Thursday, September 23, 2021
Dr John D.- Phlogiston Theory, Mass and the Birth and Demise of Gravity.
INTERTRONICS - How does plasma surface treatment improve bond strength?
Open Readings - "Move into Nano-World by Femtosecond Lasers", Wolfgang Kautek | Open Rea...
Anders Hagfeldt "Baking your own solar cells"
Fedor Jelezko "Quantum sensing with diamonds "
Boaz Almog - QuantumLevitation (dry ice nitrogen dipped/etc. superconductor coated (with quantum?? or femto-yocto or engineered or natural gold) sapphire wafer and a permanent magnet)
09-23-2021-0850 - Levitating Strip & Aluminum and Mercury Continuum Mechanics
The Royal Institution Levitating Superconductor on a Möbius strip
Nile Red aluminum and mercury
09-23-2021-0823 - minimum total potential energy principle
The minimum total potential energy principle is a fundamental concept used in physics and engineering. It dictates that at low temperatures a structure or body shall deform or displace to a position that (locally) minimizes the total potential energy, with the lost potential energy being converted into kinetic energy (specifically heat).
Some examples[edit]
- A free proton and free electron will tend to combine to form the lowest energy state (the ground state) of a hydrogen atom, the most stable configuration. This is because that state's energy is 13.6 electron volts (eV) lower than when the two particles separated by an infinite distance. The dissipation in this system takes the form of spontaneous emission of electromagnetic radiation, which increases the entropy of the surroundings.
- A rolling ball will end up stationary at the bottom of a hill, the point of minimum potential energy. The reason is that as it rolls downward under the influence of gravity, friction produced by its motion transfers energy in the form of heat of the surroundings with an attendant increase in entropy.
- A protein folds into the state of lowest potential energy. In this case, the dissipation takes the form of vibration of atoms within or adjacent to the protein.
Structural mechanics[edit]
The total potential energy, , is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, associated to the applied forces:[1]
(1)
This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy:[1]
(2)
The principle of minimum total potential energy may be derived as a special case of the virtual work principle for elastic systems subject to conservative forces.
The equality between external and internal virtual work (due to virtual displacements) is:
(3)
where
- = vector of displacements
- = vector of distributed forces acting on the part of the surface
- = vector of body forces
In the special case of elastic bodies, the right-hand-side of (3) can be taken to be the change, , of elastic strain energy U due to infinitesimal variations of real displacements. In addition, when the external forces are conservative forces, the left-hand-side of (3) can be seen as the change in the potential energyfunction V of the forces. The function V is defined as:[2]
where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, (3) becomes:
This leads to (2) as desired. The variational form of (2) is often used as the basis for developing the finite element method in structural mechanics.
https://en.wikipedia.org/wiki/Minimum_total_potential_energy_principle
https://en.wikipedia.org/wiki/Tendril_perversion
09-23-2021-0822 - tendril perversion chirality darwin 1865
Tendril perversion, often referred to in context as simply perversion, is a geometric phenomenon found in helicalstructures such as plant tendrils, in which a helical structure forms that is divided into two sections of opposite chirality, with a transition between the two in the middle.[1] A similar phenomenon can often be observed in kinked helical cables such as telephone handset cords.[2]
The phenomenon was known to Charles Darwin,[3] who wrote in 1865,
The term "tendril perversion" was coined by Goriely and Tabor in 1998 based on the word perversion found in the 19th Century science literature. "Perversion" is a transition from one chirality to another and was known to James Clerk Maxwell, who attributed it to the topologist J. B. Listing.[3][5]
Tendril perversion can be viewed as an example of spontaneous symmetry breaking, in which the strained structure of the tendril adopts a configuration of minimum energy while preserving zero overall twist.[1]
Tendril perversion has been studied both experimentally and theoretically. Gerbode et al. have made experimental studies of the coiling of cucumber tendrils.[6][7] A detailed study of a simple model of the physics of tendril perversion was made by MacMillen and Goriely in the early 2000s.[1] Liu et al. showed in 2014 that "the transition from a helical to a hemihelical shape, as well as the number of perversions, depends on the height to width ratio of the strip's cross-section."[2]
Generalized tendril perversions were put forward by Silva et al., to include perversions that can be intrinsically produced in elastic filaments, leading to a multiplicity of geometries and dynamical properties.[8]
https://en.wikipedia.org/wiki/Tendril_perversion
09-23-2021-0820 - horosphere horopter
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle.
A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface).
A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb
https://en.wikipedia.org/wiki/Horosphere
The horopter was originally defined in geometric terms as the locus of points in space that make the same angle at each eye with the fixation point, although more recently in studies of binocular vision it is taken to be the locus of points in space that have the same disparity as fixation. This can be defined theoretically as the points in space that project on corresponding points in the two retinas, that is, on anatomically identical points. The horopter can be measured empirically in which it is defined using some criterion.
The concept of horopter can then be extended as a geometrical locus of points in space where a specific condition is met:
- the binocular horopter is the locus of iso-disparity points in space;
- the oculomotor horopter is the locus of iso-vergence points in space.
As other quantities that describe the functional principles of the visual system, it is possible to provide a theoretical description of the phenomenon. The measurement with psycho-physical experiments usually provide an empirical definition that slightly deviates from the theoretical one. The underlying theory is that this deviation represents an adaptation of the visual system to the regularities that can be encountered in natural environments.[1][2]
https://en.wikipedia.org/wiki/Horopter
09-23-2021-0817 - Space Cardioid
The space cardioid is a 3-dimensional curve derived from the cardioid. It has a parametric representation using trigonometric functions.
Definition[edit]
The general form of the equation is most easily understood in parametric form, as follows:
See also[edit]
09-23-2021-0816 - plane stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2 × 2 matrix rather than 3 × 3). [1] A related notion, plane strain, is often applicable to very thick members.
Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure. In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it.[1]
In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor at each point must be complemented with bending terms.
https://en.wikipedia.org/wiki/Plane_stress
09-23-2021-0815 - hemihelix
A hemihelix is a curved geometric shape consisting of a series of heliceswith alternating chirality, connected by a perversion at the reversals.[1][2]
The formation of hemihelices with periodic distributions of perversions in slender structures is understood in terms of competing buckling instabilities generated by in-plane stresses.[3]
https://en.wikipedia.org/wiki/Hemihelix
09-23-2021-0810 - Dinostratus 390 320
Dinostratus (Greek: Δεινόστρατος; c. 390 – c. 320 BCE) was a Greek mathematician and geometer, and the brother of Menaechmus. He is known for using the quadratrix to solve the problem of squaring the circle.
Dinostratus | |
|---|---|
| Born | c. 390 BCE |
| Died | c. 320 BCE |
| Nationality | Greek |
| Known for | Quadratrix of Dinostratus Dinostratus' theorem |
| Scientific career | |
| Fields | Mathematics |
Life and work[edit]
Dinostratus' chief contribution to mathematics was his solution to the problem of squaring the circle. To solve this problem, Dinostratus made use of the trisectrix of Hippias, for which he proved a special property (Dinostratus' theorem) that allowed him the squaring of the circle. Due to his work the trisectrix later became known as the quadratrix of Dinostratus as well.[1] Although Dinostratus solved the problem of squaring the circle, he did not do so using ruler and compass alone, and so it was clear to the Greeks that his solution violated the foundational principles of their mathematics.[1] Over 2,200 years later Ferdinand von Lindemann would prove that it is impossible to square a circle using straight edge and compass alone.
https://en.wikipedia.org/wiki/Dinostratus
Ancient Greek and Hellenistic mathematics (Euclidean geometry)
Mathematicians
(timeline)
Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus
Treatises
Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner
Problems
Constructible numbers Angle trisection Doubling the cube Squaring the circle Problem of Apollonius
Concepts/definitions
Angle Central Inscribed Chord Circles of Apollonius Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center
Results
In Elements
Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon
Apollonius
Apollonius's theorem
Other
Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Law of sines Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus
Centers
Cyrene Library of Alexandria Platonic Academy
Other
Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction
Authority control Edit this at Wikidata
Integrated Authority File (Germany) VIAF 1 WorldCat
In mathematics, a quadratrix (from the Latin word quadrator, squarer) is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
https://en.wikipedia.org/wiki/Quadratrix
In geometry, a strophoid is a curve generated from a given curve C and points A (the fixed point) and O(the pole) as follows: Let L be a variable line passing through O and intersecting C at K. Now let P1 and P2 be the two points on L whose distance from K is the same as the distance from A to K. The locus of such points P1 and P2 is then the strophoid of C with respect to the pole O and fixed point A. Note that AP1 and AP2 are at right angles in this construction.
In the special case where C is a line, A lies on C, and O is not on C, then the curve is called an oblique strophoid. If, in addition, OA is perpendicular to C then the curve is called a right strophoid, or simply strophoid by some authors. The right strophoid is also called the logocyclic curve or foliate.
https://en.wikipedia.org/wiki/Strophoid
09-23-2021-0800 - center of curvature osculating circle concave mirror with light rays example curve plane osculate
In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve.[1] The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding to study of lenses and mirrors.
It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens/mirror itself.[2]
https://en.wikipedia.org/wiki/Center_of_curvature
In mathematics, osculate, meaning to touch (from the Latin osculum meaning kiss), may refer to:
- osculant, an invariant of hypersurfaces
- osculating circle
- osculating curve
- osculating plane
- osculating orbit
The obsolete Quinarian system of biological classification attempted to group creatures into circles which could touch or overlap with adjacent circles, a phenomenon called 'osculation'.
https://en.wikipedia.org/wiki/Osculate
In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold.
The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
https://en.wikipedia.org/wiki/Osculating_plane