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Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and angles.

There are precisely five Platonic solids (shown below):

The name of each figure is derived from its number of faces: respectively 4, 6, 8, 12, and 20.

The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. They are named for the ancient Greek philosopherPlato who theorized that the classical elements were constructed from the regular solids.

History

The Platonic solids have been known since antiquity. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato (Atiyah and Sutcliffe 2003). Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.

The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests he may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.

The Platonic solids feature prominently in the philosophy of Plato for whom they are named. Plato wrote about them in the dialogue Timaeusc.360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. Moreover, the solidity of the Earth was believed to be due to the fact that the cube is the only regular solid that tesselatesEuclidean space. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

Euclid gave a complete mathematical description of the Platonic solids in the Elements; the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in "Elements". Much of the information in Book XIII is probably derived from the work of Theaetetus.

In the 16th century, the GermanastronomerJohannes Kepler attempted to find a relation between the five extraterrestrial planets known at that time and the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler laid out a model of thesolar system in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came the recognition that heat transfer, conduction (or heat conduction) is the transfer of thermal energy between regions of matter due to a temperature gradient. Heat always flows from a region of higher temperature to a region of lower temperature, and results in the elimination of temperature differences by establishing thermal equilibrium. Conduction takes place in all forms of matter, viz. solids, liquids, gases and plasmas, but does not require any bulk motion of matter. In solids, it is due to the combination of vibrations of the molecules in a lattice or phonons with the energy transported by free electrons. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion.

In the engineering sciences, heat transfer includes the processes of thermal radiation, convection, and sometimes mass transfer and often more than one of these processes occurs in a given situation.

Overview

On a microscopic scale, conduction occurs as rapidly moving or vibrating atoms and molecules interact with neighboring particles, transferring some of their kinetic energy. Heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. Conduction is greater in solids because the network of relatively fixed spacial relationships between atoms helps to transfer energy between them by vibration.

As density decreases so does conduction. Therefore, fluids (and especially gases) are less conductive. This is due to the large distance between atoms in a gas: fewer collisions between atoms means less conduction. Conductivity of gases increases with temperature. Conductivity increases with increasing pressure from vacuum up to a critical point that the density of the gas is such that molecules of the gas may be expected to collide with each other before they transfer heat from one surface to another. After this point conductivity increases only slightly with increasing pressure and density.

Thermal contact conductance is the study of heat conduction between solid bodies in contact. A temperature drop is often observed at the interface between the two surfaces in contact. This phenomenon is said to be a result of a thermal contact resistance existing between the contacting surfaces. Interfacial thermal resistance is a measure of an interface's resistance to thermal flow. This thermal resistance differs from contact resistance, as it exists even at atomically perfect interfaces. Understanding the thermal resistance at the interface between two materials is of primary significance in the study of its thermal properties. Interfaces often contribute significantly to the observed properties of the materials.

The inter-molecular transfer of energy could be primarily by elastic impact as in fluids or by free electron diffusion as in metals or phonon vibration as in insulators. In insulators the heat flux is carried almost entirely by phonon vibrations.

Metals (e.g. copper, platinum, gold,etc.) are usually the best conductors of thermal energy. This is due to the way that metals are chemically bonded: metallic bonds (as opposed to covalent or ionic bonds) have free-moving electrons which are able to transfer thermal energy rapidly through the metal. The "electron fluid" of a conductive metallic solid conducts nearly all of the heat flux through the solid. Phonon flux is still present, but carries less than 1% of the energy. Electrons also conduct electric current through conductive solids, and the thermal and electrical conductivities of most metals have about the same ratio. A good electrical conductor, such as copper, usually also conducts heat well. The Peltier-Seebeck effect exhibits the propensity of electrons to conduct heat through an electrically conductive solid. Thermoelectricity is caused by the relationship between electrons, heat fluxes and electrical currents. Heat conduction within a solid is directly analogous to diffusion of particles within a fluid, in the situation where there are no fluid currents.

To quantify the ease with which a particular medium conducts, engineers employ the thermal conductivity, also known as the conductivity constant or conduction coefficient, k. In thermal conductivityk is defined as "the quantity of heat, Q, transmitted in time (t) through a thickness (L), in a direction normal to a surface of area (A), due to a temperature difference (ΔT) [...]." Thermal conductivity is a material propertythat is primarily dependent on the medium'sphase, temperature, density, and molecular bonding. Thermal effusivity is a quantity derived from conductivity which is a measure of its ability to exchange thermal energy with its surroundings.

Steady-state conduction

Steady state conduction is the form of conduction which happens when the temperature difference driving the conduction is constant so that after an equilibration time, the spatial distribution of temperatures (temperature field) in the conducting object does not change a

Heat engine

In thermodynamics, a heat engine is a system that performs the conversion of heatenergy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates heat that brings the working substance in the high temperature state. The working substance generates work in the "working body" of the engine while transferring heat to the colder "sink" until it reaches a low temperature state. During this process some of the heat is converted into work by exploiting the properties of the working substance. The working substance can be any system with a non-zero heat capacity, but it usually is a gas or liquid.

Heat engines are often confused with the cycles they attempt to mimic. Typically when describing the physical device the term 'engine' is used. When describing the model the term 'cycle' is used.

Overview

In thermodynamics, heat engines are often modeled using a standard engineering model such as the Otto cycle. The theoretical model can be refined and augmented with actual data from an operating engine, using tools such as an indicator diagram. Since very few actual implementations of heat engines exactly match their underlying thermodynamic cycles, one could say that a thermodynamic cycle is an ideal case of a mechanical engine. In any case, fully understanding an engine and its efficiency requires gaining a good understanding of the (possibly simplified or idealized) theoretical model, the practical nuances of an actual mechanical engine, and the discrepancies between the two.

In general terms, the larger the difference in temperature between the hot source and the cold sink, the larger is the potential thermal efficiency of the cycle. On Earth, the cold side of any heat engine is limited to being close to the ambient temperature of the environment, or not much lower than 300 Kelvin, so most efforts to improve the thermodynamic efficiencies of various heat engines focus on increasing the temperature of the source, within material limits. The maximum theoretical efficiency of a heat engine (which no engine ever obtains) is equal to the temperature difference between the hot and cold ends divided by the temperature at the hot end, all expressed in absolute temperature or kelvins.

The efficiency of various heat engines proposed or used today ranges from 3 percent (97 percent waste heat) for the OTEC ocean power proposal through 25 percent for most automotive engines, to 45 percent for a supercritical coal plant, to about 60 percent for a steam-cooled combined cyclegas turbine.

All of these processes gain their efficiency (or lack thereof) due to the temperature drop across them.

Power

Heat engines can be characterized by their specific power, which is typically given in kilowatts per litre of engine displacement (in the U.S. also horsepower per cubic inch). The result offers an approximation of the peak-power output of an engine. This is not to be confused with fuel efficiency, since high-efficiency often requires a lean fuel-air ratio, and thus lower power density. A modern high-performance car engine makes in excess of 75 kW/L (1.65 hp/in³).

Everyday examples

Examples of everyday heat engines include the steam engine, the diesel engine, and the gasoline (petrol) engine in an automobile. A common toy that is also a heat engine is a drinking bird. Also the stirling engine is a heat engine. All of these familiar heat engines are powered by the expansion of heated gases. The general surroundings are the heat sink, providing relatively cool gases which, when heated, expand rapidly to drive the mechanical motion of the engine.

Examples of heat engines

It is important to note that although some cycles have a typical combustion location (internal or external), they often can be implemented as the other combustion cycle. For example, John Ericsson developed an external heated engine running on a cycle very much like the earlier Diesel cycle. In addition, the externally heated engines can often be implemented in open or closed cycles.

What this boils down to is there are thermodynamic cycles and a large number of ways of implementing them with mechanical devices called engines.

Phase change cycles

In these cycles and engines, the working fluids are gases and liquids. The engine converts the working fluid from a gas to a liquid, from liquid to gas, or both, generating work from the fluid expansion or compression.

Light scattering

Light scattering is a form of scattering in which light is the form of propagating energy which is scattered. Light scattering can be thought of as the deflection of a ray from a straight path, for example by irregularities in the propagation medium, particles, or in the interface between two media. Deviations from the law of reflection due to irregularities on a surface are also usually considered to be a form of scattering. When these irregularities are considered to be random and dense enough that their individual effects average out, this kind of scattered reflection is commonly referred to as diffuse reflection.

Most objects that one sees are visible due to light scattering from their surfaces. Indeed, this is our primary mechanism of physical observation. Scattering of light depends on the wavelength or frequency of the light being scattered. Since visible light has wavelength on the order of a micron, objects much smaller than this cannot be seen, even with the aid of a microscope. Colloidal particles as small as 1 Âµm have been observed directly in aqueous suspension.

The transmission of various frequencies of light is essential for applications ranging from window glass to fiber optic transmission cables and infrared (IR) heat-seeking missile detection systems. Light propagating through an optical system can be attenuated by absorption, reflection and scattering.

Introduction

The interaction of light with matter can shed light on important information about the structure and dynamics of the material being examined. If the scattering centers are in motion, then the scattered radiation is Doppler shifted. An analysis of the spectrum of scattered light can thus yield information regarding the motion of the scattering center. Periodicity or structural repetition in the scattering medium will cause interference in the spectrum of scattered light. Thus, a study of the scattered light intensity as a function of scattering angle gives information about the structure, spatial configuration, or morphology of the scattering medium. With regards to light scattering in liquids and solids, primary material considerations include:

  • Crystalline structure: How close-packed its atoms or molecules are, and whether or not the atoms or molecules exhibit the long-range order evidenced in crystalline solids.
  • Glassy structure: Scattering centers include fluctuations in density and/or composition.
  • Microstructure: Scattering centers include internal surfaces in liquids due largely to density fluctuations, and microstructural defects in solids such as grains, grain boundaries, and microscopic pores.

In the process of light scattering, the most critical factor is the length scale of any or all of these structural features relative to the wavelength of the light being scattered.

An extensive review of light scattering in fluids has covered most of the mechanisms which contribute to the spectrum of scattered light in liquids, including density, anisotropy, and concentration fluctuations. Thus, the study of light scattering by thermally driven density fluctuations (or Brillouin scattering) has been utilized successfully for the measurement of structural relaxation and viscoelasticity in liquids, as well as phase separation, vitrification and compressibility in glasses. In addition, the introduction of dynamic light scattering and photon correlation spectroscopy has made possible the measurement of the time dependence of spatial correlations in liquids and glasses in the relaxation time gap between 10−6 and 10−2 s in addition to even shorter time scales – or faster relaxation events. It has therefore become quite clear that light scattering is an extremely useful tool for monitoring the dynamics of structural relaxation in glasses on various temporal and spatial scales and therefore provides an ideal tool for quantifying the capacity of various glass compositions for guided light wave transmission well into the far infrared portions of the electromagnetic spectrum.

Types of scattering