## map

**-**

## map

map, conventionalized representation of spatial phenomena on a plane surface. Unlike photographs, maps are selective and may be prepared to show various quantitative and qualitative facts, including boundaries, physical features, patterns, and distribution. Each point on such a map corresponds to a geographical position in accordance with a definite scale and projection (see map projection). Maps may also represent such comparative data as industrial power, population density, and birth and death rates. The earliest European printed maps (2d half of the 15th cent.) were made from woodcuts; maps are now reproduced by several processes, including photoengraving, wax engraving, and lithography. See also chart.

**Ancient Mapmaking**

Cartography, or mapmaking, antedates even the art of writing. Diagrams of areas familiar to them were made by Marshall Islanders, Eskimo, Native Americans, and many other preliterate peoples. Maps drawn by ancient Babylonians, Egyptians, and Chinese have been found. The oldest known map, now on exhibition in the Semitic Museum of Harvard, is a Babylonian clay tablet dating from c.2500 BC Our present system of cartography was established by the Greeks, who remained unexcelled until the 16th cent. Scientific measurements of earth distances by means of meridians and parallels were first made by Eratosthenes (3d cent. BC). Of the ancient scholars, the mathematician and geographer Ptolemy (2d cent. AD), expounded on the principles of cartography; his system was followed for many centuries, although his basic error in underestimating the earth's size was not corrected until the age of Mercator. Only the Mediterranean world was represented with any accuracy in early maps. During the Middle Ages, while European cartographers produced artistic, idealized maps, Arabic mapmakers, notably Idrisi (12th cent.), carried on the work of Ptolemy, and the Chinese produced the first printed maps.

**Cartography in the Sixteenth through Eighteenth Centuries**

Three major events contributed to the spectacular renaissance of cartography in Europe around 1500—the rediscovery and translation into Latin of Ptolemy's *Geographia,* the invention of printing and engraving, and the great voyages of discovery. This renaissance was manifested by the work of Gerardus Mercator in the first modern world atlas, published in 1570 by Abraham Ortelius, and by the decorative, paintinglike maps of the French Sanson family (17th cent.). Improvements in the methods of surveying and increased emphasis on accuracy led to the noted work in the 18th cent. of the Frenchmen Guillaume Delisle and J. B. B. d'Anville, the founders of modern cartography. After 1750 many European governments undertook the systematic mapping of their countries. The first important national survey was made in France (published 1756), followed by the Ordnance Survey of Great Britain (published 1801) and the topographic survey of Switzerland (organized 1832). In the United States the U.S. Geological Survey (established 1879) has mapped much of the country on varying scales.

**During the Nineteenth and Twentieth Centuries**

During the 19th cent. the demand for national maps was fulfilled, and famous world atlases were published. But with the advent of the 20th cent. the need arose for an international map of the world on a uniform scale. Accordingly, at several meetings of the International Geographical Congress (1891, 1909, 1913), the German Albrecht Penck presented and perfected plans for a world map on a scale of 1:1,000,000, to consist of about 1,500 sheets, each covering four degrees of latitude and six degrees of longitude in a modified conic projection. Uniformity of lettering and the use of layer tints to indicate relief were agreed upon. However, only part of the work has been completed. The greatest single contribution to the map of the world was made by the American Geographical Society of New York, which completed (1945) its 107-sheet *Map of Hispanic America.*

During World Wars I and II the science and art of mapping were greatly advanced. Modern technology, using remote sensing by airborne and satellite radar, as well as devices called multispectral scanners, has made it possible to quickly collect and update information for mapmaking. Computerized geographic information systems, first developed in the 1960s, now are used to link information stored in databases to maps, increasing and varying the amount of information a map can display. Such systems are used to produce maps for business use, law enforcement, natural-disaster prediction, and many other purposes. In recent years the critical cartography movement, led by a group of British scholars, notably the late J. B. Harley, has studied maps as sociopolitical constructs that interpret reality and reflect the historical power structure as well as their makers' ideas about the world.**Bibliography**

See T. W. Birch, *Maps: Topographical and Statistical* (2d ed. 1964); D. Greenhood, *Mapping* (rev. ed. 1964); F. J. Monkhouse and H. R. Wilkinson, *Maps and Diagrams* (1971); N. J. W. Thrower, *Maps and Man* (1972); G. R. Crone, *Maps and Their Makers* (5th ed. 1978); L. Bagrow and R. A. Skelton, *History of Cartography* (enl. 2d ed. 1985, repr. 2010); M. Monmonier, *How to Lie with Maps* (1991); A. H. Robinson et al., *Elements of Cartography* (6th ed. 1995); J. Black, *Maps and Politics* (1997); M. H. Edney, *Mapping an Empire* (1997); J. B. Harley and D. Woodward, ed., *History of Cartography* (2 vol., 1987–); J. B. Harley, *The New Nature of Maps* (2001); J. Black, *Maps and Politics* (2001); S. Schulten, *The Geographical Imagination in America, 1880–1950* (2001); P. Whitfield, *The Image of the World* (upd. ed. 2010); J. Brotton, *A History of the World in 12 Maps* (2013).

## Map

# Map

A map, or mapping, is a rule, often expressed as an equation, that specifies a particular element of one set for each element of another. To help understand the notion of map, it is useful to picture the two sets schematically, and map one onto the other, by drawing connecting arrows from members of the first set to the appropriate members of the second set.

For instance, let the set mapped from be well-known cities in Texas, specifically, let *A* = {Abilene, Amarillo, Dallas, Del Rio, El Paso, Houston, Lubbock, Pecos, San Antonio}. We will map this onto the set containing whole numbers of miles. The rule is that each city maps onto its distance from Abilene. The map can be shown as a diagram in which an arrow points from each city to the appropriate distance (Figure 1).

A relation is a set of ordered pairs for which the first and second elements of each ordered pair are associated or related. A function, in turn, is a relation for which every first element of an ordered pair is associated with one, and only one, second element. Thus, no two ordered pairs of a function have the same first element. However, there may be more than one ordered pair with

the same second element. The set, or collection, of all the first elements of the ordered pairs is called the domain of the function. The set of all second elements of the ordered pairs is called the range of the function. A function is a set, so it can be defined by writing down all the ordered pairs that it contains. This is not always easy, however, because the list may be very lengthy, even infinite (that is, it may go on forever).

When the list of ordered pairs is too long to be written down conveniently, or when the rule that associates the first and second elements of each ordered pair is so complicated that it is not easily guessed by looking at the pairs, then it is common practice to define the function by writing down the defining rule. Such a rule is called a map, or mapping, which, as the name suggests, provides directions for superimposing each member of a function’s domain onto a corresponding member of its range. In this sense, a map is a function. The words “map” and “function” are often used interchangeably. In addition, because each member of the domain is associated with one and only one member of the range, mathematicians also say that a function maps its domain onto its range, and refer to members of the range as values of the function.

The concept of map or mapping is useful in visualizing more abstract functions, and helps to remind us that a function is a set of ordered pairs for which a well defined relation exists between the first and second elements of each pair. The concept of map is also useful in defining what is meant by composition of functions. Given three sets *A, B,* and *C,* suppose that *A* is the domain of a function *f,* and that *B* is the range of *f.* Further, suppose that *B* is also the domain of a second function *g,* and that *C* is the range of *g.* Let the symbol o

### KEY TERMS

**Domain** —The set, or collection, of all the first elements of the ordered pairs of a function is called the domain of the function.

**Function** —A function is a set of ordered pairs or which the first and second elements of each ordered pair are related, and for which every first element of an ordered pair is associated with one, and only one, second element.

**Range** —The set containing all the values of the function.

represent the operation of composition that is defined to be the process of mapping *A* onto *B* and then mapping *B* onto *C.* The result is equivalent to mapping *A* directly onto *C* by a third function, call it *h.* This is writte *n g* so *f* = h, and read “the composition of *f* and *g* equals *h.* ”

## Resources

### BOOKS

Christian, Robert R. *Introduction to Logic and Sets.* Waltham, MA: Blaisdell Publishing Co., 1965.

Gowar, Norman. *An Invitation to Mathematics.* New York: Oxford University Press, 1979.

Kyle, James. *Mathematics Unraveled.* Blue Ridge Summit, PA: Tab Book, 1976.

Peterson, Ivars. *Islands of Truth, A Mathematical Mystery Cruise.* New York: W. H. Freeman, 1990.

### OTHER

*Wolfram MathWorld.* “Map” <http://mathworld.wolfram.com/Map.html> (accessed December 3, 2006).

J. R. Maddocks

## Map

# Map

A map, or mapping, is a rule, often expressed as an equation, that specifies a particular element of one set for each element of another set. To help understand the notion of map, it is useful to picture the two sets schematically, and map one onto the other, by drawing connecting arrows from members of the first set to the appropriate members of the second set. For instance, let the set mapped from be well-known cities in Texas, specifically, let A = {Abilene, Amarillo, Dallas, Del Rio, El Paso, Houston, Lubbock, Pecos, San Antonio}. We will map this onto the set containing whole numbers of miles. The rule is that each city maps onto its distance from Abilene. The map can be shown as a diagram in which an arrow points from each city to the appropriate distance.

A **relation** is a set of ordered pairs for which the first and second elements of each ordered pair are associated or related. A **function** , in turn, is a relation for which every first element of an ordered pair is associated with one, and only one, second element. Thus, no
two ordered pairs of a function have the same first element. However, there may be more than one ordered pair with the same second element. The set, or collection, of all the first elements of the ordered pairs is called the **domain** of the function. The set of all second elements of the ordered pairs is called the range of the function. A function is a set, so it can be defined by writing down all the ordered pairs that it contains. This is not always easy, however, because the list may be very lengthy, even infinite (that is, it may go on forever). When the list of ordered pairs is too long to be written down conveniently, or when the rule that associates the first and second elements of each ordered pair is so complicated that it is not easily guessed by looking at the pairs, then it is common practice to define the function by writing down the defining rule. Such a rule is called a map, or mapping, which, as the name suggests, provides directions for superimposing each member of a function's domain onto a corresponding member of its range. In this sense, a map is a function. The words map and function are often used inter-changeably. In addition, because each member of the domain is associated with one and only one member of the range, mathematicians also say that a function maps its domain onto its range, and refer to members of the range as values of the function.

The concept of map or mapping is useful in visualizing more abstract functions, and helps to remind us that a function is a set of ordered pairs for which a well defined relation exists between the first and second elements of each pair. The concept of map is also useful in defining what is meant by composition of functions. Given three sets A, B, and C, suppose that A is the domain of a function f, and that B is the range of f. Further, suppose that B is also the domain of a second function g, and that C is the range of g. Let the symbol o represent the operation of composition which is defined to be the process of mapping A onto B and then mapping B onto C. The result is equivalent to mapping A directly onto C by a third function, call it h. This is written g o f = h, and read "the composition of f and g equals h."

## Resources

### books

Christian, Robert R. *Introduction to Logic and Sets.* Waltham, MA: Blaisdell Publishing Co., 1965.

Gowar, Norman. *An Invitation to Mathematics.* New York: Oxford University Press, 1979.

Kyle, James. *Mathematics Unraveled.* Blue Ridge Summit, PA: Tab Book, 1976.

Peterson, Ivars. *Islands of Truth, A Mathematical Mystery**Cruise.* New York: W. H. Freeman, 1990.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Domain**—The set, or collection, of all the first elements of the ordered pairs of a function is called the domain of the function.

**Function**—A function is a set of ordered pairs or which the first and second elements of each ordered pair are related, and for which every first element of an ordered pair is associated with one, and only one, second element.

**Range**—The set containing all the values of the function.

## map

map
/ map/
•
n.
1.
a diagrammatic representation of an area of land or sea showing physical features, cities, roads, etc.:
*a street map* |
fig.
*expansion of the service sector is reshaping the map of employment.*
∎
a two-dimensional representation of the positions of stars or other astronomical objects.
∎
a diagram or collection of data showing the spatial arrangement or distribution of something over an area:
*an electron density map.*
∎
Biol.
a representation of the sequence of genes on a chromosome or of bases in a DNA or RNA molecule.
∎
Math.
another term for mapping.
2. inf., dated
a person's face.
•
v.
(mapped
, map·ping
) [tr.]
represent (an area) on a map; make a map of:
*inaccessible parts will be mapped from the air.*
∎
record in detail the spatial distribution of (something):
*the project to map the human genome.*
∎ [tr.]
associate (a group of elements or qualities) with an equivalent group, according to a particular formula or model:
*the transformational rules map deep structures into surface structures.*
∎
Math.
associate each element of (a set) with an element of another set.
∎ [intr.]
be associated or linked to something:
*it is not obvious that the subprocesses of language will map onto individual brain areas.*
PHRASES:
off the map
(of a place) very distant or remote:
*just a hick town, right off the map.*
put something on the map
bring something to prominence:
*the exhibition put Cubism on the map.*
wipe something off the map
obliterate something totally.PHRASAL VERBS:
map something out
plan a route or course of action in detail:
*I mapped out a route over familiar country near home.*DERIVATIVES:
map·less
adj.
map·pa·ble
adj.
map·per
n.

## MAP

**MAP Acronym for Manufacturing Automation Protocol.** A set of protocols originally devised by a group of US manufacturers of mechanical engineering products. This original group has been expanded to include other parties, and the protocols have become ISO OSI (open systems interconnection) standards. The protocols are intended to facilitate the exchange of data relevant to mechanical-engineering design and manufacture. They cover not only the problems of process control and assembly within a single manufacturing plant, but also the exchange of design and manufacturing data between a main contractor and his subcontractors. See also TOP, STEP.

## map

**map** •**bap**, cap, chap, clap, crap, dap, entrap, enwrap, flap, frap, gap, giftwrap, hap, Jap, knap, lap, Lapp, map, nap, nappe, pap, rap, sap, schappe, scrap, slap, snap, strap, tap, trap, wrap, yap, zap
•stopgap • mayhap • mishap • madcap
•blackcap • redcap • kneecap
•handicap
•**nightcap**, whitecap
•**snowcap**, toecap
•foolscap • hubcap • skullcap
•dunce cap • handclap • dewlap
•mudflap • thunderclap • burlap
•bitmap • catnap • kidnap • Saranwrap
•mantrap • claptrap • deathtrap
•chinstrap • jockstrap • mousetrap
•bootstrap • suntrap • firetrap
•heeltap

## map

**map** **1.** (**mapping**) See function.

**2.** See memory map.

**3.** See bitmap, pixmap.

**4.** See Karnaugh map.

## MAP

**MAP** major air pollutant

• Computing Manufacturing Automation Protocol

• maximum average price

• Med. mean arterial (blood) pressure

• medical aid post

• Member of the Association of Project Managers

• Ministry of Aircraft Production

• modified American plan (payment system in US hotels)

## map

**map** sb. XVI. — medL. *mappa*, short for *mappa mundī* ‘sheet of the world’, i.e. *mappa* (in classL. table-cloth, NAPKIN), *mundī* g. of *mundus* world.

Hence vb. XVI.

## map

**map** Graphic representation of part or all of the Earth's surface. Maps are usually printed on a flat surface using various kinds of projections based on land surveys, aerial photographs and other sources.