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Ratio

Ratio

: For the use of ratio as a human capacity, see reason. In algebra, a ratio is the relationship between two quantities. It is expressed as the quotient of two numbers, or as two numbers separated by a colon (pronounced "to"). A number that can be written as a ratio of two integers is a rational number. In physics, a ratio between two magnitudes of the same type of quantity gives a positive real number when the magnitudes are expressed relative to an absolute or natural zero. The ratio between a difference of two magnitudes to a third magnitude, such as a unit, gives a real number (i.e. positive or negative).

Examples

- If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
- The ratio of heights of the Eiffel Tower (300 m) and the Great Pyramid (137 m) is 300:137, so one structure is more than twice the height of the other (or more precisely, 2.2 times).
- The ratio of the mass of Jupiter to the mass of the Earth is approximately 317.8:1.
- The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.
- If two axles are connected by gear wheels, the number of times one axle turns for each turn of the other is known as the gear ratio. The best example being the number of turns of the pedals of a bicycle compared with number of turns of the bicycle's rear wheel.
- The ratio of hydrogen atoms to oxygen in water is 2:1, or two parts to one. Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are unitless; the units cancel out of the ratio. For example, the ingredients in a recipe that required 500 grams and 300 grams of each, would be in the ratio of 5:3, with no units. Note also the difference between ratios and vulgar fractions. For example, if there are three raspberry candies and five blackcurrant candies, then the ratio of raspberry candies to blackcurrant candies is 3:5. This indicates that there are three fifths as many raspberry candies as blackcurrant candies. However the fraction of all the candies that are raspberry is three out of a total of all eight candies or 3/(3+5) = 3/8. Thus the chances of a randomly selected candy being raspberry are three in eight.

Reason

:For alternate senses see reason (disambiguation). Reason is a term used in philosophy and other human sciences to refer to the higher cognitive faculties of the human mind. It describes a type of thought or aspect of thought, especially abstract thought, and the ability to think abstractly, which is felt to be especially human. The concept of reason is connected to language, as reflected in the meanings of the Greek word "logos", later to be translated by Latin "ratio" and then French "raison", from which the English word is derived. Indeed it has often been held that handicaped people lacking speech would fail to master reasoning; thus in some languages, a single word can still denote both 'mute' and 'stupid', e.g. stumm in German, stom (but also in full doofstom for a mute) in Dutch. The 1913 edition of Webster's Revised Unabridged Dictionary defines Reason as:

The faculty or capacity of the human mind by which it is distinguished from the intelligence of the inferior animals; the higher as distinguished from the lower cognitive faculties, sense, imagination, and memory, and in contrast to the feelings and desires. Reason comprises conception, judgment, reasoning, and the intuitional faculty. Specifically, it is the intuitional faculty, or the faculty of first truths, as distinguished from the understanding, which is called the discursive or ratiocinative faculty.

However, there is much disagreement between philosophical schools about the nature and function of reason, as well as about the extent to which it is unique to human beings, and the above definition is not universally accepted.

Reason and logic

The debate about the relationship of reason to logic extends back to the time of Plato and Aristotle. Plato made a distinction between reason and logic, whereas for Aristotle, the terms were essentially synonymous. The debate between these two viewpoints has continued down through the ages. Heinrich Heine, in On the History of Religion and Philosophy in Germany, wrote: "Plato and Aristotle! These are not merely two systems, but rather two types of human nature, that stand, since time immemorial, in hostile opposition. Across the entire middle ages, to the greatest degree, and up to the present day, this battle was waged, and this battle is the essential content of Christian church history. Plato and Aristotle are always the issue, though other names may be used." The Webster's definition cited above may be said to reflect the Platonic outlook, in that it refers to reason as "the faculty of first truths, as distinguished from the understanding, which is called the discursive or ratiocinative faculty." In the Aristotelean or neo-Aristotelean camp, reason is narrowly defined as the faculty or process of drawing logical inferences. Such types of reasoning have traditionally been classified as either deductive reasoning, meaning "from the general to the particular", or inductive reasoning, meaning "from the particular to the general". In the 19th century, Charles Peirce, an American philosopher, named a third classification related to the second, abductive reasoning, by which he meant to include guessing or hypothesising. (In modern usage, "inductive reasoning" sometimes includes almost all non-deductive reasoning, including what Peirce would call "abductive". See also logic, term logic.) The question of reason as the "faculty of first truths" is related to the topic of metaphysics, and in modern history there has been an ongoing debate between the proponents of metaphysics, and the empiricists such as David Hume, who deny the existence of a "faculty of first truths," and argue that all we know of the universe is gleaned from the application of logic to sensory data (the Platonists, in turn, argue that sensory information provides only imprecise reflections or "shadows" of reality, as in Plato's allegory of the cave, or reality as seen "through a glass darkly.") Bertrand Russell embarked on an ambitious project to produce a system of science that was completely devoid of metaphysics.

Reason versus logic

Plato made a distinction between logic, i.e. reasoning that proceeds via Syllogism from a premise (which Plato calls understanding,) and reason, in this passage, part of what is sometimes referred to as the divided line: Edgar Allan Poe makes a similar point in his story, Mellonta Tauta. Here, "creeping" and "crawling" refer to induction and deduction:
Reason as logic
Since the aforementioned Webster's definition was published in 1913, the prevailing views of reason have changed. A contemporary definition from the Merriam-Webster Online Dictionary goes as follows: :(1) : the power of comprehending, inferring, or thinking especially in orderly rational ways : INTELLIGENCE (2) : proper exercise of the mind (3) : SANITY b : the sum of the intellectual powers The definition of reason as the faculty of drawing logical inferences has gained considerable ground, to the point of being almost hegemonic, following the early 20th Century "revolt against idealism" and "revolt against metaphysics" (see Bertrand Russell: Analytic philosophy). The concept of reason as a type of thought that is "especially human" has been somewhat blurred, since animals (and computers -- see artificial intelligence) are capable of some forms of logical operations. While deductive reasoning can logically result in a definite conclusion, it requires as a starting point for human investigation, that there are generalizations which can not be questioned. But in the sense that animals, and humans, can unconsciously, associate different perceptions as causes and effects and then make decisions or even plans, (if these words may be used for the sake of this discussion), it is felt by many people that reason is more than just the ability to draw inferences. So reason has also been conceived more broadly. In Charles Peirce's terms, humans have "thirdness" while abductive reasoning is only "firstness". The neurologist Terrence Deacon, following the tradition of Peirce, has recently distinguished the type of thinking which is most essential to human rational thinking as a type of associative thinking. Reason by his account therefore requires associating perceptions in a way which may be arbitrary (or nominal, conventional or "formal") - not just associating the image or "icon" of smoke and the image of fire, but, for example, the image of smoke and the English word "smoke", or indeed any made-up "index" or "symbol" (not necessarily a spoken word). What is essential is however not the arbitrariness of symbols, but how they used. This fits into an older tradition which makes reason connected to language, but more specifically the ability to create language deliberately. Deacon and Peirce continue the English philosophical tradition: Thomas Hobbes describes the creation of “Markes, or Notes of remembrance” (Leviathan Ch.4) as “speech” (allowing by his definition that it is not necessarily a means of communication or speech in the normal sense; he was presumably thinking of "speech" as an English version of "logos" in this description). In the context of a language, these marks or notes are called "Signes" by Hobbes. David Hume, following John Locke (and Berkeley), who followed Hobbes, emphasized the importance of associations in thinking. George Lakoff and Mark Johnson explicate reason and its scope in this manner: :Reason includes not only our capacity for logical inference, but also our ability to conduct inquiry, to solve problems, to evaluate, to criticize, to deliberate about how we should act, and to reach an understanding of ourselves, other people, and the world. (Lakoff and Johnson 1999, pp. 3-4) Proponents of the logical definition of reason differ as to their preference for either induction or deduction. While the English tradition is strongly empiricist, an influential example of the opposite is Immanuel Kant. For him, reason (Vernunft in Kant's German language) is the power of synthesizing into unity, by means of comprehensive principles, the concepts provided by the intellect (Verstand). The reason which gives a priori principles Kant calls "Pure Reason" (as in his A Critique of Pure Reason), as distinguished from the "Practical Reason" which is specially concerned with the performance of particular actions. Modern proponents of a priori reasoning, at least with regards to language, are Noam Chomsky and Stephen Pinker, to whom Merlin Donald and Terrence Deacon can be very usefully contrasted.
Reason and faith
In theology, reason, as distinguished from faith, is the human critical faculty exercised upon religious truth whether by way of discovery or by way of explanation. Some commentators have claimed that Western civilization can be almost defined by its serious testing of the limits of tension between reason and faith - Jerusalem and Athens. Leo Strauss spoke of a "Greater West" which included all areas under the influence of the tension between Greek and Abrahamic thinking, including the Moslem lands. He was particularly influenced by the great moslem philosopher Al-Farabi. In order to consider to what extent Eastern philosophy might have partaken of these important tensions, it is perhaps best to consider whether dharma or tao may be equivalent to Nature (by which we mean physis (Greek). The limits within which reason may be used have been laid down differently in different churches and periods of thought: on the whole, modern religion tends to allow to reason a wide field, reserving, however, as the sphere of faith the ultimate (supernatural) truths of theology. For a critique of reason's preeminent position within western culture since the Renaissance, see Voltaire's Bastards by John Ralston Saul.
References

- George Lakoff and Mark Johnson (1999). Philosophy In The Flesh. Basic Books.
External links

- [http://www.sense-think-act.org Reasoning Exercises] a Mediawiki project Category:Epistemology Category:Belief Category:Logic ja:理性

Quantity

:For the use in linguistics, see length (phonetics). Quantity is a general term used to refer to any type of quantitative property or attribute, such as mass, length, or time. A particular quantity is a magnitude of a scalar or vector quantity. The term quantity is also often used to refer to denumerable (countable) collections of objects. A given quantity is usually represented either as a number of units, together with the type of those units, or a number of objects with a referent defining the type of object. Thus, scalar quantities such as mass, and vector quantities such as force, are continuous quantities and are usually represented as a multiple of a real number and a unit of continuous quantity, such as a gram or newton. A count of a denumerable collection of entities is represented as an integer and the type of object or entity, such as an apple or a set. A number, including a particular measurement, is not by itself a quantity. Examples are
- 1.76 litres (liters) of milk, which is continuous quantity
-

2 \pi r

metres, where r is the length of a radius of a circle expressed in metres (or meters)
- one apple, two apples, three apples, where the number is an integer representing the count of a denumerable collection of objects (apples)
- 500 people (also involving a count) Where the count is one then the indefinite article may be used (for example, a car) and similar alternatives exist for other particular counts (for example, a brace of pheasant, a dozen eggs). Quantification in its very simplest sense can be found in statements such as "A is greater than B". In the example cited, an expression is made that A has a greater quantity of something (such as volume or charisma) than B; and that if A and B were placed in an ordered set, then A would come after B if the order is arranged on an increasing (rather than decreasing) scale.

Quotient

In mathematics, a quotient is the end result of a division problem. For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor. A quotient can also mean just the integral part of the result of dividing two integers. For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3. For more, see the division algorithm. In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure. See:
- quotient set
- quotient group
- quotient ring
- quotient space (linear algebra)
- quotient space of a topological space
- quotient object
- right quotient and left quotient (operations on formal languages) The Quotient rule is a method for finding derivatives in calculus. Quotients also come up in certain tests, like the IQ test, which stands for Intelligence Quotient. In this case, your quotient is basically your score. In recent decades, as people begin to emphasize on full personal development, other similar quotients appeared. These include Emotional Quotient, Adversity Quotient, Creativity Quotient, etc. Category:Real numbers zh-tw:商數

Number

: This article is about numbers such as counting numbers and measurements. For other uses of the term, see Number (disambiguation). A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as the square root of minus one. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples

The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: . Others do not: . In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is N. If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (from the German word "zahlen"). (Some authors use W for the whole numbers, but other authors use W for the natural numbers, so the W symbol is ambiguous.) Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal. Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face Q (for "quotient"). The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a dot (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is R. All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever. There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete ordered field. Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers C. This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients. The above symbols are often written in blackboard bold, thus: :

\mathbb\sub\mathbb\sub\mathbb\sub\mathbb\sub\mathbb

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Numerals and numbering

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. Numbers are often used to give objects unique names. Examples are telephone numbers, social security numbers, and ISBNs.

Extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left, with digits in base p, where p is prime. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

External links

- [http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]

References

- Erich Friedman, [http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]
- [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] at cut-the-knot Category:Group theory Category:Numbers __NOTOC__ ko:수 (수학) ja:数 simple:Number th:จำนวน

Colon (punctuation)

A colon is a punctuation mark, with one dot above another: ":".

Uses

- A colon can be used to start off a list when not using is or are and often with the following. :The major cities of the US include the following: New York, Chicago, and Los Angeles.
- A colon may be used to emphasize a word or phrase acting as an appositive. :John moved to a new state: Missouri.
- A colon can introduce a phrase which restates a previous statement. :The road was never ending: it seemed to go on forever.
- A colon can introduce a long quotation set off by indentation (this is not quoted). :John Smith stated: ::I could never live in the same place for more than a couple of years. I have wandering feet and I like to keep them happy.
- A colon can introduce a shorter quotation not set off and replaced a comma when such a quote is to be emphasized or is lengthy. Quotation marks are used. :The sign read: "Do not enter."
- Colons are also used after the salutation in a formal letter, though in the US this is falling out of favor.
- A colon is used between chapters and verses in many religious scriptures :John 3:16; The Quran, Sura 5:18
- A colon is used between the hour and the minutes when telling time (though a full stop is sometimes used instead). :The time is 10:45.
- A colon occurs between titles and subtitles :Star Wars Episode IV: A New Hope
- Note that a colon is never preceded by: "namely," "for example," "e.g.," or "that is." The original definiton for colon on this page read, "A colon is a punctuation mark, with one dot above another, e.g.: ":"." This definition is redundant because a colon implies "e.g." within its definition.

Mathematics

The colon is also used in mathematics to indicate ratio, and is also the standard sign for division in most non-English-speaking countries. In mathematical logic the colon is often used to represent "such that" in a relational phrase from predicate calculus. Unicode provides ratio U+2236 (∶, ∶) for such mathematical usage if the distinction is required.

Linguistics

A special triangular colon symbol is used in IPA to indicate a preceding long vowel. It is available in Unicode as Modifier letter triangular colon Unicode U+02D0 (). A regular colon is often used as a fallback when this character is not available.

Computer representation

In computer science, the colon character corresponds to the decimal value 58 (hexadecimal value 3A) in Unicode and ASCII character encodings.

The colon in foreign languages

In , the colon can appear inside words in a manner similar to the English apostrophe, between a word (or abbreviation) and its grammatical suffixes.

Other meanings

See Colon, the disambiguation page. Category:Punctuation Category:Typography ja:コロン (記号)

Rational number

In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. Each rational number can be written in infinitely many forms, for example

3/6 = 2/4 = 1/2

. The simplest form is when

a

and

b

have no common divisors, and every non-zero rational number has exactly one simplest form of this type with positive denominator. The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not rational is called an irrational number. In mathematics, the term "rational something" means that the underlying field considered is the field

\mathbb

of rational numbers. For example, rational polynomials or rational prime ideals. The set of all rational numbers is denoted by Q, or in blackboard bold

\mathbb

. Using the set-builder notation

\mathbb

is defined as such: :

\mathbb = \left\

Arithmetic

\frac + \frac = \frac

\frac \cdot \frac = \frac

Two rational numbers

\frac

and

\frac

are equal if and only if

ad =  bc

Additive and multiplicative inverses exist in the rational numbers. :

- \left( \frac \right) = \frac

\left(\frac\right)^ = \frac \mbox a \neq 0

History

Egyptian fractions

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance,

\frac = \frac + \frac + \frac

For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.

Formal construction

Mathematically we may define them as an ordered pair of integers

\left(a, b\right)

, with

b

not equal to zero. We can define addition and multiplication of these pairs with the following rules: :

\left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right)

\left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right)

To conform to our expectation that

2/4 = 1/2

, we define an equivalence relation

\sim

upon these pairs with the following rule: :

\left(a, b\right) \sim \left(c, d\right) \mbox ad = bc

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain, see quotient field.) We can also define a total order on Q by writing :

\left(a, b\right) \le \left(c, d\right) \mbox (bd>0\mbox ad \le bc)\mbox(bd<0\mbox ad \ge bc)

Properties

The set

\mathbb

, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers

\mathbb

. The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of

\mathbb

. The algebraic closure of

\mathbb

, i.e. the field of roots of rational polynomials, is the algebraic numbers. The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. As a totally ordered set, the rationals are uniquely characterized by being countable, dense (in the above sense), and having no least or greatest element.

Real numbers

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expressions of continued fraction. By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric

d\left(x, y\right) = |x - y|

, and this yields a third topology on

\mathbb

. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metric space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of

\mathbb

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn

\mathbb

into a topological field: let

p

be a prime number and for any non-zero integer

a

let

|a|_p = p^

, where

p^n

is the highest power of

p

dividing

a

; in addition write

|0|_p = 0

. For any rational number

\frac

, we set

\left|\frac\right|_p = \frac

. Then

d_p\left(x, y\right) = |x - y|_p

defines a metric on

\mathbb

. The metric space

\left(\mathbb, d_p\right)

is not complete, and its completion is the p-adic number field

\mathbb_p

. Category:Elementary mathematics Category:Field theory Category:Fractions Category:Real numbers Category:Set theory ko:유리수 ja:有理数 simple:Rational number th:จำนวนตรรกยะ

Quantity

2 \pi r

Eiffel Tower

The Eiffel Tower (French: Tour Eiffel) is an iron tower built on the Champ de Mars, beside the River Seine, in Paris, France. It is the most famous French landmark and is used as a symbol of France. At the time of its construction in 1889 it was the tallest structure in the world, and remained so until 1930. Named after its designer, engineer Gustave Eiffel, it is a premier tourist destination, with over 5.5 million visitors per year. Pronunciation (IPA): English: , similar to "eye-full"; French: , similar to "a-fell" The tower stands 300 meters (986 feet) high, not including the 24 meter (72 feet) antenna.

Background

French The structure was built between 1887 and 1889 as the entrance arch for the Exposition Universelle (1889), a World's fair marking the centennial celebration of the French revolution. It is located at geographic coordinates . The tower was inaugurated on March 31, 1889, and opened on May 6. Three hundred workers joined together 18,038 pieces of puddled iron, using two and a half million rivets, in a structural design by Maurice Koechlin. The risk of accident was great, for unlike modern skyscrapers the tower is an open frame without any intermediate floors except the two platforms. Yet, because Eiffel took good care of his workers with movable stagings, guard-rails and screens, only one man died (during the installation of Otis Elevator's lifts). The tower is 300 meters (986 feet) tall, not including the 24-meter television antenna on top. The metal structure weighs 7,300 metric tons, the total weight is 10,100 metric tons. According to the official website for the tower, the summit is reached by 1,665 steps and not, as popularly believed, by 1,792 steps (the same as the year of the First French Republic). First French Republic] Depending on the ambient temperature, the top of the Eiffel Tower may shift away from the sun by up to eight centimetres (3 and 1 quarter inches), due to expansion of the metal on the side facing the sun. Maintenance on the tower includes applying 50 metric tons of three graded tones of paint every 7 years to protect it from rust. On occasion, the colour of the paint is changed (the tower is currently painted a shade of brown). On the first floor, there are interactive consoles hosting a poll for the colour to use for a future session of painting. The tower was met with resistance from the public when it was first built, many thinkng it an eyesore. Today, it is widely considered to be one of the most striking pieces of structural art in the world. One of the great Hollywood movie clichés is that the view from a Parisian window always includes the Eiffel Tower. In reality, the Tower is not visible from a large part of Paris. Originally, Eiffel had a permit to leave the tower standing for 20 years, more than recouping his expenses, but, as it proved valuable for communication purposes, it was allowed to stay after the end of the permit.

Installations

communication Since the beginning of the 20th century, the Eiffel Tower has been used for radio transmission. Until the 1950s, there was an aerial running from the top to anchor points on the Champ de Mars. This aerial was fed by long-wave transmitters which were in small housings on the Champ de Mars. Since 1957, the Eiffel Tower has been used as transmission tower for FM and television. The Eiffel Tower has [http://www.tour-eiffel.fr/teiffel/uk/pratique/resto/index.html two restaurants]: Altitude 95, on the first floor (95 m above sea level); and the Jules Verne, an expensive gastronomical restaurant on the second floor, with a private elevator. This restaurant has one star in the Michelin Red Guide.

Events

Michelin Red Guide Father Theodor Wulf in 1910 took observations of radiant energy radiating at the top and bottom of the Eiffel Tower, discovering more than was expected at the top, and thereby detecting what are today known as cosmic rays. In 1925, the con artist Victor Lustig twice "sold" the Eiffel Tower for scrap. In 1930, the Tower lost the title of the World's tallest structure when the Chrysler Building was completed in New York. From 1925 to 1934, illuminated signs for Citroën adorned three of the tower's four sides, making it the tallest billboard in the world at the time. Citroën When the Nazis occupied Paris in 1940, the lift cables were cut by the French so that Hitler would have to climb the 1,665 steps to the summit - the part to repair them was allegedly impossible to obtain because of the war, though it was working again within hours of the departure of the Nazis. He chose to stay on the ground. A Frenchman also scaled the tower during the German occupation to hang the French flag. In August 1944, when the Allies were nearing on Paris, Hitler ordered general Dietrich von Choltitz, the military governor of the Paris, to burn down the tower along with the rest of the city. He disobeyed the order. On January 3, 1956 a fire damaged the top of the tower. In 1959 the present radio antenna was added to the top. In the 1980s an old restaurant and its supporting iron scaffolding midway up the tower was dismantled; this was purchased and reconstructed in New Orleans, Louisiana, originally as the Tour Eiffel Restaurant, more recently known as the Red Room. In the year 2000, flashing lights and four high-power searchlights were installed on the tower. Since then the light show has become a nightly event. The searchlights on top of the tower make it a beacon in Paris' night sky. The tower received its 200,000,000th guest on November 28, 2002. At 19:20 on July 22, 2003, a fire occurred at the top of the tower in the broadcasting equipment room. The entire tower was evacuated; the fire was extinguished after forty minutes, and there were no reports of injuries.

The 72 names

broadcasting gardens and the Palais de Chaillot. A pleasure boat cruises on the river]] On the tower, the 72 names of French scientists and engineers are engraved in recognition of their contributions. This engraving was overpainted at the beginning of the 20th century and restored in 1986-1987 by SNTE ("Société Nouvelle d'Exploitation de la Tour Eiffel"), a company contracted to operate business related to the Tower (the Tower is owned by the City of Paris.

Image copyright

Images of the Eiffel Tower have long been in the public domain; however in 2003, the operating company SNTE installed a new lighting display on the tower, the design of which they then copyrighted. The effect is to put the night-time image of the tower under copyright. It follows that it is no longer legal to publish contemporary photographs of the tower without permission. The imposition of copyright is not without some controversy. The Director of Documentation for SNTE, Stéphane Dieu, commented in January 2005 "It is really just a way to manage commercial use of the image, so that it isn't used in ways we don't approve". However, it also potentially has the effect of prohibiting tourist photographs of the tower at night from being published. [http://blog.fastcompany.com/archives/2005/02/02/eiffel_tower_repossessed.html] In a recent decision, the Court of Cassation ruled that an architect could not claim copyright over images including one building the design of which they held the copyright of if the photograph encompasses a larger area. This seems to indicate that SNTE cannot claim copyright on photographs of Paris incorporating the lighted tower at night. It should be noted that under [http://www.copyright.gov/title17/92chap1.html#120 American law], copyright does not go so far, and photography of a publicly visible building is freely permitted, whether or not the photograph encompasses a larger area or just the building itself. However, copyright infringers could potentially be sued for civil tort before French courts, and damages recouped by asking the execution of the decision from American courts.

Appearance in film

Court of Cassation
- 1923: René Clair's Paris dort starts, ends and has many scenes on the Eiffel Tower.
- 1949: In the film The Man On The Eiffel Tower the Tower plays a central role, and the climactic scene involves a climbing chase that predates the Mount Rushmore scene of North by Northwest.
- 1951: The Lavender Hill Mob - models of the tower are central to the plot, and the climax takes place on the real tower.
- 1953: In the end of The War of the Worlds, the tower is seen destroyed.
- 1958: At the beginning of Francois Truffaut's The 400 Blows the tower is seen between Parisian apartment blocks
- 1965: At the end of the Blake Edwards movie The Great Race, starring Tony Curtis and Jack Lemmon, [http://www.imdb.com/title/tt0059243/] the tower is blown up by a misfired cannon shot from Professor Fate's car.
- 1970: The tower is shown in the classic animated film The Aristocats.
- 1980: The tower (and the rest of Paris) were almost blown up by a terrorist nuclear bomb and Lois Lane almost plunged to her death under its elevator in Superman II.
- 1981: Condorman attempts to fly off of the tower in the movie by the same name.
- 1985: The James Bond film A View to a Kill contains a scene in the Eiffel Tower including scenes in a fictional restaurant there.
- 1985: In the film National Lampoon's European Vacation, Rusty throws his beret off the Tower. A dog, thinking it is a frisbe, jumps after it. Because they sought a PG-13 rating, however, the dog is not seen dying, but rather, lands in a pond at the bottom of the tower.
- 1991: In Star Trek VI: The Undiscovered Country the Eiffel Tower is shown as still standing in the 23rd Century and is visible from the office of the Federation President. The Eiffel Tower was also seen in the Star Trek franchise in 24th Century Paris in the episode of "We'll Always Have Paris"(1988) of Star Trek: The Next Generation and in two episodes of Star Trek: Deep Space Nine, "Homefront" and "Paradise Lost" (1996).
- 1995: In La Haine, the main protagonists lament the fact that they cannot switch the lights of the Eiffel Tower off like people can in the movies. The lights switch off just after they have given up and turned their backs on the tower.
- 1996: The Eiffel Tower can be seen on TV in Independence Day (and is destroyed in the French movie version).
- 1996: In Mars Attacks!, the Eiffel Tower is destroyed by Martians.
- 1998: The tower is destroyed in Armageddon.
- 2000: In Rugrats in Paris: The Movie, the babies are atop the tower while using the giant Reptar invention.
- 2001: In Moulin Rouge!, an object thrown from Montmartre by Christian (Ewan McGregor) bounces off the tower underneath the smiling moon during the finale.
- 2003: In The Real World Paris television show on the US MTV network, the tower is seen.
- 2003: The Tower is featured in Looney Tunes: Back In Action.
- 2004: In Van Helsing, the Eiffel Tower is under construction.
- 2004: In Team America: World Police, a rocket blows the tower up, then the tower falls on the Arc de Triomphe.
- 2004: The tower flew and moved around Paris in the puppet version of Without a Paddle, in a scene that starts only after the credits end.
- 2004: In Godzilla: Final Wars, Kamacuras attacks the tower.
- 2004: The tower is seen in Eurotrip. Eurotrip (Montparnasse Tower) in the distance]]

Imitations and reproductions

Several reproductions/models of the Eiffel Tower (often smaller-scale) exist.

Imitations (similar towers, not scale models)

In order of decreasing height:
- Kiev TV Tower, Kiev, Ukraine - At 385m, the world's tallest lattice tower, with similarities to the Eiffel Tower, although with no observation deck for visitors.
- Riga Radio and TV Tower, Riga, Latvia - 368.5 m concrete tower on three legs, in similar style to the Eiffel Tower.
- Dragon Tower, Harbin - a 336 metre high lattice tower at Harbin, China.
- Tokyo Tower, Tokyo, Japan - 9m higher than the original (33 m if the TV antenna is included)).
- TV Tower Yerevan, Yerevan, Armenia - 311.7 m high lattice tower built from 1974 to 1977.
- St. Petersburg TV Tower, St. Petersburg, Russia - 310 m lattice tower without observation deck, resembling the Eiffel Tower.
- Star Tower, Cincinnati, Ohio - 291.4 m transmission tower from a similar design, but without observation deck.
- Qingdao TV Tower, China - 232 m TV tower with observation deck.
- Crystal Palace Transmitter, London, England - 222 m TV tower without observation deck, nicknamed London's Eiffel Tower.
- Brasilia TV Tower, Brasilia, Brazil - 218 m lattice tower with an observation deck at a height of 75 m.
- Guangzhou TV Tower, Guangzhou, China - A 217 metre high TV tower of lattice steel at Guangzhou
- Guangdong TV Tower, Guangdong, China - A 200 metre high TV tower of lattice steel at Guangdong
- Nagoya TV Tower, Nagoya, Japan - 180 m
- Odinstårnet, Odense, Denmark - A 177 metre high lattice tower, destroyed in 1944
- Blackpool Tower, Blackpool, England - 158 m (519 ft); it is not quite a free-standing structure, it stands above the Tower Circus complex, where the four "legs" can be seen.
- Mesquite Tower, Mesquite, Texas - 155.3 m transmission tower from a similar design, but without observation deck.
- Croydon Transmitter - A 152 metre high transmission tower at London without observation deck
- Radio Tower Berlin, Berlin, Germany - 150m transmission tower with observation deck. Sometimes nicknamed as a copy of the Eiffel Tower, although the two structures are not too similar. The Radio Tower Berlin is the only observation tower whose feet are insulated against ground.
- Sapporo TV Tower, Sapporo, Japan - 147 m.
- Beppu Tower, Beppu, Japan - 100 m, [http://www2.odn.ne.jp/yoko-tower/list1-e.htm]
- Zendstation Zwollerkerspel - 90 m high radio tower remembering to Eiffel Tower
- Tour métallique de Fourvière, Lyon, France - 85.7 m lattice tower built from 1892 to 1894. Used until 1953 as an observation tower, but is now a TV Tower closed to visitors.
- Torre del Reformador, Guatemala City, Guatemala - 75 m.
- Transmitter Brookmans Park - two 60.96 metre high lattice towers, insulated against ground
- Petrinska rozhledna, Prague, Czech Republic - 60 m, built in 1891.
- Watkins' Tower, Wembley Park, London, England - never completed, demolished in 1907.
- Joseph's Cross, Stollberg/Harz, Germany - 38 m observation tower in form of a double-cross, resembling the Eiffel Tower.
- Lemberg Tower, Lemberg Mountain, Germany - 33 m observatio tower of lattice steel, built in 1899
- Tour du Belvédère - a small observation tower in Mulhouse, Alsace, France resembling to Eiffel Tower.
- Woodwards Building, Vancouver, Canada - A small reproduction on the roof of the building is topped by a signature neon "W". This building is now being converted into social housing. [http://www.downtowneastside.ca/images/woody/wo08.jpg]

Reproductions (scale models)

In order of decreasing height:
- In front of the Paris Las Vegas hotel/casino on the Las Vegas Strip, Paradise, Nevada, near Las Vegas, USA - 165 m (540 ft, scale 1:2). [http://www.timshell.com/pics/lasvegas/EiffelTower2.jpg]
- Shenzhen, China - ~100 m (~328 ft, scale 1:3)
- Paramount's Kings Island, Ohio, USA - ~100 m (~328 ft, scale 1:3)
- Paramount's Kings Dominion, Virginia, USA - 84 m (275 ft, scale 1:3.59)
- Slobozia, Romania - 54 m (177 ft)
- In Parizh, Chelyabinsk Oblast, Nagaybaksky District, Chelyabinsk Oblast, Russia. Built by South Ural Cell Telephone company, used as a cell phone tower. - 50 m (164 ft)
- Walt Disney World's Epcot theme park in Lake Buena Vista, Florida, USA (at the France Pavilion in World Showcase) - 23 m (76 ft, scale 1:13) [http://www.disneyworldtrivia.com/trivia/epcot/worldshowcase/france.php (information)]
- Paris, Texas - 20 m (65 ft)
- As a Meccano model, housed at the Technology Museum of Georgia (Atlanta, Georgia). - 11m (36 ft) [http://www.dalefield.com/mwes/history/eiffel_tower.html]
- On a roof of an industrial building in Satteldorf, Germany -(height unknown)
- Centerpiece of the Falconcity of Wonders, a planned new development project in Dubai. UAE, featuring seven modern wonders of the world (planned). [http://www.falconcity.com/]
- Model of the Eiffel Tower in Paris, Tennessee, about 25 feet (7.6 m) tall.
- Eiffel Tower reproduction (4 total) used as rooks in a chess set owned by Martin Scobowski of Whittier, New York; about 2.5 inches (64 mm) tall.

Access

- Metro: Trocadéro (9) or Bir-Hakeim (6)
- RER: Champs-de-Mars - Tour-Eiffel (C)

External links

- [http://www.tour-eiffel.fr/ Official website of the Eiffel Tower] - [http://www.tour-eiffel.fr/teiffel/uk/ English version]
- [http://en.structurae.de/structures/data/index.cfm?id=s0000021 Structurae: Eiffel Tower]
- [http://www.cbsforum.com/cgi-bin/articles/partners/cbs/search.cgi?template=display&dbname=cbsarticles&key2=eiffel&action=searchdbdisplay The story of Eiffel Tower] - by [http://www.cbsforum.com/ CBS Forum]
- [http://www.abcparislive.com 3 Live Webcams of the Eiffel Tower]
- [http://paris.tourismeville.wanadoo.fr/index.php?theme=affwebcam&id=21&arrdt=7&nom=TourEiffel&titre=Tour+Eiffel Live webcam showing the Eiffel tower]
- [http://www.insecula.com/musee/M0054.html/ Eiffel Tower at insecula.com] (site in French)
- [http://www.discoverfrance.net/France/Paris/Monuments-Paris/Eiffel.shtml Discover France - Eiffel Tower]
- [http://www.panoramas.dk/fullscreen/fullscreen32.html Panoramic photo of the Eiffel Tower] in QuickTime VR format
- [http://www.photoglobe.info/spc_eiffel_tower.html Eiffel Tower from Space]
- [http://maps.google.com/maps?ll=48.858197,2.294490&spn=0.005412,0.005759&t=k&hl=en Satellite view of the Eiffel Tower] (Google Maps)
- [http://www.googleearthhacks.com/dlfile66/Eiffel-Tower---3d.htm 3D render of the Eiffel Tower for use in Google Earth]
- [http://www.frommers.com/destinations/paris/A25288.html Frommer's Review of the Eiffel Tower] Category:Buildings and structures in Paris Category:Tourism in Paris Category:Towers in France Category:Landmarks Category:Historic civil engineering landmarks als:Eiffelturm ja:エッフェル塔 simple:Eiffel Tower th:หอไอเฟล

Great Pyramid

The Great Pyramid of Giza () is the oldest and last remaining of the Seven Wonders of the World. It is generally believed the Great Pyramid was built as the tomb of Fourth dynasty Egyptian king Khufu (also known under his Greek name Cheops and believed to have reigned from 2606-2583 BC), after whom it is sometimes called Khufu's Pyramid or the Pyramid of Khufu. Traditionally, the architect of the pyramid was HemInwo, a relative of Khufu.

Age and location

Believed by mainstream egyptologists to have been constructed in approximately 20 years, the most widely accepted estimate for its date of completion is c. 2580 BC. This date is confirmed by archæological findings, since extensive excavations have revealed no civilisation older than the fourth dynasty in the area. In 1984 the Edgar Cayce Foundation tried to support their claim the pyramids were about 10,000 years old, so they funded the "David H. Koch Pyramids Radiocarbon Project". The project took organic material from several places of the core of the pyramid and dated them with radiocarbon. This yielded results that averaged 374 years earlier than the accepted date by egyptologists, but much more recent than 10,000 years. A second dating in 1995 with new material obtained a date about one hundred years earlier than the historic record, but with dates scattered by 400 years. This put forward interesting questions on the origin of the wood; massive quantities of wood were used and burned, and possibly old wood was used. Most of the pyramids of the old kingdom have this anomaly. Dating of more short-lived material around the pyramid (cloth, small fires, etc) yield dates nearer historical records. 2580 BC card photo.]] An astronomical study, by Kate Spence (see below), suggest a date of 2467 BC. Alternative hypotheses suggest much of the Giza complex dates to a far earlier time period, circa 10,000 BC, attributing its construction to an ancient civilisation that was destroyed with the abrupt end of the last ice age, long before pharonic Egyptian civilisation. In recent years, authors such as Graham Hancock, John Anthony West, Robert Buval, and Boston University geology professor Robert Shoch in particular, have offered as compelling proof, among other things, the astronomical orientation of the Giza plâteau corroborating this earlier date as well as water erosion studies, namely of the Great Sphinx and surrounding enclosure, which predate pharonic Egypt. The Great Pyramid is the oldest and largest of the three pyramids in the Giza Necropolis adjacent to the outskirts of modern Cairo, Egypt in Africa. It is the main part of a complex setting of buildings that included a special walkway, two temples, three small pyramids (called the queens' pyramids), boat pits (with boats buried inside) and the mastabas for the nobles. Also there was a town for the workers along with their cemetery, bakeries, a beer factory and a copper smelting complex. More buildings and complexes are being discovered by the The Giza Mapping Project. A few hundred metres south-west of Khufu's Great Pyramid lie the slightly smaller Pyramid of Khafre, one of Khufu's successors who is believed to have built the Great Sphinx, and a few hundred metres further south-west is the Pyramid of Menkaure, Khafre's successor, which is about half as tall. Khafre's pyramid appears the tallest on some photographs as it is somewhat steeper and built on higher terrain.

Quick Facts

•The Pyramid stands 146 meters high (480 feet.) That is about 50 stories high. It covers an area of 13 acres. •The platform on which this pyramid is built (made of limestone) is ½ an inch from being perfectly level. •After many surveys it has been proven that the Great Pyramid is built along the four Cardinal Points with extreme accuracy. •Even the golden ratio (2:3) is manifested in the great pyramid design. •As many have observed the calculation for pi is also seen in the Pyramid. (The height of the pyramid multiplied by 2 pi equals the perimeter of the pyramid. Thus the height is elegantly equal to two diameters of a circle and the base equal to the circumference of the circle.) •It is a perfect square having same side lengths and 90 degree angles – these ancient people used the geometry that we study in school today. •The Great Pyramid contains some three million huge blocks of stone, some of which weight about 15 tons. •The pyramid is built at 300N and 300S.

Construction

:See also: Egyptian Pyramid construction techniques At construction, the Great Pyramid was 280 Egyptian Old Royal Cubits tall (146.5 metres or 481 feet), but due to erosion and the theft of its topmost stone (the so-called pyramidion) its current height is 455.21 ft, approximately 138.75 m. As has been proven by papyrus documents, each base side measured in antiquity 440 (20.63-inch) royal cubits. Thus, the Great Pyramid base was originally 231 m on a side and covered approximately 5.3 hectares. Today each side has an approximate length of about 230.36 meters, well within the precision of that measurement. The reduction in size and area of the structure into its current rough-hewn appearance is due to the absence of its original polished casing stones, some of which were up to two and a half meters thick and weighing upwards of 15 tonnes. In the 14th century (1301 AD) a massive earthquake loosened many of the outer casing stones of which much was carted away by Bahri Sultan An-Nasir Nasir-ad-Din al-Hasan in 1356 in order to build mosques and fortresses in nearby Cairo; the stones can still be seen as part of these structures to this day. Later explorers reported massive piles of rubble at the base of the pyramids left over from the continuing collapse of the casing stones which were subsequently cleared away during continuing excavations of the site. Nevertheless, many of the casing stones around the base of the Great Pyramid can be seen to this day in situ displaying the same workmanship and precision as has been reported for centuries. The first precision mesasurements of the pyramid were done by Sir Flinders Petrie in 1880–82 and published as " The Pyramids and Temples of Gizeh". Almost all reports are based on his measurements. Petrie found the pyramid is oriented 4' West of North and the second pyramid is similarly oriented. Petrie also found a different orientation in the core and in the casing ( – 5' 16" ± 10"). Petrie suggested a redetermination of North was made after the contruction of the core, but a mistake was made, and the casing was built with a different orientation. This deviation from the north in the core, corresponding to the position of the stars b-Ursae Minoris and z-Ursae Majoris about 3,000 years ago, takes into account the precession of the axis of the Earth. A study by egyptologist Kate Spence , shows how the changes in orientation of 8 pyramids corresponds with changes of position of those stars through time. This would date the start of the construction of the pyramid at 2467 BC. For four millennia it was the world's tallest building, unsurpassed until the 160-metre tall spire of Lincoln Cathedral was completed c. 1300 AD. The accuracy of the pyramid's workmanship is such that the four sides of the base have a mean error of only 50 mm in length, and 12 seconds in angle from a perfect square. The sides of the square are closely aligned to the four ordinal compass points to within 3 minutes of arc and is based not on magnetic north, but true north. true north has some of its smooth outer casing limestones intact.]] The pyramid was constructed of cut and dressed blocks of limestone, basalt or granite. The core was made mainly of rough blocks of low quality limestone taken from a quarry at the south of Khufu’s Great Pyramid. Those blocks weighed from two to four tonnes on average, with the heaviest used at the base of the pyramid, an estimated 2.4 million blocks were used in the construction. High quality limestone was used for the outer casing, with some of the blocks weighing up to 15 tonnes. This limestone came from Tura, about 8 miles away on the other side of the Nile. Granite was used for the portcullis and the relieving chambers, weighing as much as 60-80 tonnes, the granite blocks were quarried nearly 500 miles away in Aswan. Total mass of the pyramid is estimated at 5.9 million tonnes. Volume (including an internal hillock) is believed to be 2,600,000 cubic metres. The pyramid is the largest in Egypt and the tallest in the world and is surpassed only by the Great Pyramid of Cholula in Puebla, Mexico, which, although much lower in height, occupies a greater volume. At completion, the Great Pyramid was surfaced by white 'casing stones' – slant-faced, but flat-topped, blocks of highly polished stone. These caused the monument to shine brightly in the sun and even in the evening under moonlight being visible from mountains in the south of Egypt as far away as 200 miles (322 km). Visibly all that remains is the underlying step-pyramid core structure seen today, but several of the casing stones can still be found around the base. The casing stones of the Great Pyramid and Khafre's Pyramid (constructed directly beside it) were cut to such optical precision as to be off of true plane over their entire surface area by only as little as 1/50th of an inch. They were fit together so perfectly that the tip of a knife cannot be inserted between the joints along any edge. The Great Pyramid differs in its internal arrangement from the other pyramids in the area. The greater number of passages and chambers, the high finish of parts of the work, and the accuracy of construction all distinguish it. The walls throughout the pyramid are totally bare and uninscribed, but there are inscriptions — or to be more precise, graffiti — made by the workers on the stones before they were assembled. All the five relieving chambers are inscribed. The most famous inscription is one of the few that mentions the name of Khufu; it says "year 17 of Khufu's reign". Although alternative theorists have suggested otherwise, given its precarious location it is hard to believe it could have been inscribed after construction; even Graham Hancock accepted this, after Dr Hawass let him examine the inscription. Another inscription refers to "the friends of Khufu", and probably was the name of one of the gangs of workers. Though this doesn't offer indisputable proof Khufu originated the construction of the Great Pyramid or when building began, it does however clear any doubt he at least took part in some phase of its construction during his reign. There are three chambers inside the Great Pyramid. These are arranged centrally, on the vertical axis of the pyramid. The lowest chamber is cut into the bedrock upon which the pyramid was built. This chamber is the largest of the three, but totally unfinished, only rough-cut into the rock. The middle chamber, or Queen's Chamber, is the smallest, measuring approximately 5.74 by 5.23 metres, and 4.57 metres in height. Its eastern wall has a large angular doorway or niche, and two narrow shafts, about 20 centimeters wide, extending from the chamber to the outer surface of the pyramid, but blocked by limestone "doors" at several points. Egyptologist Mark Lehner believes that the Queen's chamber was intended as a serdab—a structure found in several other Egyptian pyramids—and that the niche would have contained a statue of the interred. The Ancient Egyptians believed that the statue would serve as a "back up" vessel for the Ka of the Pharaoh, should the original mummified body be destroyed. The true purpose of the chamber, however; remains a mystery.[http://www.touregypt.net/featurestories/greatpyramid3.htm] At the end of the lengthy series of entrance ways leading into the pyramid interior is the structure's main chamber, the King's Chamber. This chamber was originally 10 x 20 x 5V5 cubits, or about 17 x 34 x 19 ft, roughly a double cube.

Labor

Ka Many varied estimates have been made regarding the labor force needed to construct the Great Pyramid. Herodotus, the Greek historian in the 5th century BC, estimated that construction may have required the labor of 100,000 slaves for 20 years. Polish architect Wieslaw Kozinski believed that it took as many as 25 men to transport a 1.5-ton stone block; based on this, he estimated the workforce to be 300,000 men on the construction site, with an additional 60,000 off-site. 19th century Egyptologist William Flinders Petrie proposed that the labor force was largely composed not of slaves but of the rural Egyptian population, working during periods when the Nile river was flooded and agricultural activity suspended. Egyptologist Miroslav Verner posited that the labor was organized into a hierarchy, consisting of two gangs of 1000 men, divided into five zaa or phyle of 200 men each, which may have been further divided according to the skills of the workers. Some research suggests alternate estimates to the aforementioned labor size. For instance, mathematician Kurt Mendelssohn calculated that the labor force may have been 50,000 men at most, while Ludwig Borchardt and Louis Croon placed the number at 36,000. According to Verner, a labor force of no more than 30,000 was needed in the Great Pyramid's construction. A construction management study carried out by the firm Daniel, Mann, Johnson, & Mendenhall in association with Mark Lehner and other egyptologists(JUNE 1999 CIVIL ENGINEERING MAGAZINE), estimates that the total project required an average workforce of 13,200 persons and a peak workforce of 40,000 and was completed from start to finish in approximately 10yrs. The study estimates the number of blocks used in construction was between 2-2.8 million (an average of 2.4 million), but settles on a much reduced finished total of 2 million subtracting the estimated area of the hollow spaces of the chambers and galleries. Their calculations suggest the workforce could have sustained a rate of 180 blocks per hour (3 blocks every 60 seconds)with ten man-days for putting each individual block in place. They derived these estimates from contruction projects in the third world that did not use modern machinery. Regardless of how many workers were required for construction, to use the following equation: 2,400,000 (total stones used in construction) ÷ 20 years (estimated time of completion) ÷ 365 days in a year ÷ 10 work hours in a day ÷ 60 minutes in one hour, the resulting answer is 0.55 stones/minute. What this means is that no matter how many workers were used or in what configuration, to complete the construction of the Great Pyramid within this time frame 1.1 blocks would have to be put in place every 2 minutes, ten hours a day, 365 days a year for twenty years. To use the same equation, but instead assuming the time of completion to be one hundred years instead of twenty, it would require 1.1 blocks to be set every ten minutes, ten hours a day, 365 days a year. These equations, however, do not include the time and labor required to design, plan, survey, and level the 13 acre site which the Great Pyramid sits on. Nor does it include construction time for the two other main pyramids on the site, the Sphinx, the temples (which feature stones weighing upwards of 200 tonnes), networks of causeways, several square miles of paving stones (which originally covered the entire Giza plateau), the leveling of the entire Giza plateau, the 35 boat pits carved out of solid bedrock (some of which are nearly 150ft long and 30ft deep), or several other highly laborious features. When considering the time it would have taken to build the Great Pyramid alone, it is worth noting that the construction of the entire Giza plateau is believed to have been accomplished by three pharaohs in less than a hundred years starting with Khufu who reigned from 2606-2583 BC and ending with Menkaure 2548-2530 BC (76 years). To apply the Great Pyramid labor formula (which only provides for the physical act of dropping the stones in place) to the entire Giza plateau would require stones, even the 80-200 tonne variety some of which were quarried over 500 miles away in Aswan, to be placed ten hours a day, 365 days a year for approximately 76 years - not every few minutes, but every few seconds. This feat becomes even more impessive given beginning with king Snefru who ruled from 2630-2606 BC (leaving a span of 100 years between the beginning of his reign and the end of Menkaure's in 2530 BC), three other massive pyramids were built: the Step Pyramid of Saqqara (believed to be the first egyptian pyramid), the Bent Pyramid, and the Red Pyramid of Dashur. Herodotus speculated that the stone blocks used in the Great Pyramid's construction were maneuvered into place by raising them up a succession of short wooden scaffolds. Another possibility proposed by the ancient scholar Diodorus Siculus was that the giant blocks were dragged along a system of ramps to the necessary height. More recently, Mark Lehner speculated that a spiralling ramp, beginning in the stone quarry to the southeast and continuing around the exterior of the pyramid, may have been used. In Lehner's model, the stone blocks may have been drawn on sleds lubricated by water. Another source claims milk was a lubricant. The most precisely cut stone blocks were reserved for the outside. Once in place their corners were smoothed to give an almost shiny outer appearance of the pyramid. For the inner core, the blocks were cut with less precision, since there are gaps big enough to introduce an arm. These gaps were filled with rubble, mixed with gypsum. Recent studies by Gilles Dormion and Jean Patrice Goidin suggest the existence of cavities filled with sand, that could amount to 10 to 15% of the volume of the pyramid. This could reduce the amount of work required of the contruction. The idea of using rollers to move stone blocks was made popular in Hollywood movies, but as of today, whether it be ramp, roller, or otherwise, there are few historical records to demonstrate how ground transportation was done. If a ramp were used to push the top-most blocks of the pyramid into place, the incline would contain more material than the pyramid itself and this material would have had to be removed after construction was completed. Excavation on the area south of the Great Pyramid revealed evidence of the remains of a ramp consisting of two walls built of stone rubble and mixed with Tafla. The area in between was filled with sand and gypsum forming the bulk of the ramp. They were discovered during the work of relocating the Sound and Light Show cables at Giza (Hawass, The Pyramids of Ancient Egypt). Given the mass required to build a ramp of such magnitude to contruct the Great Pyramid as ramp theories suggest, it is unknown what purpose this much smaller newly discovered ramp may have served. According to the theory of materials scientist Joseph Davidovits, the blocks that form the pyramid are not strictly carved stone, but mostly a form of limestone concrete (not moved, but) 'cast', as with modern cement blocks, except -- because of the blocks huge 2.5-15+ tons size -- each in situ. It has also been suggested that Egyptians might have moved the stones with wind power, relying on kites and pulleys rather than huge numbers of slaves. On June 23, 2001, Caltech aeronautics professor Mory Gharib and a small team of undergraduates raised a 3000kg, 3m-tall obelisk into vertical position in 22mph winds in a California desert in under 25 seconds, using a 10m kite connected to a pulley system and support frame, to demonstrate that wind power can be harnessed to create large lifting forces. The originator of this idea, business consultant Maureen Clemmons, recalled seeing a building frieze now displayed in a Cairo museum, showing a wing pattern in bas relief that did not resemble any living bird, directly below which were several men standing near vertical objects that could be ropes. However, though the engineering may have been feasible, Egyptian experts point out there is no evidence that ancient Egyptians used either kites or pulleys as we know them today.

External links

Archeology
- [http://guardians.net/hawass/sphinx-pyramid-main.htm Dr. Zahi Hawass on Sphinx and Pyramids] (Dr. Zahi Hawass is Egypt's Secretary General of the Supreme Council of Antiquities & Director of the Giza Pyramids Excavation project
- [http://www.pbs.org/wgbh/nova/pyramid/ PBS/NOVA on Pyramids]
- [http://www.archaeology.org/cgi-bin/perlfect/search/search.pl?q=pyramid&mode=all Archeology.org and Archaeology Magazine]
- [http://www.gizapyramids.org/ Giza Archives Project] Museum of Fine Art Boston's repository for archaeological records dealing with MFA excavations at Giza
- [http://oi.uchicago.edu/OI/PROJ/GIZ/Giza.html The Giza Mapping Project] Exploration
- [http://www.travellersinegypt.org/archives/2005/03/inside_the_great_pyramid.html Inside the Great Pyramid by Lieutenant-Colonel Fitzclarence] Other theories
- Wall, John, "[http://www.hallofmaat.com/maat/article.php?sid=17 The Wrong Question (or: The Myth of the Mystery of the Missing Messages)]". In the Hall of Maat.
- World-Mysteries.com - Mystic Places : [http://www.world-mysteries.com/mpl_2.htm The Great Pyramid]
- [http://www.aiwaz.net/giza/ Composition of Giza Plateau]
- Ottar Vendel's [http://www.nemo.nu/ibisportal/0egyptintro/3egypt/3main.htm Age of the Pyramids]
- [http://members.aol.com/aditt48670/pyramid.html Pyramid construction theory]
- Joseph Davidovits' "[http://www.geopolymer.org/science_archaeology/pyramids_egypt/index.html Ari-Kat Technology]" - Geopolymer theory of pyramid construction
- Maureen Clemmons' "[http://pr.caltech.edu/periodicals/CaltechNews/articles/v35/obelisk.html How Many Caltechers Does It Take to Raise An Egyptian Obelisk?]" - Wind power construction theory
- Chris Dunn "[http://www.gizapower.com/]" - The Theory that the Giza Pyramid was a giant Maser News
- Guardian's [http://www.guardians.net/egypt/pyramids.htm Pyramids of Egypt]
- [http://www.guardian.co.uk/international/story/0,3604,1293377,00.html Secret chamber may hold key to mystery of the Great Pyramid] (The Guardian, August 30 2004.)
- [http://www.abc.net.au/news/newsitems/200408/s1188387.htm Amateur archaeologists track lost tomb of Cheops] (Australian Broadcasting Corporation, August 30 2004.)
- [http://www.guardians.net/hawass/pbuildrs.htm Pyramid Construction]: Ancient ramp leading to the Great Pyramid discovered, but only of maximal height approximately 100 feet (30 m). Pyramid's original height was 481 feet. Also, the heaviest stone blocks were discovered to have holes bored on opposite sides, indicating the use of cranes (or other mechanical means) to raise and precisely position them. Images
- [http://www.gizapyramid.com/newtour1.htm A Picture Tour of The Great Pyramid] at the [http://www.gizapyramid.com/ Great Pyramid of Giza Research Association].
- Fullscreen Quicktime VR Panorama' [http://www.panoramas.dk/fullscreen/fullscreen38.html Pyramids of Giza]
- Google Satellite maps of the Pyramids [http://maps.google.com/maps?ll=29.977140,31.131649&spn=0.013282,0.017896&t=k&hl=en 29°58'51"N 31°09'00"E]
- [http://www.pyramidcam.com Pyramidcam!] Category:Ancient Egypt Category:Pyramids ja:ギザの大ピラミッド ms:Piramid Besar Kufu

Perfect fifth

The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. The prefix perfect identifies it as belonging to the group of perfect intervals, so called because of their extremely simple pitch relationships resulting in a high degree of consonance. It occurs most commonly built on the root of all major and minor triads and their extensions. However, due to its high level of consonance in this position, the perfect fifth contributes very little to the overall harmonic effect of any chords containing it (power chords excepted). In any situation that necessitates the omission of notes from a chord, such as for practical reasons of fingering, for example, the note forming the perfect fifth above the chord's root can often be safely omitted, its absence being barely, if at all, noticeable. The 'perfect fifth is abbreviated as P5 and its inversion is the perfect fourth. A perfect fifth in just intonation, a just fifth, corresponds to a pitch ratio of 3:2, while in 12-tone equal temperament, a perfect fifth is equal to seven semitones, a ratio of 1:2^7/12 (approximately 1.4983), or 700 cents, about two cents smaller. The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. The circle of fifths is a model of

Ratio

Examples

See also

Reason

Reason and logic

Reason versus logic

Reason as logic

Reason and faith

References

External links

Quantity

See also:

Quotient

Number

Examples

Further generalizations

Numerals and numbering

Extensions

See also

External links

References

Colon (punctuation)

Uses

Mathematics

Linguistics

Computer representation

The colon in foreign languages

Other meanings

Rational number

Arithmetic

History

Egyptian fractions

Formal construction

Properties

Real numbers

p-adic numbers

Quantity

See also:

Eiffel Tower

Background

Installations

Events

The 72 names

Image copyright

Appearance in film

Imitations and reproductions

Imitations (similar towers, not scale models)

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External links

Great Pyramid

Age and location

Quick Facts

Construction

Labor

Further reading

See also

External links

Perfect fifth