The law of the prolonged sine was observed when measuring strength of the reaction of the plant stems and roots in response to turning from their usual vertical orientation. Such organisms maintained their usual vertical growth, and, if turned, start bending back toward the vertical. The prolonged sine law was observed when measuring the dependence of the bending speed from the angle of reorientation.
It was observed that deviation from the desired growth direction by more than the 90 degrees causes further increase of the bending speed. After turning the 135 degrees the reoriented plant or fungi understands that it is placed "head down" and bends faster than turned by just 45 degrees. Poul Larsen in 1962 [1] [2] proposed, that the intensity of the gravitropic reaction (bending rate) is proportional to
where α is the angle of reorientation, g - gravity vector and constants a and b are determined experimentally. [3]
Following the popular hypothesis of the mechanism of the plant spatial orientation, the bending from the horizontal position is caused by some small heavy particles that after turning put the pressure on the side cell (statocyte) wall, irritating some system and activating the bending process. The pressure of such particle to the cell was would be proportional to the sine of the reorientation angle, being maximal at 90° reorientation. It would be equal for the reorientations by both 45° and 135°. The prolonged sine law concludes that there are very significant deviations from such a predicted reaction.
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
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 .
Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation, in contrast to rotation around a fixed axis.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of Newtonian mechanics.
In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of the flavour-changing weak interaction. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violation. This matrix was introduced for three generations of quarks by Makoto Kobayashi and Toshihide Maskawa, adding one generation to the matrix previously introduced by Nicola Cabibbo. This matrix is also an extension of the GIM mechanism, which only includes two of the three current families of quarks.
Gravitropism is a coordinated process of differential growth by a plant in response to gravity pulling on it. It also occurs in fungi. Gravity can be either "artificial gravity" or natural gravity. It is a general feature of all higher and many lower plants as well as other organisms. Charles Darwin was one of the first to scientifically document that roots show positive gravitropism and stems show negative gravitropism. That is, roots grow in the direction of gravitational pull and stems grow in the opposite direction. This behavior can be easily demonstrated with any potted plant. When laid onto its side, the growing parts of the stem begin to display negative gravitropism, growing upwards. Herbaceous (non-woody) stems are capable of a degree of actual bending, but most of the redirected movement occurs as a consequence of root or stem growth outside. The mechanism is based on the Cholodny–Went model which was proposed in 1927, and has since been modified. Although the model has been criticized and continues to be refined, it has largely stood the test of time.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).
In optics, a dispersive prism is an optical prism that is used to disperse light, that is, to separate light into its spectral components. Different wavelengths (colors) of light will be deflected by the prism at different angles. This is a result of the prism material's index of refraction varying with wavelength (dispersion). Generally, longer wavelengths (red) undergo a smaller deviation than shorter wavelengths (blue). The dispersion of white light into colors by a prism led Sir Isaac Newton to conclude that white light consisted of a mixture of different colors.
In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as times of day, and fractional parts of real numbers.
Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and , the law of cosines states:
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron.
Isaac Newton's sine-squared law of air resistance is a formula that implies the force on a flat plate immersed in a moving fluid is proportional to the square of the sine of the angle of attack. Although Newton did not analyze the force on a flat plate himself, the techniques he used for spheres, cylinders, and conical bodies were later applied to a flat plate to arrive at this formula. In 1687, Newton devoted the second volume of his Principia Mathematica to fluid mechanics.
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