A week is a time unit equal to seven days.

The English word week continues an Old English ''wice'', ultimately from a Common Germanic '''', from a root '''' "turn, move, change". The Germanic word probably had a wider meaning prior to the adoption of the Roman calendar, perhaps "succession series", as suggested by Gothic ''wikō'' translating ''taxis'' "order" in Luke 1:8.

The term "week" is sometimes expanded to refer to other time units comprising a few days. Such "weeks" of between four and ten days have been used historically in various places. Intervals longer than 10 days are not usually termed "weeks" as they are closer in length to the fortnight or the month than to the seven-day week.

Seven-day week

Evidence of continuous use of a seven-day week appears with the Jews during the Babylonian Captivity of the 6th century BC. Both Judaism (based on the Creation narrative in the Torah/Bible) and ancient Babylonian religions used a seven-day week. Other cultures adopted the seven-day week at different times. Between the 1st and 3rd centuries the Roman Empire gradually replaced the eight-day Roman nundinal cycle with the seven-day week. Hindus may have adopted a seven day week as early as the 1st century BC. There is evidence of some Chinese groups using a seven day week as early as 4th century AD.

Systems derived from the seven-day week

Soviet Union

Between 1929 and 1931 the USSR changed from the seven-day week to a five-day week. There were 72 weeks and an additional five national holidays inserted within three of them totaling a year of 365 days.

In 1931, after the Soviet Union's five-day week they changed to a six-day week. Every sixth day (6th, 12th, 18th, 24th and 30th) of the Gregorian Calendar was a state rest day. The five additional national holidays in the earlier five-day week remained and did not fall on the state rest day.

But as January, March, May, July, August, October and December have 31 days, the week after the state rest day of the 30th was seven days long (31st–7th). This extra day was a working day for most or an extra holiday for others.

Also as February is only 28 or 29 days depending on whether it is a leap year or not, the first of March was also made a state rest day, although not every enterprise conformed to this.

To clarify, the week after the state rest day, 24/25 February to 1 March, was only five or six days long, depending on whether it was a leap year or not. The week after that, 2 to 6 March, was only five days long.

The calendar was abandoned 26 June 1940 and the seven-day week reintroduced the day after.

Decimal calendar

A 10-day week, called ''décade'', was used in France for nine and a half years from October 1793 to April 1802; furthermore, the Paris Commune adopted the Revolutionary Calendar for 18 days in 1871.

Christian "eighth day"

For early Christians, Sunday, as well as being the first day of the week, was also the spiritual eighth day, as it symbolised the new world created after Christ's resurrection. The concept of the eighth day was symbolic only and had no effect on the use of the seven-day week for calendar purposes. Justin Martyr wrote: "the first day after the Sabbath, remaining the first of all the days, is called, however, the eighth, according to the number of all the days of the cycle, and [yet] remains the first". This does not set up an eight-day week, since the eighth day is also considered to be the first day of the next cycle (i.e., not the following day).

A period of eight days, starting and ending on a Sunday or starting on a major feast day and finishing on the same day of the week a (seven-day) week later, is called an octave. For centuries these were a major feature of the liturgical calendar, particularly of the Catholic Church, and some are still observed, though the number of such octaves has now been radically reduced. Some modern Church uses also preserve the idea of an eight-day period, starting and finishing on the same day of the week, and retain the name "octave" for them; for example, many churches observe an annual "Octave of Prayer for Christian Unity" on 18–25 January or in the week that begins with Pentecost Sunday. Organizations such as 8th Day Center for Justice, based out of Chicago, Illinois, use the concept in terms of social justice as well.

Hermetic lunar week

The Hermetic Lunar Week Calendar is one of many proposed reforms to the Gregorian Calendar. The lunation is divided into the four moon phases and has six, seven, eight, or nine days depending on the actual time difference between the full moon, first quarter, new moon and last quarter.

"Weeks" in other calendars

Periods termed "weeks" in calendars unrelated to the Judeo-Christian tradition.

Three day

The names of the days of the week (''aste'') in Guipuscoan Basque point to an earlier three-day week. #''astelehena'' ("week-first", Monday) #''asteartea'' ("week-between", Tuesday) #''asteazkena'' ("week-last", Wednesday)

Four-day

The Igbo of Nigeria have a traditional calendar with a four-day week. This "market week" features prominently in the fiction of Chinua Achebe.

Five-day

The Javanese people of Indonesia have a five-day week known as the Pasaran cycle. This is still in use today and superimposed with seven-day week of the Gregorian calendar and Islamic calendar to become what is known as the 35-day Wetonan Cycle.

Six-day

The Akan people of West Africa have a 42-day cycle known as ''Adaduanan''. The ''Adaduanan'' cycle appears to be based on an older six-day week, still existent in some northern Guan communities such as the Nchumuru, on which is superimposed a seven-day week which may have been brought south with itinerant traders from the Savannah. The six-day week is referred to as ''Nanson'' (literally seven-days) and reflects the lack of zero in the numbering systems; the last day and the first day are both included when counting the days of a week.

Eight-day

Nundinal cycle

The ancient Etruscans developed an eight-day market week known as the nundinal cycle around the 8th or 7th century BC. This was passed on to the Romans no later than the 6th century BC. As Rome expanded, it encountered the seven-day week and for a time attempted to include both. The popularity of the seven-day rhythm won and the eight-day week disappeared forever.

The cycle of seven days, named after the sun, the moon, and the five planets visible to the naked eye, was already customary in the time of Justin Martyr, who wrote of the Christians meeting on the Day of the Sun (Sunday).

Emperor Constantine eventually established the seven-day week in the Roman calendar in AD 321.

Celts

It is believed the ancient celts of the British Isles used a nine-''night'' week. The moon was used to measure one day from another so nights were more significant. The nine nights divided nicely into a sidereal month of 27 nights. Each week of nine nights had eight days.

Nine-day

Baltic calendars

In the 14th century, the Grand Duchy of Lithuania used a solar-lunar calendar. The structure of this calendar was understood with the help of the so-called Gediminas Sceptre discovered in 1680.

Historical records give evidence that the week of the ancient Balts was nine days long. Thus, the sidereal month must have been divided into three parts.

Ten-day

China

The Chinese ten-day week went as far back as the Shang Dynasty (1200-1045 BC). The law in the Han Dynasty (206 BC – AD 220) required officials of the empire to rest every five days, called ''mu'' (沐), while it was changed into 10 days in the Tang Dynasty (AD 618 – 907), called ''huan'' or ''xún'' (旬). Months were almost three weeks long (alternating 29 and 30 days to keep in line with the lunation). The weeks were labelled ''shàng xún'' (上旬), ''zhōng xún'' (中旬), and ''xià xún'' (下旬) which mean roughly "upper", "middle" and "lower" week.

Egypt

Ancient Egypt had a ten-day week, three weeks per month with ten extra days at the end of the year.

Revolutionary France

The French Republican Calendar consisted of three ten-day weeks in a month called ''décades''

Other calendar intervals

Aztecs

The Aztecs divided a ritual cycle of 260 days, known as Tonalpohualli, into 20 weeks of 13 days known as Trecena.

The Aztecs divided a solar year of 365 days, Xiuhpohualli, into 18 periods of 20 days and five nameless days known as ''Nemontemi''. Although some call this 20-day division or grouping a month, it has no relation to a lunation and therefore the word "week" is more appropriate.

Maya

The Maya divided a 260-day ritual cycle known Tzolk'in into 20 weeks of 13 days known as Trecena.

The Maya also divided the year, Haab', into 18 periods of 20 days, ''Uinal'', and five nameless days known as ''Wayeb'''.

Bali, Indonesia

The Pawukon is a 210-day calendar consisting of 10 different concurrent weeks of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 days.

References

Further reading

External links

Englisc Rímbóc: The Anglo-Saxon Calendar

Category:Units of time

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In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.

In physical terms, ''dimension'' refers to the constituent structure of all space (cf. volume) and its position in time (perceived as a scalar dimension along the ''t''-axis), as well as the spatial constitution of objects within – structures that have correlations with both particle and field conceptions, interact according to relative properties of mass, and which are fundamentally mathematical in description. These or other axes may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and occurrences. Physical theories that incorporate time, such as general relativity, are said to work in 4-dimensional "spacetime", (defined as a Minkowski space). Modern theories tend to be "higher-dimensional" including quantum field and string theories. The state-space of quantum mechanics is an infinite-dimensional function space.

The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

In mathematics

In mathematics, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This ''intrinsic'' notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.

The dimension of Euclidean -space is . When trying to generalize to other types of spaces, one is faced with the question “what makes -dimensional?" One answer is that to cover a fixed ball in by small balls of radius , one needs on the order of such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension. But there are also other answers to that question. For example, one may observe that the boundary of a ball in looks locally like and this leads to the notion of the inductive dimension. While these notions agree on , they turn out to be different when one looks at more general spaces.

A tesseract is an example of a four-dimensional object. Whereas outside of mathematics the use of the term "dimension" is as in: "A tesseract ''has four dimensions''", mathematicians usually express this as: "The tesseract ''has dimension 4''", or: "The dimension of the tesseract ''is'' 4".

Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schlafi's 1852 ''Theorie der vielfachen Kontinuität'', Hamilton's 1843 discovery of the quaternions and the construction of the Cayley Algebra marked the beginning of higher-dimensional geometry.

The rest of this section examines some of the more important mathematical definitions of the dimensions.

Dimension of a vector space

The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the ''Hamel dimension'' or ''algebraic dimension'' to distinguish it from other notions of dimension.

=== Manifolds === A connected topological manifold is locally homeomorphic to Euclidean -space, and the number is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold.

The theory of manifolds, in the field of geometric topology, is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases }} are simplified by having extra space in which to "work"; and the cases and are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.

Lebesgue covering dimension

For any normal topological space , the Lebesgue covering dimension of is defined to be n if ''n'' is the smallest integer for which the following holds: any open cover has an open refinement (a second open cover where each element is a subset of an element in the first cover) such that no point is included in more than elements. In this case dim . For a manifold, this coincides with the dimension mentioned above. If no such integer exists, then the dimension of is said to be infinite, and one writes dim . Moreover, has dimension −1, i.e. dim if and only if is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".

Inductive dimension

An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a ''new direction'', one obtains a 2-dimensional object. In general one obtains an ()-dimensional object by dragging an dimensional object in a ''new'' direction.

The inductive dimension of a topological space could refer to the ''small inductive dimension'' or the ''large inductive dimension'', and is based on the analogy that )-dimensional}} balls have dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

Hausdorff dimension

For sets which are of a complicated structure, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values. The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe many natural objects and phenomenon.

Hilbert spaces

Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the above dimensions coincide.

In physics

Spatial dimensions

Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies; ''i.e.'', moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Space and Cartesian coordinate system.)

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Time

A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction.

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space.

Additional dimensions

Theories such as string theory and M-theory predict that physical space in general has in fact 10 and 11 dimensions, respectively. The extra dimensions are spatial. We perceive only three spatial dimensions, and no physical experiments have confirmed the reality of additional dimensions. A possible explanation that has been suggested is that space acts as if it were "curled up" in the extra dimensions on a subatomic scale, possibly at the quark/string level of scale or below.

Up to now, no experimental or observational signs of extra dimensions have been officially reported. An analysis of results from the Large Hadron Collider in December 2010 severely constrains theories with large extra dimensions.

Networks and dimension

Some complex networks are characterized by fractal dimensions. The concept of dimension can be generalized to include networks embedded in space. The dimension characterize their spatial constraints.

Literature

Perhaps the most basic way in which the word ''dimension'' is used in literature is as a hyperbolic synonym for ''feature'', ''attribute'', ''aspect'', or ''magnitude''. Frequently the hyperbole is quite literal as in ''he's so 2-dimensional'', meaning that one can see at a glance what he ''is''. This contrasts with 3-dimensional objects which have an interior that is hidden from view, and a back that can only be seen with further examination.

Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones.

One of the most heralded science fiction novellas regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novel ''Flatland'' by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described ''Flatland'' as "The best introduction one can find into the manner of perceiving dimensions."

The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's ''The Appendix and the Spectacles'' (1928) and Murray Leinster's ''The Fifth-Dimension Catapult'' (1931); and appeared irregularly in science fiction by the 1940s. Some of the classic stories involving other dimensions include Robert A. Heinlein's 1941 ''—And He Built a Crooked House'', in which a California architect designs a house based on a three-dimensional projection of a tesseract, and Alan E. Nourse's ''Tiger by the Tail'' and ''The Universe Between'', both from 1951. Another reference would be Madeleine L'Engle's novel ''A Wrinkle In Time'' (1962) which uses the 5th Dimension as a way for "tesseracting the universe", or in a better sense, "folding" space in half to move across it quickly.

The fourth and fifth dimensions were also a key component of the book ''The Boy Who Reversed Himself'', by William Sleator.

Philosophy

In 1783, Kant wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition ''a priori'' because it is apodictically (demonstrably) certain."

More dimensions

See also

Degrees of freedom

Dimension (data warehouse) and dimension tables

Dimensional analysis

Fractal dimension

Hyperspace (disambiguation page)

Space-filling curve

Intrinsic dimension

A list of topics indexed by dimension

Zero dimensions:

*Point

*Zero-dimensional space

*Integer

One dimension:

*Line

*Graph (combinatorics)

*Real number

Two dimensions:

*Complex number

*Cartesian coordinate system

*List of uniform tilings

*Surface

Three dimensions

*Platonic solid

*Stereoscopy (3-D imaging)

*Euler angles

*3-manifold

*Knot (mathematics)

Four dimensions:

*Spacetime

*Fourth spatial dimension

*Convex regular 4-polytope

*Quaternion

*4-manifold

High-dimensional topics from mathematics:

*Octonion

*Vector space

*Manifold

*Calabi–Yau spaces

High-dimensional topics from physics:

*Kaluza–Klein theory

*String theory

*M-theory

Infinitely many dimensions:

*Hilbert space

*Function space

References

Further reading

Edwin A. Abbott, (1884) ''Flatland: A Romance of Many Dimensions'', Public Domain. Online version with ASCII approximation of illustrations at Project Gutenberg.

Thomas Banchoff, (1996) ''Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Second Edition'', Freeman.

Clifford A. Pickover, (1999) ''Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons'', Oxford University Press.

Rudy Rucker, (1984) ''The Fourth Dimension'', Houghton-Mifflin.

Michio Kaku, (1994) ''Hyperspace, a Scientific Odyssey Through the 10th Dimension'', Oxford University Press.

Category:Physics Category:Abstract algebra Category:Algebra Category:Linear algebra

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