mathbf{x}\text{ in }S\}.}$

A sesquilinear form φ is reflexive if φ(u, x) = 0 implies φ(x, u) = 0.

A duality is a polarity if and only if the (nondegenerate) sesquilinear form defining it is reflexive.

Polarities have been classified, a result of Birkhoff & von Neumann (1936) that has been reproven several times. Let V be a (left) vector space over the skewfield K and φ be a reflexive nondegenerate sesquilinear form on V with companion anti-automorphism σ. If φ is the sesquilinear form associated with a polarity then either:

1. σ = id (hence, K is a field) and φ(u, x) = φ(x, u) for all u, x in V, that is, φ is a bilinear form. In this case, the polarity is called orthogonal (or ordinary). If the characteristic of the field K is two, then to be in this case there must exist a vector z with φ(z, z) ≠ 0, and the polarity is called a pseudo polarity.
2. σ = id (hence, K is a field) and φ(u, u) = 0 for all u in V. The polarity is called a null polarity (or a symplectic polarity) and can only exist when the projective dimension n is odd.
3. σ² = id ≠ σ (here K need not be a field) and φ(u, x) = φ(x, u)^σ for all u, x in V. Such a polarity is called a unitary polarity (or a Hermitian polarity).

A point P of PG(n, K) is an absolute point (self-conjugate point) with respect to polarity ⊥ if P I P^⊥. Similarly, a hyperplane H is an absolute hyperplane (self-conjugate hyperplane) if H^⊥ I H. Expressed in other terms, a point x is an absolute point of polarity π with associated sesquilinear form φ if φ(x, x) = 0 and if φ is written in terms of matrix G, x^T G x^σ = 0.

The set of absolute points of each type of polarity can be described. We again restrict the discussion to the case that K is a field.

1. If K is a field whose characteristic is not two, the set of absolute points of an orthogonal polarity form a nonsingular quadric (if K is infinite, this might be empty). If the characteristic is two, the absolute points of a pseudo polarity form a hyperplane.
2. All the points of the space PG(2s + 1, K) are absolute points of a null polarity.
3. The absolute points of a Hermitian polarity form a Hermitian variety, which may be empty if K is infinite.

When composed with itself, the correlation φ(x_P) = x_H (in any dimension) produces the identity function, so it is a polarity. The set of absolute points of this polarity would be the points whose homogeneous coordinates satisfy the equation:

x_H ⋅ x_P = x₀x₀ + x₁x₁ + ... + x_nx_n = x₀² + x₁² + ... + x_n² = 0.

Which points are in this point set depends on the field K. If K = R then the set is empty, there are no absolute points (and no absolute hyperplanes). On the other hand, if K = C the set of absolute points form a nondegenerate quadric (a conic in two-dimensional space). If K is a finite field of odd characteristic the absolute points also form a quadric, but if the characteristic is even the absolute points form a hyperplane (this is an example of a pseudo polarity).

Under any duality, the point P is called the pole of the hyperplane P^⊥, and this hyperplane is called the polar of the point P. Using this terminology, the absolute points of a polarity are the points that are incident with their polars and the absolute hyperplanes are the hyperplanes that are incident with their poles.

Polarities in finite projective planes

By Wedderburn's theorem every finite skewfield is a field and an automorphism of order two (other than the identity) can only exist in a finite field whose order is a square. These facts help to simplify the general situation for finite Desarguesian planes. We have:

If π is a polarity of the finite Desarguesian projective plane PG(2, q) where q = p^e for some prime p, then the number of absolute points of π is q + 1 if π is orthogonal or q^3/2 + 1 if π is unitary. In the orthogonal case, the absolute points lie on a conic if p is odd or form a line if p = 2. The unitary case can only occur if q is a square; the absolute points and absolute lines form a unital.

In the general projective plane case where duality means plane duality, the definitions of polarity, absolute elements, pole and polar remain the same.

Let P denote a projective plane of order n. Counting arguments can establish that for a polarity π of P:

The number of non-absolute points (lines) incident with a non-absolute line (point) is even.

Furthermore,

The polarity π has at least n + 1 absolute points and if n is not a square, exactly n + 1 absolute points. If π has exactly n + 1 absolute points then;

1. if n is odd, the absolute points form an oval whose tangents are the absolute lines; or
2. if n is even, the absolute points are collinear on a non-absolute line.

An upper bound on the number of absolute points in the case that n is a square was given by Seib and a purely combinatorial argument can establish:

A polarity π in a projective plane of square order n = s² has at most s³ + 1 absolute points. Furthermore, if the number of absolute points is s³ + 1, then the absolute points and absolute lines form a unital (i.e., every line of the plane meets this set of absolute points in either 1 or s + 1 points).

Poles and polars

Main article: Pole and polar

Pole and polar with respect to circle C. P and Q are inverse points, p is the polar of P, P is the pole of p.

Reciprocation in the Euclidean plane

A method that can be used to construct a polarity of the real projective plane has, as its starting point, a construction of a partial duality in the Euclidean plane.

In the Euclidean plane, fix a circle C with center O and radius r. For each point P other than O define an image point Q so that OP ⋅ OQ = r². The mapping defined by P → Q is called inversion with respect to circle C. The line p through Q which is perpendicular to the line OP is called the polar of the point P with respect to circle C.

Let q be a line not passing through O. Drop a perpendicular from O to q, meeting q at the point P (this is the point of q that is closest to O). The image Q of P under inversion with respect to C is called the pole of q. If a point M is on a line q (not passing through O) then the pole of q lies on the polar of M and vice versa. The incidence preserving process, in which points and lines are transformed into their polars and poles with respect to C is called reciprocation.

In order to turn this process into a correlation, the Euclidean plane (which is not a projective plane) needs to be expanded to the extended euclidean plane by adding a line at infinity and points at infinity which lie on this line. In this expanded plane, we define the polar of the point O to be the line at infinity (and O is the pole of the line at infinity), and the poles of the lines through O are the points of infinity where, if a line has slope s (≠ 0) its pole is the infinite point associated to the parallel class of lines with slope −1/s. The pole of the x-axis is the point of infinity of the vertical lines and the pole of the y-axis is the point of infinity of the horizontal lines.

The construction of a correlation based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane). The correlations constructed in this manner are of order two, that is, polarities.

Algebraic formulation

Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair.

We shall describe this polarity algebraically by following the above construction in the case that C is the unit circle (i.e., r = 1) centered at the origin.

An affine point P, other than the origin, with Cartesian coordinates (a, b) has as its inverse in the unit circle the point Q with coordinates,

${\displaystyle \left(\frac{a}{a^2+b^2},\frac{b}{a^2+b^2}\right).}$

The line passing through Q that is perpendicular to the line OP has equation ax + by = 1.

Switching to homogeneous coordinates using the embedding (a, b) ↦ (a, b, 1), the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:

${\displaystyle \pi :\mathbb{R}{P}^2\to \mathbb{R}{P}^2}$

such that

${\displaystyle \pi \left((x,y,z{)}^{\mathsf{T}}\right)=(x,y,-z).}$

Or, using the alternate notation, π((x, y, z)_P) = (x, y, −z)_L. The matrix of the associated sesquilinear form (with respect to the standard basis) is:

${\displaystyle G=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right).}$

The absolute points of this polarity are given by the solutions of:

${\displaystyle 0=P^{\mathsf{T}}GP=x^2+y^2-z^2,}$

where P^T= (x, y, z). Note that restricted to the Euclidean plane (that is, set z = 1) this is just the unit circle, the circle of inversion.

Synthetic approach

Diagonal triangle P, Q, R of quadrangle A, B, J, K on conic. Polars of diagonal points are colored the same as the points.

The theory of poles and polars of a conic in a projective plane can be developed without the use of coordinates and other metric concepts.

Let C be a conic in PG(2, F) where F is a field not of characteristic two, and let P be a point of this plane not on C. Two distinct secant lines to the conic, say AB and JK determine four points on the conic (A, B, J, K) that form a quadrangle. The point P is a vertex of the diagonal triangle of this quadrangle. The polar of P with respect to C is the side of the diagonal triangle opposite P.

The theory of projective harmonic conjugates of points on a line can also be used to define this relationship. Using the same notation as above;

If a variable line through the point P is a secant of the conic C, the harmonic conjugates of P with respect to the two points of C on the secant all lie on the polar of P.

Properties

There are several properties that polarities in a projective plane have.

Given a polarity π, a point P lies on line q, the polar of point Q if and only if Q lies on p, the polar of P.

Points P and Q that are in this relation are called conjugate points with respect to π. Absolute points are called self-conjugate in keeping with this definition since they are incident with their own polars. Conjugate lines are defined dually.

The line joining two self-conjugate points cannot be a self-conjugate line.

A line cannot contain more than two self-conjugate points.

A polarity induces an involution of conjugate points on any line that is not self-conjugate.

A triangle in which each vertex is the pole of the opposite side is called a self-polar triangle.

A correlation that maps the three vertices of a triangle to their opposite sides respectively is a polarity and this triangle is self-polar with respect to this polarity.

History

The principle of duality is due to Joseph Diaz Gergonne (1771−1859) a champion of the then emerging field of Analytic geometry and founder and editor of the first journal devoted entirely to mathematics, Annales de mathématiques pures et appliquées. Gergonne and Charles Julien Brianchon (1785−1864) developed the concept of plane duality. Gergonne coined the terms "duality" and "polar" (but "pole" is due to F.-J. Servois) and adopted the style of writing dual statements side by side in his journal.

Jean-Victor Poncelet (1788−1867) author of the first text on projective geometry, Traité des propriétés projectives des figures, was a synthetic geometer who systematically developed the theory of poles and polars with respect to a conic. Poncelet maintained that the principle of duality was a consequence of the theory of poles and polars.

Julius Plücker (1801−1868) is credited with extending the concept of duality to three and higher dimensional projective spaces.

Poncelet and Gergonne started out as earnest but friendly rivals presenting their different points of view and techniques in papers appearing in Annales de Gergonne. Antagonism grew over the issue of priority in claiming the principle of duality as their own. A young Plücker was caught up in this feud when a paper he had submitted to Gergonne was so heavily edited by the time it was published that Poncelet was misled into believing that Plücker had plagiarized him. The vitriolic attack by Poncelet was countered by Plücker with the support of Gergonne and ultimately the onus was placed on Gergonne. Of this feud, Pierre Samuel has quipped that since both men were in the French army and Poncelet was a general while Gergonne a mere captain, Poncelet's view prevailed, at least among their French contemporaries.

Chronobiology

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Chronobiology

Overview, including some physiological parameters, of the human circadian rhythm ("biological clock").

Chronobiology is a field of biology that examines timing processes, including periodic (cyclic) phenomena in living organisms, such as their adaptation to solar- and lunar-related rhythms. These cycles are known as biological rhythms. Chronobiology comes from the ancient Greek χρόνος (chrónos, meaning "time"), and biology, which pertains to the study, or science, of life. The related terms chronomics and chronome have been used in some cases to describe either the molecular mechanisms involved in chronobiological phenomena or the more quantitative aspects of chronobiology, particularly where comparison of cycles between organisms is required.

Chronobiological studies include but are not limited to comparative anatomy, physiology, genetics, molecular biology and behavior of organisms related to their biological rhythms. Other aspects include epigenetics, development, reproduction, ecology and evolution.

The subject

Chronobiology studies variations of the timing and duration of biological activity in living organisms which occur for many essential biological processes. These occur (a) in animals (eating, sleeping, mating, hibernating, migration, cellular regeneration, etc.), (b) in plants (leaf movements, photosynthetic reactions, etc.), and in microbial organisms such as fungi and protozoa. They have even been found in bacteria, especially among the cyanobacteria (aka blue-green algae, see bacterial circadian rhythms). The best studied rhythm in chronobiology is the circadian rhythm, a roughly 24-hour cycle shown by physiological processes in all these organisms. The term circadian comes from the Latin circa, meaning "around" and dies, "day", meaning "approximately a day." It is regulated by circadian clocks.

The circadian rhythm can further be broken down into routine cycles during the 24-hour day:

• Diurnal, which describes organisms active during daytime
• Nocturnal, which describes organisms active in the night
• Crepuscular, which describes animals primarily active during the dawn and dusk hours (ex: domestic cats, white-tailed deer, some bats)

While circadian rhythms are defined as regulated by endogenous processes, other biological cycles may be regulated by exogenous signals. In some cases, multi-trophic systems may exhibit rhythms driven by the circadian clock of one of the members (which may also be influenced or reset by external factors). The endogenous plant cycles may regulate the activity of the bacterium by controlling availability of plant-produced photosynthate.

Many other important cycles are also studied, including:

• Infradian rhythms, which are cycles longer than a day. Examples include circannual or annual cycles that govern migration or reproduction cycles in many plants and animals, or the human menstrual cycle.
• Ultradian rhythms, which are cycles shorter than 24 hours, such as the 90-minute REM cycle, the 4-hour nasal cycle, or the 3-hour cycle of growth hormone production.
• Tidal rhythms, commonly observed in marine life, which follow the roughly 12.4-hour transition from high to low tide and back.
• Lunar rhythms, which follow the lunar month (29.5 days). They are relevant e.g. for marine life, as the level of the tides is modulated across the lunar cycle.
• Gene oscillations – some genes are expressed more during certain hours of the day than during other hours.

Within each cycle, the time period during which the process is more active is called the acrophase. When the process is less active, the cycle is in its bathyphase or trough phase. The particular moment of highest activity is the peak or maximum; the lowest point is the nadir.

History

A circadian cycle was first observed in the 18th century in the movement of plant leaves by the French scientist Jean-Jacques d'Ortous de Mairan. In 1751 Swedish botanist and naturalist Carl Linnaeus (Carl von Linné) designed a flower clock using certain species of flowering plants. By arranging the selected species in a circular pattern, he designed a clock that indicated the time of day by the flowers that were open at each given hour. For example, among members of the daisy family, he used the hawk's beard plant which opened its flowers at 6:30 am and the hawkbit which did not open its flowers until 7 am.

The 1960 symposium at Cold Spring Harbor Laboratory laid the groundwork for the field of chronobiology.

It was also in 1960 that Patricia DeCoursey invented the phase response curve, one of the major tools used in the field since.

Franz Halberg of the University of Minnesota, who coined the word circadian, is widely considered the "father of American chronobiology." However, it was Colin Pittendrigh and not Halberg who was elected to lead the Society for Research in Biological Rhythms in the 1970s. Halberg wanted more emphasis on the human and medical issues while Pittendrigh had his background more in evolution and ecology. With Pittendrigh as leader, the Society members did basic research on all types of organisms, plants as well as animals. More recently it has been difficult to get funding for such research on any other organisms than mice, rats, humans and fruit flies.

The role of Retinal Ganglion cells

Melanopsin as a circadian photopigment

In 2002, Hattar and his colleagues showed that melanopsin plays a key role in a variety of photic responses, including pupillary light reflex, and synchronization of the biological clock to daily light-dark cycles. He also described the role of melanopsin in ipRGCs. Using a rat melanopsin gene, a melanopsin-specific antibody, and fluorescent immunocytochemistry, the team concluded that melanopsin is expressed in some RGCs. Using a Beta-galactosidase assay, they found that these RGC axons exit the eyes together with the optic nerve and project to the suprachiasmatic nucleus (SCN), the primary circadian pacemaker in mammals. They also demonstrated that the RGCs containing melanopsin were intrinsically photosensitive. Hattar concluded that melanopsin is the photopigment in a small subset of RGCs that contributes to the intrinsic photosensitivity of these cells and is involved in their non-image forming functions, such as photic entrainment and pupillary light reflex.

Melanopsin cells relay inputs from rods and cones

Light enters the eye and hits the retinal pigmented epithelium (maroon). This excites rods (grey) and cones (blue/red). These cells synapse onto bipolar cells (pink), which stimulate ipRGCs (green) and RGCs (orange). Both RGCs and ipRGCs transmit information to the brain through the optic nerve. Furthermore, light can directly stimulate the ipRGCs through its melanopsin photopigment. The ipRGCs uniquely project to the superchiasmatic nucleus, allowing the organism to entrain to light-dark cycles.

Hattar, armed with the knowledge that melanopsin was the photopigment responsible for the photosensitivity of ipRGCs, set out to study the exact role of the ipRGC in photoentrainment. In 2008, Hattar and his research team transplanted diphtheria toxin genes into the mouse melanopsin gene locus to create mutant mice that lacked ipRGCs. The research team found that while the mutants had little difficulty identifying visual targets, they could not entrain to light-dark cycles. These results led Hattar and his team to conclude that ipRGCs do not affect image-forming vision, but significantly affect non-image forming functions such as photoentrainment.

Distinct ipRGCs

Further research has shown that ipRGCs project to different brain nuclei to control both non-image forming and image forming functions. These brain regions include the SCN, where input from ipRGCs is necessary to photoentrain circadian rhythms, and the olivary pretectal nucleus (OPN), where input from ipRGCs control the pupillary light reflex. Hattar and colleagues conducted research that demonstrated that ipRGCs project to hypothalamic, thalamic, stratal, brainstem and limbic structures. Although ipRGCs were initially viewed as a uniform population, further research revealed that there are several subtypes with distinct morphology and physiology. Since 2011, Hattar's laboratory has contributed to these findings and has successfully distinguished subtypes of ipRGCs.

Diversity of ipRGCs

Hattar and colleges utilized Cre-based strategies for labeling ipRGCs to reveal that there are at least five ipRGC subtypes that project to a number of central targets. Five classes of ipRGCs, M1 through M5, have been characterized to date in rodents. These classes differ in morphology, dendritic localization, melanopsin content, electrophysiological profiles, and projections.

Diversity in M1 cells

Hattar and his co-workers discovered that, even among the subtypes of ipRGC, there can be designated sets that differentially control circadian versus pupillary behavior. In experiments with M1 ipRGCs, they discovered that the transcription factor Brn3b is expressed by M1 ipRGCs that target the OPN, but not by ones that target the SCN. Using this knowledge, they designed an experiment to cross Melanopsin-Cre mice with mice that conditionally expressed a toxin from the Brn3b locus. This allowed them to selectively ablate only the OPN projecting M1 ipRGCS, resulting in a loss of pupil reflexes. However, this did not impair circadian photo entrainment. This demonstrated that the M1 ipRGC consist of molecularly distinct subpopulations that innervate different brain regions and execute specific light-induced functions. This isolation of a 'labeled line' consisting of differing molecular and functional properties in a highly specific ipRGC subtype was an important first for the field. It also underscored the extent to which molecular signatures can be used to distinguish between RGC populations that would otherwise appear the same, which in turn facilitates further investigation into their specific contributions to visual processing.

Psychological impact of light exposure

Previous studies in circadian biology have established that exposure to light during abnormal hours leads to sleep deprivation and disruption of the circadian system, which affect mood and cognitive functioning. While this indirect relationship had been corroborated, not much work had been done to examine whether there was a direct relationship between irregular light exposure, aberrant mood, cognitive function, normal sleep patterns and circadian oscillations. In a study published in 2012, the Hattar Laboratory was able to show that deviant light cycles directly induce depression-like symptoms and lead to impaired learning in mice, independent of sleep and circadian oscillations.

Effect on mood

ipRGCs project to areas of the brain that are important for regulating circadian rhythmicity and sleep, most notably the SCN, subparaventricular nucleus, and the ventrolateral preoptic area. In addition, ipRGCs transmit information to many areas in the limbic system, which is strongly tied to emotion and memory. To examine the relationship between deviant light exposure and behavior, Hattar and his colleagues studied mice exposed to alternating 3.5-hour light and dark periods (T7 mice) and compared them with mice exposed to alternating 12-hour light and dark periods (T24 mice). Compared to a T24 cycle, the T7 mice got the same amount of total sleep and their circadian expression of PER2, an element of the SCN pacemaker, was not disrupted. Through the T7 cycle, the mice were exposed to light at all circadian phases. Light pulses presented at night lead to expression of the transcription factor c-Fos in the amygdala, lateral habenula, and subparaventricular nucleus further implicating light's possible influence on mood and other cognitive functions.

Mice subjected to the T7 cycle exhibited depression-like symptoms, exhibiting decreased preference for sucrose (sucrose anhedonia) and exhibiting more immobility than their T24 counterparts in the forced swim test (FST). Additionally, T7 mice maintained rhythmicity in serum corticosterone, however the levels were elevated compared to the T24 mice, a trend that is associated with depression. Chronic administration of the antidepressant Fluoxetine lowered corticosterone levels in T7 mice and reduced depression-like behavior while leaving their circadian rhythms unaffected.

Effect on learning

The hippocampus is a structure in the limbic system that receives projections from ipRGCs. It is required for the consolidation of short-term memories into long-term memories as well as spatial orientation and navigation. Depression and heightened serum corticosterone levels are linked to impaired hippocampal learning. Hattar and his team analyzed the T7 mice in the Morris water maze (MWM), a spatial learning task that places a mouse in a small pool of water and tests the mouse's ability to locate and remember the location of a rescue platform located just below the waterline. Compared to the T24 mice, the T7 mice took longer to find the platform in subsequent trials and did not exhibit a preference for the quadrant containing the platform. In addition, T7 mice exhibited impaired hippocampal long-term potentiation (LTP) when subjected to theta burst stimulation (TBS). Recognition memory was also affected, with T7 mice failing to show preference for novel objects in the novel object recognition test.

Necessity of ipRGCs

Mice without (Opn4^aDTA/aDTA mice) are not susceptible to the negative effects of an aberrant light cycle, indicating that light information transmitted through these cells plays an important role in regulation of mood and cognitive functions such as learning and memory.

Research developments

Light and melatonin

More recently, light therapy and melatonin administration have been explored by Alfred J. Lewy (OHSU), Josephine Arendt (University of Surrey, UK) and other researchers as a means to reset animal and human circadian rhythms. Additionally, the presence of low-level light at night accelerates circadian re-entrainment of hamsters of all ages by 50%; this is thought to be related to simulation of moonlight.

In the second half of 20th century, substantial contributions and formalizations have been made by Europeans such as Jürgen Aschoff and Colin Pittendrigh, who pursued different but complementary views on the phenomenon of entrainment of the circadian system by light (parametric, continuous, tonic, gradual vs. nonparametric, discrete, phasic, instantaneous, respectively).

Chronotypes

Humans can have a propensity to be morning people or evening people; these behavioral preferences are called chronotypes for which there are various assessment questionnaires and biological marker correlations.

Mealtimes

There is also a food-entrainable biological clock, which is not confined to the suprachiasmatic nucleus. The location of this clock has been disputed. Working with mice, however, Fuller et al. concluded that the food-entrainable clock seems to be located in the dorsomedial hypothalamus. During restricted feeding, it takes over control of such functions as activity timing, increasing the chances of the animal successfully locating food resources.

Diurnal patterns on the Internet

In 2018 a study published in PLoS ONE showed how 73 psychometric indicators measured on Twitter Content follow a diurnal pattern. A followup study appeared on Chronobiology International in 2021 showed that these patterns were not disrupted by the 2020 UK lockdown.

Modulators of circadian rhythms

In 2021, scientists reported the development of a light-responsive days-lasting modulator of circadian rhythms of tissues via Ck1 inhibition. Such modulators may be useful for chronobiology research and repair of organs that are "out of sync".

Other fields

Chronobiology is an interdisciplinary field of investigation. It interacts with medical and other research fields such as sleep medicine, endocrinology, geriatrics, sports medicine, space medicine and photoperiodism.