dinsdag 31 mei 2011

Representation (politics)

From Wikipedia, the free encyclopedia
In politics, representation describes how some individuals stand in for others or a group of others, for a certain time period. Representation usually refers to representative democracies, where elected officials nominally speak for their constituents in the legislature. Generally, only citizens are granted representation in the government in the form of voting rights; however, some democracies have extended this right further.

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[edit] Theories of representation

The most groundbreaking work on this subject was done by Hanna Fenichel Pitkin who established four theories of representation[1]:
  1. Formalistic Representation, including:
    1. Authorization
    2. Accountability
  2. Symbolic Representation
  3. Descriptive Representation, and
  4. Substantive Representation
  5. Interest Model of Representation

[edit] Burke

British politician Edmund Burke in his 1774 Speech to the Electors at Bristol at the Conclusion of the Poll was noted for his articulation of the principles of representation against the notion that elected officials should be delegates who exactly mirror the opinions of the electorate:
It ought to be the happiness and glory of a representative to live in the strictest union, the closest correspondence, and the most unreserved communication with his constituents. Their wishes ought to have great weight with him; their opinion, high respect; their business, unremitted attention. It is his duty to sacrifice his repose, his pleasures, his satisfactions, to theirs; and above all, ever, and in all cases, to prefer their interest to his own. But his unbiased opinion, his mature judgment, his enlightened conscience, he ought not to sacrifice to you, to any man, or to any set of men living. These he does not derive from your pleasure; no, nor from the law and the constitution. They are a trust from Providence, for the abuse of which he is deeply answerable. Your representative owes you, not his industry only, but his judgment; and he betrays, instead of serving you, if he sacrifices it to your opinion.[2]
Pitkin points out that Burke linked the district's interest with the proper behaviour of its elected official, explaining, "Burke conceives of broad, relatively fixed interest, few in number and clearly defined, of which any group or locality has just one. These interests are largely economic or associated with particular localities whose livelihood they characterize, in his over-all prosperity they involve."[3]

[edit] Representation by population

In this method, elected representatives will be chosen by more or less numerically equivalent blocks of voters (See also Proportional Representation). This is not always practical for historical and current political reasons, and sometimes is impractical purely on the basis of logistics, as in regions where travel is difficult and distances are long. The shortened term "rep-by-pop" is used in Britain but is relatively uncommon in U.S.
Historically rep-by-pop is the alternative to rep-by-area. However, in the colonial countries, the geographic realities made a necessity of low-population electoral districts in order to give meaningful representation to remote communities, and only in urban and suburban areas has there been any success with applying rep-by-pop more or less evenly.
In the United States and other democracies, typically the lower house of a bicameral (two-chamber) system is based on population—more or less—while the upper House is based on area. Or, as it might be put in the United Kingdom, on title to land, as was originally the case with the old pre-Reforms House of Lords. In the Senate or the Lords, it does not matter how many people are living in your jurisdiction, it matters that you have the jurisdiction (by election, heredity or appointment—the US, the UK and Canada respectively).

[edit] Representation by area

The principle of rep-by-pop, when brought in and promoted publicly, removed many archaic seats in the British House of Commons although some northern and rural counties necessarily still have variably lower populations than most urban ridings. Former British colonies like Canada and Australia also have rural and wilderness areas spanning tens of thousands of square miles, with fewer voters in them than a tiny urban-core riding. In the most extreme case, one riding of the Canadian parliament covers more than 2 million square kilometres, Nunavut, yet has less than one third the average number of voters for a riding, with a population of about 30,000. Making the riding larger would be difficult for the elected member, as well as for campaigning and also unfair to remotely rural constituents, whose concerns are radically different from those of the medium-sized towns that typically dominate the electorate in such ridings.
The American Constitution has built into it a series of compromises between rep-by-pop and rep-by-area: two Senators per state, at least one Representative per state, and representation in the electoral college. In Canada, provinces such as Prince Edward Island have unequal representation in Parliament (in the Commons as well as the Senate) relative to Ontario, British Columbia, and Alberta, partly for historical reasons, partly because those electoral allotments are constitutionally guaranteed, and partly because governments have simply chosen to under-represent certain voters and over-represent others. In the United States, Baker v. Carr (1962) established the "one-person/one vote" standard, that each individual had to be weighted equally in legislative apportionment.
In Canada, until recent reforms, there were still many federal and provincial electoral districts in British Columbia and other provinces that had less than a few thousand votes cast, notably Atlin, covering the province's far northwest, with no more than 1,500. The area of the riding was about the size of New Brunswick and Nova Scotia combined, and larger than many American states. In practicality, the voters of the tiny communities scattered across the subarctic landscape, less than the population of a city block, had as much electoral clout as two Fraser Valley municipalities totaling up to 60,000 in population. The population imbalance between largely rural areas and overwhelmingly urban areas is one reason why the realities of representation by area still have sway against the ideal of representation by population.

[edit] Descriptive representation

Descriptive representation is the idea that elected representatives in democracies should represent not only the expressed preferences of their constituencies (or the nation as a whole) but also those of their descriptive characteristics that are politically relevant, such as geographical area of birth, occupation, ethnicity, or gender. According to this idea, an elected body should resemble a representative sample of the voters they are meant to represent concerning outward characteristics—a constituency of 50% women and 20% blacks, for example, should have 50% female and 20% black legislators.
Sometimes voting systems that obtain proportional representation may achieve descriptive representation as well. However this can be guaranteed only to the extent that voting patterns reflect descriptive characteristics of the voters. If a particular trait is not a concern for voters or prospective candidates (for instance, eye color), then, if the system does not introduce other biases, an elected body will resemble a random sampling of the voters instead.
Some [Ulbig 2005] argue that cynicism and distrust towards government of disadvantaged minorities is partly due to not having representatives with similar characteristics. Supporters of this argument point out that as descriptive representation increases, distrust decreases. This can be the basis of laws imposing that half the candidates on a given list be women (for example in France since 2001) or of voluntary measures (Spain's current government has eight women and eight men). Opponents of such logic argue that political interests as already addressed by the political system may play a larger role. For example, only 2% of African-Americans supported the Bush administration[citation needed] despite the high-profile Bush nominations of the African Americans Colin Powell and Condoleezza Rice.

[edit] Interest Model of Representation

In his masters thesis, Uno proposed what he called an Interest Model of Representation that relied less on descriptive, authorization, accountability representation or how responsive a representative was to the represented than how well a representative acted in a interest of the represented. His Interest Model used as its basis Pitkin (1980) definition of representation, "Representing here means acting in the interest of the represented, in a manner responsive to them." He posited that for representation to have substance and moral strength rather than just descriptive value, representation needed to be based on the capacity of the represented, the nature of interest, and the political context in which representation took place.

[edit] See also

[edit] References

  1. ^ Political Representation - Stanford Encyclopedia of Philosophy
  2. ^ The Works of the Right Honourable Edmund Burke. Volume I (London: Henry G. Bohn, 1854), pp. 446-8.
  3. ^ Hanna Fenichel Pitkin, The concept of representation (1972) p. 174

[edit] Bibliography

  • Mansbridge, Jane. (1999) "Should Blacks Represent Blacks and Women Represent Women? A Contingent `Yes'" Journal of Politics, vol. 61(3): 627-657.
  • Pitkin, Hannah. (1967) The Concept of Representation. University of California Press.
  • Phillips, Anne. 1995. The Politics of Presence. Oxford: Oxford University Press.
  • Smith, Michael A. Bringing Representation Home: State Legislators among Their Constituencies (2003)
  • Ulbig, Stacy G. (2005) "Political Realities and Political Trust: Descriptive Representation in Municipal Government". Southwestern Political Science Association Meeting. Retrieved from [1] on July 19, 2005.
  • Uno, Tab (1986) The Communication of Representation Between Legislators and Constituents. Masters Thesis, University of Utah.
  • Williams, Melissa S. 1998. Voice, Trust, and Memory: Marginalized Groups and the Failings of Liberal Representation. Princeton: Princeton University Press.

Representative democracy

From Wikipedia, the free encyclopedia
Representative democracy is a form of government founded on the principle of elected individuals representing the people, as opposed to autocracy and direct democracy.[1]. Two countries which use representative democracy are the United Kingdom (a constitutional monarchy) and Germany (a federal republic).
It is an element of both the parliamentary system and presidential system of government and is typically used in a lower chamber such as the House of Commons (UK) or Bundestag (Germany), and is generally curtailed by constitutional constraints such as an independent judiciary or an upper chamber.

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[edit] Characteristics

The representatives form an independent ruling body (for an election period) charged with the responsibility of acting in the people's interest, but not as their proxy representatives not necessarily always according to their wishes, but with enough authority to exercise swift and resolute initiative in the face of changing circumstances. It is often contrasted with direct democracy, where representatives are absent or are limited in power as proxy representatives. Edmund Burke was an early proponent of these principles:
...it ought to be the happiness and glory of a representative to live in the strictest union, the closest correspondence, and the most unreserved communication with his constituents. Their wishes ought to have great weight with him; their opinion, high respect; their business, unremitted attention. It is his duty to sacrifice his repose, his pleasures, his satisfactions, to theirs; and above all, ever, and in all cases, to prefer their interest to his own. But his unbiassed opinion, his mature judgment, his enlightened conscience, he ought not to sacrifice to you, to any man, or to any set of men living. These he does not derive from your pleasure; no, nor from the law and the constitution. They are a trust from Providence, for the abuse of which he is deeply answerable. Your representative owes you, not his industry only, but his judgment; and he betrays, instead of serving you, if he sacrifices it to your opinion.[2]
There is no necessity that individual liberties be respected in a representative democracy: one that does not is an illiberal democracy. A representative democracy that emphasizes individual liberty is a liberal democracy.
Today, in liberal representative democracies, representatives are usually elected in free and fair multi-party elections. Different methods of selecting representatives are described in the article on electoral systems, but often a number of representatives are elected by, and responsible to, a particular subset of the total electorate: this is called his or her constituency.

[edit] Powers of representatives

Representatives sometimes hold the power to select other representatives, presidents, or other officers of government (indirect representation)[citation needed]
The power of representatives is usually curtailed by a constitution (as in a constitutional democracy or a constitutional monarchy) or other measures to balance representative power:[citation needed]

[edit] History

A European medieval tradition of selecting representatives from the various estates (effectively, classes, but not as we know them today) to advise/control monarchs led to relatively wide familiarity with representative systems.
Representative democracy came into particular general favour in post-industrial revolution nation states where large numbers of subjects or (latterly) citizens evinced interest in politics, but where technology and population figures remained unsuited to direct democracy. As noted above, Edmund Burke in his speech to the electors of Bristol classically analysed their operation in Britain and the rights and duties of an elected representative.
Globally, a majority of the world's people live in representative democracies including constitutional monarchy with strong representative branch - the first time in history that this has been true. It has been the most successful form of civics since absolute monarchy[citation needed].

[edit] Relation to republicanism

The related term republic may have many different meanings. It normally means a state with an elected or otherwise non-monarchical head of state, such as the Islamic Republic of Iran or Republic of Korea.
Sometimes in the US it is used similarly to liberal (representative) democracy. For example:
"the United States relies on representative democracy, but its system of government is much more complex than that. It is not a simple representative democracy, but a constitutional republic in which majority rule is tempered."[3]

Mental representation

From Wikipedia, the free encyclopedia
  (Redirected from Representation (psychology))
A (mental) representation, in philosophy of mind, cognitive psychology, neuroscience, and cognitive science, is a hypothetical internal cognitive symbol that represents external reality, or else a mental process that makes use of such a symbol; "a formal system for making explicit certain entities or types of information, together with a specification of how the system does this."[1]
In contemporary philosophy, specifically in fields of metaphysics such as philosophy of mind and ontology, a mental representation is one of the prevailing ways of explaining and describing the nature of ideas and concepts.

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[edit] Representationalism and representational theories of mind

Representational theories of mind conceive of thinking as occurring within an internal system of representation. The propositional attitudes of the mind are token mental representations with semantic properties. Representationalism (also known as indirect realism)) is the view that representations are the main way we access external reality. Another major prevailing philosophical theory posits that concepts are entirely abstract objects.[2]
The representational theory of mind attempts to explain the nature of ideas, concepts and other mental content in contemporary philosophy of mind, cognitive science and experimental psychology. In contrast to theories of naive or direct realism, the representational theory of mind postulates the actual existence of mental representations which act as intermediaries between the observing subject and the objects, processes or other entities observed in the external world. These intermediaries stand for or represent to the mind the objects of that world.
For example, when someone arrives at the belief that his or her floor needs sweeping, the representational theory of mind states that he or she forms a mental representation that represents the floor and its state of cleanliness.
The original or "classical" representational theory probably can be traced back to Thomas Hobbes and was a dominant theme in classical empiricism in general. According to this version of the theory, the mental representations were images (often called "ideas") of the objects of states of affairs represented. For modern adherents, such as Jerry Fodor, Steven Pinker and many others, the representational system consists rather of an internal language of thought. The contents of thoughts are represented in symbolic structures (the formulas of Mentalese) which, analogously to natural languages but on a much more abstract level, possess a syntax and semantics very much like those of natural languages.

Representation (mathematics)

From Wikipedia, the free encyclopedia
In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yi conform in some consistent way to those existing among the corresponding represented objects xi. Somewhat more formally, for a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under an isomorphism that preserves Π. The label representation is sometimes also applied to the isomorphism itself.

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[edit] Representation theory

Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representation theory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.

[edit] Other examples

Although the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout mathematics.

[edit] Graph theory

An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A key class of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, more precisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerable families of sets. [1] One foundational result here, due to Paul Erdős and colleagues, is that every n-vertex graph may be represented in terms of intersection among subsets of a set of size no more than n2/4.[2]
Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the field of spectral graph theory. [3]

[edit] Order theory

Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set is isomorphic to a collection of sets ordered by the containment (or inclusion) relation ⊆. Among the posets that arise as the containment orders for natural classes of objects are the Boolean lattices and the orders of dimension n. [4]
Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them are the n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circle orders, the posets representable in terms of containment among disks in the plane. A particularly nice result in this field is the characterization of the planar graphs as those graphs whose vertex-edge incidence relations are circle orders. [5]
There are also geometric representations that are not based on containment. Indeed, one of the best studied classes among these are the interval orders, [6] which represent the partial order in terms of what might be called disjoint precedence of intervals on the real line: each element x of the poset is represented by an interval [x1, x2] such that for any y and z in the poset, y is below z if and only if y2 < z1.

[edit] Polysemy

Under certain circumstances, a single function f:XY is at once an isomorphism from several mathematical structures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y—one of the things that Y is trying to tell us—this phenomenon is called polysemy, a term borrowed from linguistics. Examples include:
  • intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneously represented by a single collection of sets Sv such that any distinct vertices u and w in V...
are adjacent in G1 if and only if their corresponding sets intersect ( SuSw ≠ Ø ), and
are adjacent in G2 if and only if the complements do ( SuCSwC ≠ Ø ).[7]
  • competition polysemy—motivated by the study of ecological food webs, in which pairs of species may have prey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competition polysemic if and only if there exists a single directed graph D on the same vertex set such that any distinct vertices u and v...
are adjacent in G1 if and only if there is a vertex w such that both uw and vw are arcs in D, and
are adjacent in G2 if and only if there is a vertex w such that both wu and wv are arcs in D.[8]
  • interval polysemy—pairs of posets P1 and P2 on a common ground set that can be simultaneously represented by a single collection of real intervals that is an interval-order representation of P1 and an interval-containment representation of P2.[9]

Multiple representations (mathematics education)

From Wikipedia, the free encyclopedia
Multiple representations are ways to symbolize, describe and refer to the same mathematical entity. They are used to understand and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds.

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[edit] Higher-order thinking

Use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills [1],[2],[3]. The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.[citation needed] Estimation, another complex task, can strongly benefit from multiple representations [4]
Curricula that support starting from conceptual understanding, then developing procedural fluency, for example, AIMS Foundation Activities [5], frequently use multiple representations.
Supporting student use of multiple representations may lead to more open-ended problems, or at least accepting multiple methods of solutions and forms of answers. Project-based learning units, such as WebQuests, typically call for several representations.[citation needed]

[edit] Motivation

Some representations, such as pictures, videos and manipulatives, can motivate because of their richness, possibilities of play, technologies involved, or connections with interesting areas of life [3]. Tasks that involve multiple representations can sustain intrinsic motivation in mathematics by supporting higher-order thinking and problem solving.
Multiple representations may also remove some of the gender biases that exist in math classrooms. Explaining probability solely through baseball statistics may potentially alienate students who have no interest in sports. When showing a tie to real-life applications, teachers should choose representations that are varied and of interest to all genders and cultures.

[edit] Assessment

Tasks that involve construction, use, and interpretation of multiple representations can lend themselves to rubric assessment [6] and to other assessment types suitable for open-ended activities. For example, tapping into visualization for math problem solving manifests multiple representations. These multiple representations arise when each student uses their knowledge base, and experience to create a visualization of the problem domain on the way toward a solution. Since visualization can be categorized into two main areas, schematic or pictorial[7], most students will provide on or the other or sometimes both methods to represent the problem domain.
Comparison of the different visualization tools created by each student is a excellent example of multiple representations. Further, the instructor may glean from these examples elements which they incoporate into their grading rubric. In this way, it is the students that provide the examles and standards against which scoring is done. This crucial factor places each student on equal footing and links them directly with their performance in class.

[edit] Special education and differentiated instruction

Students with special needs may be weaker in their use of some of the representations. For these students, it may be especially important to use multiple representations for two purposes. First, including representations that currently work well for the student ensures the understanding of the current mathematical topic. Second, connections among multiple representations within the same topic strengthens overall skills in using all representations, even those currently problematic [1].
It is also helpful to ESL/ELL (English as a Second Language/English Language Learners) to use multiple representations. The more you can bring a concept to "life" in a visual way, the more likely the students will grasp what you are talking about. This is also important with younger students who may have not had a lot of experience/prior knowledge on the topics we are teaching.
Using multiple representations can help differentiate instruction by addressing different learning styles [3],[8].

[edit] Qualitative and quantitative reasoning

Visual representations, manipulatives, gestures, and to some degree grids, can support qualitative reasoning about mathematics. Instead of only emphasizing computational skills, multiple representations can help students make the conceptual shift to the meaning and use of mathematical entities, and to develop algebraic thinking. By focusing more on the conceptual representations of algebraic problems students will become more capable problem solvers [2].

[edit] NCTM representations standard

National Council of Teachers of Mathematics has a standard dealing with multiple representations. In part, it reads [9] "Instructional programs should enable all students to do the following:
  • Create and use representations to organize, record, and communicate mathematical ideas
  • Select, apply, and translate among mathematical representations to solve problems
  • Use representations to model and interpret physical, social, and mathematical phenomena"

[edit] Four most frequent school mathematics representations

While there are many representations used in mathematics, the secondary curricula heavily favor numbers (often in tables), formulas, graphs and words [10].

[edit] Systems of manipulatives

Several curricula use extensively developed systems of manipulatives and the corresponding representations. For example, Cuisinaire rods [11], Montessori beads[citation needed], and Algebra Tiles[citation needed], Base-10 blocks, counters

[edit] Use of technology

Use of computer tools to create and to share mathematical representations opens several possibilities. It allows to link multiple representations dynamically. For example, changing a formula can instantly change the graph, the table of values, and the text read-out for the function represented in all these ways. Technology use can increase accuracy and speed of data collection and allow real-time visualization and experimentation [12]. It also supports collaboration [13].
Computer tools may be intrinsically interesting and motivating to students, and provide a familiar and comforting context students already use in their daily life.
Spreadsheet software such as Excel, Open Office, Google Documents, is widely used in many industries, and showing students the use of applications can make math more realistic. Most spreadsheet programs provide dynamic links among formulas, grids and several types of graphs.
Carnegie Learning curriculum is an example of emphasis on multiple representations and use of computer tools [14]. More specifically, Carnegie learning focuses the student not only on solving the real life scenarios presented in the text, but also promotes literacy through sentence writing and explanations of student thinking. In conjunction with the scenario based text Carnegie Learning provides a web based tutoring program called the "Cognitive Tutor" which uses data collected from each question a student answers to direct the student to areas where they need more help.
GeoGebra is free software dynamically linking geometric constructions, graphs, formulas, and grids [15]. It can be used in a browser and is light enough for older or low-end computers [16].
Project Interactivate [17] has many activities linking visual, verbal and numeric representations. There are currently 159 different activities available, in many areas of math, including numbers and operations, probability, geometry, algebra, statistics and modeling.
Another helpful tool for mathematicians, scientists, engineers is LaTeX. It is a typsetting program that allows you to create tables, figures, graphs etc in order to give a precise visual of the problem being worked on. Here is more information on LaTeX http://en.wikipedia.org/wiki/LaTeX

Structural formula

From Wikipedia, the free encyclopedia
  (Redirected from Representation (chemistry))
The structural formula of a chemical compound is a graphical representation of the molecular structure, showing how the atoms are arranged. The chemical bonding within the molecule is also shown, either explicitly or implicitly. There are several common representations used in publications. These are described below. Also several other formats are used, as in chemical databases, such as SMILES, InChI and CML.
Unlike chemical formulas or chemical names, structural formulas provide a representation of the molecular structure. Chemists nearly always describe a chemical reaction or synthesis using structural formulas rather than chemical names, because the structural formulas allow the chemist to visualize the molecules and the changes that occur.
Many chemical compounds exist in different isomeric forms, which have different structures but the same overall chemical formula. A structural formula indicates the arrangements of atoms in a way that a chemical formula cannot.

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[edit] Lewis structures

Representation of molecules by molecular formula
Lewis structures (or "Lewis dot structures") are flat graphical formulas that show atom connectivity and lone pair or unpaired electrons, but not three-dimensional structure. This notation is mostly used for small molecules. Each line represents the two electrons of a single bond. Two or three parallel lines between pairs of atoms represent double or triple bonds, respectively. Alternatively, pairs of dots may used to represent bonding pairs. In addition, all non-bonded electrons (paired or unpaired) and any formal charges on atoms are indicated.

[edit] Condensed formulas

In early organic-chemistry publications, where use of graphics was severely limited, a typographic system arose to describe organic structures in a line of text. Although this system tends to be problematic in application to cyclic compounds, it remains a convenient way to represent simple structures:
CH3CH2OH (ethanol)
Parentheses are used to indicate multiple identical groups, indicating attachment to the nearest non-hydrogen atom on the left when appearing within a formula, or to the atom on the right when appearing at the start of a formula:
(CH3)2CHOH or CH(CH3)2OH (2-propanol)
In all cases, all atoms are shown, including hydrogen atoms.

[edit] Skeletal formulas

Skeletal formulas are the standard notation for more complex organic molecules. First used by the organic chemist Friedrich August Kekulé von Stradonitz the carbon atoms in this type of diagram are implied to be located at the vertices (corners) and termini of line segments rather than being indicated with the atomic symbol C. Hydrogen atoms attached to carbon atoms are not indicated: each carbon atom is understood to be associated with enough hydrogen atoms to give the carbon atom four bonds. The presence of a positive or negative charge at a carbon atom takes the place of one of the implied hydrogen atoms. Hydrogen atoms attached to atoms other than carbon must be written explicitly.

[edit] Indication of stereochemistry

Several methods exist to picture the three-dimensional arrangement of atoms in a molecule (stereochemistry).

[edit] Stereochemistry in skeletal formulas

Chirality in skeletal formulas is indicated by the Natta projection method. Solid or dashed wedged bonds represent bonds pointing above-the-plane or below-the-plane of the paper, respectively.

[edit] Unspecified stereochemistry

Wavy single bonds represent unknown or unspecified stereochemistry or a mixture of isomers. For example the diagram below shows the fructose molecule with a wavy bond to the HOCH2- group at the left. In this case the two possible ring structures are in chemical equilibrium with each other and also with the open-chain structure. The ring continually opens and closes, sometimes closing with one stereochemistry and sometimes with the other.

[edit] Perspective drawings

[edit] Newman projection and sawhorse projection

The Newman projection and the sawhorse projection are used to depict specific conformers or to distinguish vicinal stereochemistry. In both cases, two specific carbon atoms and their connecting bond are the center of attention. The only difference is a slightly different perspective: the Newman projection looking straight down the bond of interest, the sawhorse projection looking at the same bond but from a somewhat oblique vantage point. In the Newman projection, a circle is used to represent a plane perpendicular to the bond, distinguishing the substituents on the front carbon from the substituents on the back carbon. In the sawhorse projection, the front carbon is usually on the left and is always slightly lower:

[edit] Cyclohexane conformations

Certain conformations of cyclohexane and other small-ring compounds can be shown using a standard convention. For example, the standard chair conformation of cyclohexane involves a perspective view from slightly above the average plane of the carbon atoms and indicates clearly which groups are axial and which are equatorial. Bonds in front may or may not be highlighted with stronger lines or wedges.

[edit] Haworth projection

The Haworth projection is used for cyclic sugars. Axial and equatorial positions are not distinguished; instead, substituents are positioned directly above or below the ring atom to which they are connected. Hydrogen substituents are typically omitted.

[edit] Fischer projection

The Fischer projection is mostly used for linear monosaccharides. At any given carbon center, vertical bond lines are equivalent to stereochemical hashed markings, directed away from the observer, while horizontal lines are equivalent to wedges, pointing toward the observer. The projection is totally unrealistic, as a saccharide would never adopt this multiply eclipsed conformation. Nonetheless, the Fischer projection is a simple way of depicting multiple sequential stereocenters that does not require or imply any knowledge of actual conformation:
DGlucose Fischer.svg
Fischer projection of D-Glucose