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Dihedral Angle

Dihedral angle

:In aerospace engineering, the dihedral is the angle between the two wings; see dihedral. ---- In geometry, the angle between two planes is called their dihedral angle. It can be defined as the angle between two lines normal to the planes. Every polyhedron, regular and nonregular, convex and concave, has a dihedral angle at every edge. A dihedral angle (also called the face angle) is the angle at which two adjacent faces meet. Every dihedral angle in a particular Platonic solid has the same value, called "the" dihedral angle. Thus, the dihedral angle of a cube is 90°, while the dihedral angle of a dodecahedron is 116° 34′. "The" dihedral angle of each Platonic solid is:

Name	exact dihedral angle (in radians)	approximate dihedral angle (in degrees)
Tetrahedron	arccos(1/3)	70.53°
Hexahedron or Cube	π/2	90°
Octahedron	π − arccos(1/3)	109.47°
Dodecahedron	2·arctan(φ)	116.56°
Icosahedron	2·arctan(φ + 1)	138.19°

where φ = (1 + √5)/2 is the golden mean.

External links

- [http://kjmaclean.com/Geometry/Platonic.html Analysis of the 5 Regular Polyhedra] gives a step-by-step derivation of these exact values. category:Euclidean solid geometry Category:Angle

Aerospace engineering

Aerospace engineering is the branch of engineering that concerns aircraft, spacecraft and related topics. It is often called aeronautical engineering, particularly when referring solely to aircraft, and astronautical engineering, when referring to spacecraft. Some of the elements of aerospace engineering are:
- Aerodynamics - the study of fluid flow around objects such as wings or through objects such as wind tunnels (see also lift and aeronautics)
- Propulsion - the energy to move a vehicle through the air (or in outer space) is provided by internal combustion engines, jet engines, or rockets (see also propeller and Spacecraft Propulsion)
- Control engineering - the study of mathematical modelling of systems and designing them in order that they behave in the desired way
- Structures - design of the physical configuration of the craft to withstand the forces encountered during flight. Aerospace engineering aims very much at keeping structures lightweight.
- Materials science - related to structures, aerospace engineering also studies the materials of which the aerospace structures are to be built. New materials with very specific properties are invented, or existing ones are modified to improve their performance.
- Aeroelasticity - the interaction of aerodynamic forces and structural flexibility, potentially causing flutter, divergence, etc
- Computer science - specifically concerning the design and programming of any computer systems on board an aircraft or spacecraft and the simulation of systems. The basis of most of these elements lies in theoretical mathematics, such as fluid dynamics for aerodynamics or the equations of motion for flight dynamics. However, there is also a large empirical component. Historically, this empirical component was derived from testing of scale models and prototypes, either in wind tunnels or in the free atmosphere. More recently, advances in computing have enabled the use of computational fluid dynamics to simulate the behavior of fluid, reducing time and expense spent on wind-tunnel testing. Additionally, aerospace engineering addresses the integration of all components that constitute an aerospace vehicle (subsystems including power, communications, thermal control, life support, etc.) and its life cycle (design, temperature, pressure, radiation, velocity, life time...), leading to extraordinary challenges and solutions specific to the domain of aerospace systems engineering.

Aerospace Engineering Degrees

Aerospace (or Aeronautical) Engineering can be studied at the Bachelors, Masters, and Ph.D levels at several universities, although it is often regarded as a subset of Mechanical Engineering. Some universities offering Aerospace Engineering degrees are the Ecole Nationale Supérieure de l'Aéronautique et de l'Espace (also known as SUPAERO) in France, the Embry-Riddle Aeronautical University, The University of Bristol, the University of Southampton and the specialist Cranfield University in the United Kingdom, the University of New South Wales in Australia, and Carleton University in Ottawa, Canada.

Challenging Aerospace Engineering competitions

- Centennial Challenges - NASA prize contests

National Aerospace Agencies

- National Aeronautics and Space Administration (NASA)
- European Space Agency (ESA)

Renown degree granting Universites

- École Nationale Supérieure de l'Aéronautique et de l'Espace (SUPAERO)
- Embry-Riddle Aeronautical University

Aerospace Institutions and Collaborations

- American Institute of Aeronautics and Astronautics (AIAA)

Major Aerospace Corporations

- Airbus
- Boeing
- Lockheed Martin
- Northrop Grumman

External Links

- [http://www.soton.ac.uk/AeronauticsAstronautics/ Aerospace Degree] at the [http://www.soton.ac.uk University of Southampton]
- Category:Technology ja:航空工学 th:วิศวกรรมอวกาศยาน

Dihedral

:In geometry, the dihedral is the angle between two planes. See dihedral angle. ---- Dihedral is the upward angle of an aircraft's (or bird's) wings from root to tip, as viewed from directly in front of or behind the aircraft. Downward angled wings are said to have anhedral. wing The purpose of dihedral is to confer stability in the roll axis. A popular but erroneous explanation for how it works is that if the aircraft is perturbed such that one wing is lowered relative to the other, dihedral causes the lower wing to increase its surface area relative to the airflow, thus increasing its lift. This acts to oppose the original roll motion. An alternative way to visualize this is to imagine that the aircraft is sitting in the bottom of a shallow V-shaped "slot" in the air, thanks to the angle of the wings. This position is naturally stable. This explanation is often put forward in many books "explaining" aeronautical principles, but it is wholly false. The true explanation for the action of dihedral is this: If a disturbance causes an aircraft to roll away from its normal position, the aircraft will sideslip in the direction of the down-going wing (see Fig. 1). This creates an airflow component along the length of the wing from tip to root. The dihedral angle can be seen as presenting a positive angle of attack to this lateral flow, hence generating some additional lift. It is this lift which restores the aircraft to its normal attitude (Fig. 2). The apparent increase in surface area is in fact an illusion and contributes no additional lift. angle of attack angle of attack
Most aircraft in the civilian or transport sector use dihedral for stability. Military combat aircraft, in contrast, often have flat wings or anhedral. This reduces inherent stability but increases manoeuvrability. Many military aircraft are in fact inherently unstable, and only fly due to the constant vigilance of on-board computers. angle of attack A side effect of dihedral can be roll-coupling, a tendency for an aircraft to "corkscrew" through the air under certain conditions. This rolling motion, called a dutch roll, is unpleasant to experience for those flying, and can lead to loss of control or can overstress an aircraft. A certain amount of anhedral can combat this effect. Pronounced anhedral is also often seen on aircraft with a high mounted wing, such as the BAe 146, Lockheed Galaxy and others. In such designs, the high mounted wing itself confers roll stability (due to the pendulum effect of the fuselage, engines, etc), so additional dihedral is not required. In fact, such designs can be excessively stable, so the anhedral is added to cancel out some of the roll stability to ensure that the aircraft can be easily manoeuvred. Sweptback wing also increases roll stability. This is another reason for anhedral configuration on military aircraft with high sweep angle, as well as on some airliners, even the low-wing ones such as Tu-134 and Tu-154. An alternative to dihedral for the wing as a whole is to cant the wingtips or outer section of the wing upwards instead. This has the same effect. It is commonly seen on gliders, and some other aircraft. The McDonnell Douglas F-4 Phantom II is one such example, unique among fighters for having dihedral wingtips.

Dihedral

Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

External links

- [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at cut-the-knot
- [http://www.elvenkids.com/tools/geometria/Geometria.php Geometria] An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- [http://www.geogebra.at/ Geogebra] A free dynamic geometry tool, useful for exploring geometry.
- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
- [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
- Stanford Encyclopedia of Philosophy:
- [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
- [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
- [http://www.egwald.com/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens Category:Geometry ko:기하학 ja:幾何学 simple:Geometry zh-min-nan:Kí-hô-ha̍k

Angle

This article is about angles in geometry. For other articles, see Angle (disambiguation) ---- An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry. Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

Units of measure for angles

In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.
- The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The SI system of units uses radians as the (derived) unit for angles. Because of the relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians have particular analytic properties, just as do exponential functions in the base e. (As we've discovered, this is no coincidence).
- The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. 2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians. Degrees are further broken down into minutes of arc and seconds of arc, which are 1/60th and 1/3600th of a degree, respectively. Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc were also used in navigation, and a nautical mile is roughly defined as one minute of arc of the earth's surface.
- The grad, also called grade, gradian or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.
- The point is used in navigation, and is defined as 1/32 of a circle, or exactly 11.25°.
- The full circle or full turns represents the number or fraction of complete full turns. For example, π/2 radians = 90° = 1/4 full circle

Conventions on measurement

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315. In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

Types of angles

An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle. Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:
- Angles smaller than a right angle are called acute angles
- Angles larger than a right angle are called obtuse angles.
- Angles equal to two right angles are called straight angles.
- Angles larger than two right angles are called reflex angles.
- The difference between an acute angle and a right angle is termed the complement of the angle
- The difference between an angle and two right angles is termed the supplement of the angle.

Some facts

In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) × π radians or (n − 2) × 180°. If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles. If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.

A formal definition

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if

\theta

is a Euclidean angle, it is true that :

\cos \theta = \frac

and :

\sin \theta = \frac

for two numbers

x

and

y

. So an angle can be legitimately given by two numbers

x

and

y

, or by a ratio

\frac

. What's more, to any such ratio there corresponds exactly one angle, since :

\frac = \frac = \frac

(i.e., changing the ratio will necessarily change the sin and cos, which in the geometric range

0 < \theta < 2\pi

are one-to-one - one sin or cos corresponds to one

\theta

Angles in different contexts

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula :

\mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Two intersecting planes form an angle, called their dihedral angle. It is defined as the angle between two lines normal to the planes. Also a plane and an intersecting line form an angle. This angle is equal to π/2 radians minus the angle between the intersecting line and the line that goes through the point of intersection and is perpendicular to the plane.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G, :

\cos \theta = \frac
.

Angles in astronomy

In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars. Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928. The obsolete (but still commonly used) format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″

External links

- [http://www.unitconversion.org/unit_converter/angle.html Online Angle Converter - convert between various units of angle, such as degree, radian, grad, gon, minute, second, sign, mil, and so on]
- [http://www.unitconversion.org/unit_converter/angle-v.html Interactive Angle Conversion Table - convert selected unit to all other units of angle]
- [http://www.cut-the-knot.org/triangle/ABisector.shtml Angle Bisectors] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/PerpBiInQuadri.shtml Angle Bisectors and Perpendiculars in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at cut-the-knot
- [http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at cut-the-knot
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units] Category:Elementary geometry Category:Trigonometry
- ko:각도 ja:角度 simple:Angle

Plane

Plane may mean:
- Airplane, a type of fixed-wing aircraft
- Plane (mathematics), theoretical surface which has infinite width and length, zero thickness, and zero curvature
- Plane (tool), a woodworking tool for flattening surfaces
- Plane (cosmology), a theoretical region beyond the known universe, or the region containing the universe itself
- Plane (Dungeons & Dragons), a plane of existence within the role-playing game
- Plane (Magic: The Gathering), a plane of existence within the trading card game
- Planes of Existence (talker), a plane of existence within the talker
- Planing (sailing), a method of travelling quickly across water by using speed to lift the hull out of the water
- Planes Mistaken For Stars, a punk band Plane may also be:
- Platanus, a genus of trees
- Sycamore maple See also: Plain.

Polyhedron

A polyhedron is a geometric shape which in mathematics is defined by three related meanings. In the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. Further generalizing the latter, there are topological polyhedra.

Classical polyhedron

polytope In classical mathematics, a polyhedron (from Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face") is a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes, the faces meet in edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron. A polyhedron is a three-dimensional analog of a polygon. The general term for polygons, polyhedra and even higher dimensional analogs is polytope. Names of polyhedra by number of faces are tetrahedron, pentahedron, hexahedron, octahedron, decahedron, etc. Such terms are in particular used with "regular" in front or implied (in the five cases in which this is applicable) because for each there are different types which have not much in common except having the same number of faces. For a tetrahedron this applies to a much lesser extent, it is always a triangular pyramid.

Characteristics

A polyhedron is:
- Convex if the line segment joining any two points of the polyhedron is contained in the polyhedron or its interior
- Vertex-uniform if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first isometrically onto the second
- Edge-uniform if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second
- Face-uniform if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second
- Regular if it is vertex-uniform, edge-uniform and face-uniform; this implies that every face is a regular polygon
- Quasi-regular if it is vertex-uniform and edge-uniform but not face-uniform, and every face is a regular polygon
- Semi-regular if it is vertex-uniform but neither edge-uniform nor face-uniform, and every face is a regular polygon
- Uniform if it is vertex-uniform and every face is a regular polygon, i.e. it is regular, quasi-regular, or semi-regular. The Euler characteristic relates the number of edges E, vertices V, and faces F of a simply connected polyhedron: V - E + F = 2.

Uniform polyhedra

As conjectured by H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling, there are exactly 75 uniform polyhedra, plus an infinite number of prisms and antiprisms. Uniform polyhedra can be organized as follows:
- 9 regular polyhedra:
- 5 convex solids: Platonic solids
- 4 non-convex solids: Kepler-Poinsot solids
- 15 quasi-regular polyhedra:
- 2 convex solids
- 13 non-convex solids
- semi-regular polyhedra:
- Semi-regular convex polyhedra:
- An infinite number of prism and antiprisms
- 11 other convex solids
- 40 semi-regular non-convex polyhedra:
- 17 stellated Archimedean solids
- 23 other semi-regular non-convex solids Based on this organisation, except the prisms and antiprisms, there are exactly 13 convex semi-regular and quasi-regular polyhedra; they are called the Archimedean solids. As shown here, there are exactly 53 non-convex semi-regular and quasi-regular polyhedra.

Regular polyhedra

Regular polyhedra are vertex-uniform, edge-uniform and face-uniform -- this implies that every face is a regular polygon.

Platonic solids

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There are exactly five regular convex polyhedra. These have been known since ancient times, and are called the Platonic solids:
- Tetrahedron
- Hexahedron or cube
- Octahedron
- Dodecahedron
- Icosahedron

Kepler-Poinsot solids

250px There are exactly four regular non-convex polyhedra: the Kepler-Poinsot solids:
- Small stellated dodecahedron
- Great stellated dodecahedron
- Great icosahedron
- Great dodecahedron

Quasi-regular polyhedra

Quasi-regular means vertex- and edge-uniform but not face-uniform, and every face is a regular polygon; this implies that there are two "kinds" of faces, and that at every edge one of each meet; also that at every vertex four faces meet: alternatingly two of each kind.

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There are exactly two quasi-regular convex polyhedra; both are also Archimedean solids:
- Cuboctahedron (with triangles and squares)
- Icosidodecahedron (with triangles and pentagons) No other convex edge-uniform polyhedra composed of regular polygons exist than the five regular and two quasi-regular convex polyhedra, so edge uniformity and face regularity with convexity implies vertex-uniformity. (There are two other edge-uniform convex polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, but they are not face-regular and not vertex-uniform. These are the duals of the quasi-regular convex polyhedra, and are both members of the Catalan solids.) There are exactly 13 quasi-regular non-convex polyhedra (Hart):
- Dodecadodecahedron
- Great icosidodecahedron
- 3 triambic polyhedra:
- Small triambic icosidodecahedron
- Triambic dodecadodecahedron
- Great triambic icosidodecahedron
- 9 Hemihedra:
- Tetrahemihexahedron
- Octahemioctahedron
- Cubohemioctahedron
- Small icosihemidodecahedron
- Small dodecahemidodecahedron
- Great dodecahemiicosahedron
- Small dodecahemiicosahedron
- Great dodecahemidodecahedron
- Great icosihemidodecahedron

Semi-regular polyhedra

Semi-regular means vertex-uniform but not edge-uniform. There are infinitely many semi-regular convex polyhedra due to two infinite series:
- Prisms (with 2 n-gons and n squares) and

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4.4.3

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4.4.4

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4.4.5

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4.4.6

- Antiprisms (with 2 n-gons and 2n triangles)

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3.3.3.3

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3.3.3.4

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3.3.3.5

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3.3.3.6

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3.3.3.17

The others are 11 of the Archimedean solids:
- Truncated cube
- Truncated dodecahedron
- Truncated tetrahedron
- Truncated octahedron
- Truncated icosahedron
- Truncated cuboctahedron
- Truncated icosidodecahedron
- Snub cube or snub cuboctahedron
- Snub dodecahedron or
- Rhombicuboctahedron
- Rhombicosidodecahedron

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The semi-regular non-convex polyhedra include the 17 stellated Archimedean solids:
- 3 have cubic symmetry:
- Great cubicuboctahedron
- Cuboctatruncated cuboctahedron or cubitruncated cuboctahedron
- Quasitruncated hexahedron or stellated truncated cube
- 11 have icosahedral symmetry:
- Quasitruncated small stellated dodecahedron or small stellated truncated dodecahedron
- Quasitruncated great stellated dodecahedron or great stellated truncated dodecahedron
- Great truncated dodecahedron
- Great truncated icosahedron
- Rhombidodecadodecahedron
- Icosidodecatruncated icosidodecahedron
-
- Small ditrigonal icosidodecahedron
-
- Great ditrigonal icosidodecahedron
-
- Great quasitruncated icosidodecahedron or great truncated icosidodecahedron
- Great dodecicosidodecahedron
- Great ditrigonal dodecicosidodecahedron
-
- 3 have the symmetry of the snub dodecahedron:
- Snub dodecadodecahedron
- Great inverted snub icosidodecahedron or great vertisnub icosidodecahedron
- Great inverted retrosnub icosidodecahedron or great retrosnub icosidodecahedron In addition to the 17 stellated Archimedean solids, there are 23 more semi-regular non-convex polyhedra:
- ? have octahedral symmetry:
- Small cubicuboctahedron
- Small rhombicube
- Great rhombicube
- Great truncated cuboctahedron
- ? have icosahedral symmetry:
- Rhombicosahedron
- Small rhombidodecahedron
- Great rhombidodecahedron
- Small dodecicosahedron
- Great dodecicosahedron
- Small dodecicosidodecahedron
- Icositruncated dodecadodecahedron
- Great truncated dodecadodecahedron
- Vertisnub dodecadodecahedron
- Icosidodecadodecahedron
- Snub icosidodecadodecahedron
- Small dodekic icosidodecahedron ??
- Great dodekic icosidodecahedron ??
- Small icosic icosidodecahedron ??
- Great icosic icosidodecahedron ??
- Great snub icosidodecahedron
- Snub disicosidodecahedron
- Retrosnub disicosidodecahedron
- Great snub icosidisdodecahedron
- Great snub disicosidisdodecahedron TODO: Check this list for duplicates/alternate names Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron. This is a version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1900.

Polyhedron duals

75px For every polyhedron there is a dual polyhedron which can be obtained by connecting the midpoints of the faces. Face-uniformity of a polyhedron corresponds to vertex-uniformity of the dual and conversely, and edge-uniformity of a polyhedron corresponds to edge-uniformity of the dual. Thus the regular polyhedra come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself. In most duals of uniform polyhedra, faces are irregular polygons. The exceptions are:
- Cube, which is a special trapezohedron, hence dual to an antiprism (namely the octahedron).
- Triangular dipyramid, the octahedron and the pentagonal dipyramid, which are special bipyramids, hence dual to prisms.
- Icosahedron and Dodecahedron, which are dual to one another.

Quasi-regular duals

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The duals of the quasi-regular polyhedra are edge- and face-uniform. These are, correspondingly:
- Rhombic dodecahedron
- Rhombic triacontahedron and 13 other, nonconvex ones.

Semi-regular duals

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The duals of the semi-regular polyhedra are face-uniform. These are, correspondingly:
- Bipyramids
- trapezohedra
- 11 of the Catalan solids →

Other polyhedra with regular faces

:See main entries: Johnson solid and Deltahedron Norman Johnson sought which non-uniform polyhedra had regular faces. In 1966, he published a list of 92 convex solids, the Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
- 3 regular convex polyhedra (3 of the Platonic solids)
- Tetrahedron
- Octahedron
- Icosahedron
- 5 non-uniform convex polyhedra (5 of the Johnson solids)
- Triangular dipyramid
- Pentagonal dipyramid
- Snub disphenoid
- Triaugmented triangular prism
- Gyroelongated square dipyramid With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even. There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron. There exists no polyhedron whose faces are all regular polygons with six or more sides. There exist an infinite number of non-uniform non-convex polyhedra.

General polyhedron

More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. It could be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of half-spaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron. All classical polyhedra are general polyhedra, and in addition there are examples like:
- A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: . Its sides are the two positive axes.
- An octant in Euclidean 3-space,
- A prism of infinite extent. For instance a doubly-infinite square prism in 3-space, consisting of a square in the xy-plane swept along the z-axis:
- Each cell in a Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point c∈S is bounded (hence a classical polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded.

Topological polyhedron

A topological polyhedron is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way that needs better description.

Relation with graphs

Any polyhedron gives rise to a graph, called skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra:
- The Archimedean solids give rise to regular graphs: 7 Archimedean solids are degree 3, 4 solids are degree 4, and the remaining 2 are chiral pairs of degree 5.
- The octahedron gives rise to a strongly regular graph, because adjacent vertices have always two common neighbors, and non-adjacent vertices always four.
- Only the tetrahedron gives rise to a complete graph (K₄).
- Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.

External links

- [http://www.queenhill.demon.co.uk/polyhedra/ Polyhedra Index Page]
- [http://www.software3d.com/Stella.html Stella: Polyhedron Navigator]
- [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
- [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] - The Encyclopedia of Polyhedra
- [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links
- [http://www.polyedergarten.de/ Paper Models of Uniform (and other) Polyhedra]
- [http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D polyhedra in Java]
- ko:다면체 ja:多面体

Platonic solid

In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with
- all its faces being congruent regular polygons, and
- the same number of faces meeting at each of its vertices. (These are in contrast to
- the Kepler-Poinsot solids, which are not convex, and
- the Archimedean and Johnson solids, which while having regular polygons as faces, are not themselves regular.) The five Platonic solids were all known to the ancient Greeks, and a proof that they are the only regular polyhedra can be found in Euclid's Elements.

Table

The number of faces times the number of edges per face (or the number of vertices per face) is equal to the number of vertices times the number of edges meeting at a vertex (or the number of faces meeting at a vertex): twice the number of edges (corresponding to the fact that an edge connects two vertices and that two faces meet at each edge). This product is equal to the order of the rotation group: 12, 24, 24, 60, and 60, respectively. The order of the symmetry group is twice that number, i.e. four times the number of edges, hence 24, 48, 48, 120, and 120, respectively.

Limited number of Platonic polyhedra

The limitation to five such three-dimensional solids is easily demonstrated: # Each vertex of the solid must coincide with one vertex each of at least three faces. # At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°. # The angles at all vertices of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3=120°. # Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for: #
- Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively. #
- Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube. #
- Pentagonal faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.

Dual polyhedra

Pentagon Connecting the centers of all the pairs of adjacent faces of any platonic solid produces another (smaller) platonic solid. The number of faces and vertices is interchanged, while the number of edges of the two is the same. The relationship between such a pair of polyhedra is called being each other's dual.
- The dual of a tetrahedron is another tetrahedron.
- The dual of an octahedron is a cube, and vice versa.
- The dual of a dodecahedron is an icosahedron, and vice versa.

Origins of name

The Platonic solids are named after Plato, who wrote about them in Timaeus. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of Euclid's Elements. Propositions 13 through 17 describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order.

Ancient symbolism

Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids that had been discovered by the Pythagoreans (in the Timaeus). These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium fluoride) in octahedra, and pyrite in dodecahedra (see uses below). This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. 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 breaks when picked up, in stark difference to the smooth flow of water. The fifth Platonic Solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven" (Timaeus 55). He didn't really know what else to do with it. 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.

Other symbolism

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements. Two-dimensional images of each of the Platonic solids are found within Metatron's Cube, a construct which originates from joining all the centres together from the Flower of Life.

Inscribed Platonic polyhedra

When the Platonic polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:
- Tetrahedron: 12.2518%
- Cube: 36.7553%
- Octahedron: 31.8310%
- Dodecahedron: 66.4909%
- Icosahedron: 60.5461% The Platonic solids may be seen as increasingly better approximations to that sphere. (The Archimedean solids and geodesic domes are in many ways even better approximations to the sphere). However, either the dodecahedron or the icosahedron may be seen as the Platonic solid that "best approximates" a sphere. On one hand, the icosahedron has the most sides and the flattest dihedral angle. This may be the source of the common assumption that the icosahedron is the Platonic solid that gives the closest approximation to the sphere. On the other hand, the dodecahedron occupies significantly more of the sphere's volume than the apparently more spherical icosahedron. The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere. The dodecahedron is also most like the sphere in the sense that it has the smallest central angle (ratio of the strut length to the radius of the circumscribed sphere), and the greatest surface area. [http://kjmaclean.com/Geometry/Platonic.html] In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron. Both have an inscribed sphere whose radius is :

\sqrt

, about 0.795 of the radius of the circumscribed sphere.

Uses

inscribed sphere.]] The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.). The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube. In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.

External links

- [http://www.software3d.com/Stella.html Stella: Polyhedron Navigator] Tool for exploring polyhedra
- [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links
- [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
- [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
- [http://www.lsbu.ac.uk/water/platonic.html London South Bank University] Water structure and behavior
- [http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html Book XIII] of Euclid's Elements.
- [http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D Polyhedra] in Java
- ko:정다면체

Cube (geometry)

:This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation). A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. The cube is a special kind of square prism, of rectangular parallelepiped and of 3-sided trapezohedron, and is dual to the octahedron. Thus it has octahedral symmetry. __TOC__

Canonical coordinates

Canonical coordinates for the vertices of a cube centered at the origin are (±1,±1,±1), while the interior of the same consists of all points (x₀, x₁, x₂) with -1 < x_i < 1.

Area and volume

The area A and volume V of a cube of edge length a are: :

A=6a^2

V=a^3

A cube construction has the largest volume among cuboids (rectangular boxes) with a given surface area (e.g., paper, cardboard, sheet metal, etc.). Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).

Geometric relations

The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron. zonohedron zonohedron

Higher dimensions

zonohedron In the four-dimensional Euclidean space, the analogue of a cube has a special name — a tesseract or hypercube. The analog of the cube in the n-dimensional Euclidean space is called n-dimensional cube, or simply cube, if it does not lead to a confusion. The name measure polytope is also used.

Related polyhedra

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other. One such regular tetrahedron has a volume of 1/3 of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each. The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount. A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes. compound compound)]] compound] compound] The figures shown have the same symmetries as the cube (see octahedral symmetry).

Trivia

If each edge of a cube is replaced by a one ohm resistor, the resistance between opposite vertices is 5/6 ohms, and that between adjacent vertices 7/12 ohms.

External links

- [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
- [http://www.mathsisfun.com/geometry/hexahedron.html Spinning Hexahedron]
- [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
- [http://www.korthalsaltes.com/ Paper Models of Polyhedra] Many links Category:Platonic solids Category:Polyhedra Category:Uniform polyhedra Category:Prismatoid polyhedra Category:Volume ko:정육면체 ja:正六面体 simple:Cube

Platonic solid

Table

Limited number of Platonic polyhedra

Dual polyhedra

Origins of name

Ancient symbolism

Other symbolism

Inscribed Platonic polyhedra

\sqrt

, about 0.795 of the radius of the circumscribed sphere.

Uses

External links

Tetrahedron

For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids. __TOC__ image:tetrahedron flat.png Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

Area and volume

The area A and the volume V of a regular tetrahedron of edge length a are: :

A=\sqrta^2

V=\begin\end\sqrta^3

The height is

h=(a/3) \sqrt

, the angle between an edge and a face is arctan

\sqrt

(ca. 55°), and between two faces arccos (1/3) = arctan

2\sqrt

(ca. 71°). Note that with respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that in a face, from the midpoint at the base. Like for any pyramid, the volume is

V = \frac Ah

where A is the area of the base and h the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apexes to the opposite faces are inversely proportional to the areas of these faces. Also, for a tetrahedron ABCT the volume is given by

V = \frac  \cdot \sqrt

where a is angle ATB, b angle BTC, and c angle CTB. For the distance between edges, see skew line. The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.

Geometric relations

A tetrahedron is a 3-simplex. Unlike in the case of other Platonic solids, all vertices are equidistant from each other (they are in the only possible arrangement of four equidistant points). Tetrahedra are a special type of triangular pyramid and are self-dual. Canonical coordinates of the tetrahedron are (1, 1, 1), (−1, −1, 1), (−1, 1, −1) and (1, −1, −1). A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. The volume of this tetrahedron is 1/3 the volume of the cube. Taking both tetrahedra within a single cube gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron). Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra. Regular tetrahedra can't tile space by themselves, although it seems likely enough that Aristotle reported it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron which can tile space. This is one of the five Andreini tessellations, and is a limiting case of another, a tiling involving tetrahedra and truncated tetrahedra. However, there is at least one irregular tetrahedron of which copies can tile space. If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in various ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.) The tetrahedron is unique among the Platonic solids in possessing no parallel faces.

Related polyhedra

truncated tetrahedra truncated tetrahedra