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Roberto Assagioli

Roberto Assagioli (27 February 1888 – 23 August 1974) was an Italian psychiatrist and pioneer in the fields of humanistic and transpersonal psychology. Assagioli founded the psychological movement known as psychosynthesis, which is still being developed today by therapists and psychologists who practice the psychological methods and techniques he developed. His work, including two books and many monographs published as pamphlets, emphasized the possibility of progressive integration (that is, synthesis) of the personality.

Life
Assagioli was born on 27 February 1888 in Venice, Italy, from a middle-class, Jewish background. He was born Roberto Marco Grego, the son of Elena Kaula and Leone Greco. However, his biological father died when Assagioli was two years old, and his mother remarried to Alessandro Emanuele Assagioli soon afterward. Assagioli was exposed to many creative outlets at a young age, such as art and music, which were believed to have inspired his work in Psychosynthesis. By the age of 18, he had learned eight different languages, including Italian (his native tongue), English, French, Russian, Greek, Latin, German, and Sanskrit. It was at this age that he began to travel, mainly to Russia, where he learned about social systems and politics.

In 1922 he married a young woman named Nella Ciapetti, and they had one son, Ilario Assagioli.

In 1940, Assagioli was arrested and imprisoned by Benito Mussolini's Fascist government, having been accused of "praying for peace and inviting others to join him along with other international crimes." He was placed in a solitary cell in Regina Coeli prison for 27 nights, until he was released and returned to his family. During World War II, his family's farm in Florence, Italy was destroyed, and both he and his family went into hiding in the Catenaia Alps (in the province of Arezzo) and in the Upper Tiber Valley. His son died at the age of 28 from lung disease, which was accredited to severe stress from the harsh living conditions during the war. After the war ended, Assagioli returned to his work and began his legacy, known as Psychosynthesis.

The years after the war were relatively calm, and it was during this time that he founded various foundations dedicated to Psychosynthesis in Europe and North America. Assagioli lived a long and prosperous life with a happy forty-year marriage until he died at age 86 on 23 August 1974. The cause of his death was unknown.

Assagioli did not like to discuss his personal life, as he preferred to be remembered for his scientific work. Very few biographical accounts about the life of Assagioli are available, and most are not in English.

Education
Assagioli received his first degree in neurology and psychiatry at Istituto di Studii Superiori Pratici e di Perfezionamento, in Florence in 1910. During this time, he began writing articles criticizing psychoanalysis, arguing that he had a more holistic approach.

After finishing his studies in Italy, Assagioli went to Switzerland, where he was trained in psychiatry at the psychiatric hospital Burghölzli in Zürich. This led to him opening the first psychoanalytic practice in Italy, known as Istituto di Psicosintesi. However, his work in psychoanalysis left him unsatisfied with the field of psychiatry; as a whole, he felt that psychoanalysis was incomplete.

Inspiration and development
Assagioli is famous for developing and founding the science of psychosynthesis, a spiritual and holistic approach to psychology that had developed from psychoanalysis. He was largely inspired by Freud's idea of the repressed mind and Jung's theories of the collective unconscious. Trained in psychoanalysis but unsatisfied by what he regarded as its incompleteness as a whole, Assagioli felt that love, wisdom, creativity, and will all were important components that should be included in psychoanalysis. Assagioli's earliest development of Psychosynthesis started in 1911, when he began his formal education in psychology. He continued his work on Psychosynthesis right up until his death. Freud and Assagioli were known to have corresponded, although they never had the chance to meet. Assagioli said, "Psychosynthesis presupposes psychoanalysis, or rather, includes it as a first and necessary stage."

However, Assagioli disagreed with theories formulated by Sigmund Freud that he considered limiting. He refused to accept Freud's reductionism and neglect of the positive dimensions of the personality. Psychosynthesis became the first approach born of psychoanalysis that also included the artistic, altruistic and heroic potentials of the human being. Assagioli's work was more in alignment with psychologist, Carl Jung. Both Assagioli and Jung validated the importance of the spiritual level of human existence. Assagioli shared with Jung the insight that psychological symptoms can be triggered by spiritual dynamics. Assagioli considered Jung's theories to be closest to his understanding of psychosynthesis.

Assagioli accredited much of his inspiration for psychosynthesis to his month-long incarceration in solitary confinement in 1940. He used his time in prison to exercise his mental will by meditating daily. He concluded that he was able to change his punishment into an opportunity to investigate his inner-self.

Psychology Today interview
In the December 1974 issue of Psychology Today, Assagioli was interviewed by Sam Keen, in which Assagioli discussed the differences between Freudian psychoanalysis and psychosynthesis: "We pay far more attention to the higher unconscious and to the development of the transpersonal self. In one of his letters Freud said, 'I am interested only in the basement of the human being.' Psychosynthesis is interested in the whole building. We try to build an elevator which will allow a person access to every level of his personality. After all, a building with only a basement is very limited. We want to open up the terrace where you can sun-bathe or look at the stars. Our concern is the synthesis of all areas of the personality. That means psychosynthesis is holistic, global and inclusive. It is not against psychoanalysis or even behavior modification but it insists that the needs for meaning, for higher values, for a spiritual life, are as real as biological or social needs. We deny that there are any isolated human problems."

Assagioli noted that Carl Jung, "of all modern psychotherapists, is the closest in theory and practice to psychosynthesis", and further expanded on the similarities between his own and Jung's views: "In the practice of therapy we both agree in rejecting 'pathologism' that is, concentration upon morbid manifestations and symptoms of a supposed psychological 'disease'. We regard man as a fundamentally healthy organism in which there may be temporary malfunctioning. Nature is always trying to re-establish harmony, and within the psyche the principle of synthesis is dominant. Irreconcilable opposites do not exist. The task of therapy is to aid the individual in transforming the personality and integrating apparent contradictions. Both Jung and myself have stressed the need for a person to develop the higher psychic functions, the spiritual dimension." He also highlighted the differences between Jung's work and psychosynthesis: "Perhaps the best way to state our differences is with a diagram of the psychic functions. Jung differentiates four functions: sensation, feeling, thought, and intuition. Psychosynthesis says that Jung's four functions do not provide for a complete description of the psychological life. Our view can be visualized like this: We hold that outside imagination or fantasy is a distinct function. There is also a group of functions that impels us toward action in the outside world. This group includes instincts, tendencies, impulses, desires, and aspirations. And here we come to one of the central foundations of psychosynthesis: There is a fundamental difference between drives, impulses, desires, and the will. In the human condition there are frequent conflicts between desire and will. And we will place the will in a central position at the heart of self-consciousness or the ego." Assagioli asserted about the will: "The will is not merely assertive, aggressive, and controlling. There is the accepting will, yielding will, the dedicated will. You might say that there is a feminine polarity to the will – the willing surrender, the joyful acceptance of the other functions of the personality."

At the end of the interview, Keen himself concluded: "It is hard to know what counts as evidence for the validity of a world view and the therapeutic it entails. Every form of therapy has dramatic successes and just as dramatic failures. Enter as evidence in the case for psychosynthesis an ad hominem argument: in speaking about death there was no change in the tone or intensity of Assagioli's voice and the light still played in his dark eyes, and his mouth was never very far from a smile."

Spiritual work
Assagioli was also interested and active in the field of consciousness and transpersonal work. Having studied theosophy and Eastern philosophy, his written work developed different meditation techniques, including reflective, receptive and creative meditation. He also contributed to several spiritual groups in the tradition known as the "ageless wisdom." He founded two groups intended to teach meditation based on the ideas of the New Age teacher Alice Bailey: The Group for Creative Meditation and the Meditation Group for the New Age. He was also a co-founder of the School for Esoteric Studies, intended to teach the work of Alice Bailey at an advanced level.

Published works

 * 1906 – Published in Farrari's Magazine – Gli effetti del riso e le loro applicazioni pedagogiche a.k.a., Smiling Wisdom (Italian)
 * 1910 - La psicanalisi Thesis (Italian)
 * 1926 - Psychosynthesis – A new method of healing by Roberto Assagioli. Booklet. Institute of Psychosynthesis, Rome, Italy & Lucia Publishing Company, New York, USA (English)
 * 1965 - Psychosynthesis: A Collection of Basic Writings by Roberto Assagioli ISBN 0-9678570-0-7 (English)
 * 1966 - Per l'armonia della vita. La psicosintesi by Roberto Assagioli. Istituto di Psiconsintesi, Firenze, Italy (Italian)
 * 1973 - The Act of Will by Roberto Assagioli ISBN 0-670-10309-8 (English)
 * 1993 - (posthumously) Transpersonal Development: The Dimension Beyond Psychosynthesis by Roberto Assagioli ISBN 1-85538-291-1 (English)
 * 2016 - (posthumously) Freedom in Jail by Roberto Assagioli. Ed. by Catherine Ann Lombard, Istituto di Psiconsintesi, Firenze, Italy (English)
 * 2022 - (posthumously) Creating Harmony in Life by Roberto Assagioli. Ed. by Catherine Ann Lombard, Istituto di Psiconsintesi, Firenze, Italy ISBN 979-1221402742 (English)

title: Roberto Assagioli


text:
National City Lines

National City Lines, Inc. (NCL) was a public transportation company. The company grew out of the Fitzgerald brothers' bus operations, founded in Minnesota, United States, in 1920 as a modest local transport company operating two buses. Part of the Fitzgerald's operations were reorganized into a holding company in 1936, and later expanded about 1938 with equity funding from General Motors, Firestone Tire, Standard Oil of California and Phillips Petroleum for the express purpose of acquiring local transit systems throughout the United States in what became known as the General Motors streetcar conspiracy. The company formed a subsidiary, Pacific City Lines in 1937 to purchase streetcar systems in the western United States. National City Lines, and Pacific City Lines were indicted in 1947 on charges of conspiring to acquire control of a number of transit companies, and of forming a transportation monopoly for the purpose of "conspiring to monopolize sales of buses and supplies to companies owned by National City Lines." They were acquitted on the first charge and convicted on the second in 1949.

History
The company has roots back to 1920, when E. Roy Fitzgerald and his brother began operating two buses in Minnesota, transporting miners and schoolchildren. In 1936 the company was organized into a holding company. In 1938, National City Lines wished to purchase transportation systems in cities "where street cars were no longer practicable" and replace them with passenger buses. To fund this expansion the company obtained equity funding from companies seeking to increase sales of commercial buses and supplies, including General Motors, Firestone Tire, Standard Oil of California and Phillips Petroleum.

In 1936, they bought 13 transit companies in Illinois, Oklahoma and Michigan, then in 1937, they replaced streetcars in Butte, Montana and made purchases in Mississippi and Texas. Sometimes these systems were already run down, but not always. Major investment had recently been made with improvements to the streetcars systems in Beaumont, Texas. The Butte system, while sound, deliberately replaced to lower the load on the overtaxed electric system, which was primarily used for commercial uses, including electrolytic refining of copper and zinc.

In 1938 the company entered into exclusive dealing arrangements and obtained equity funding from companies seeking to increase sales of commercial buses and supplies, including General Motors, Firestone Tire, Standard Oil of California and Phillips Petroleum. The company was indicted in 1947 and was later convicted in the United States District Court for the Northern District of Illinois of conspiring to monopolize the sale of buses and related products to the local transit companies that they controlled.

Over 1938 and 1939 the company made purchases in Alabama, Indiana and Ohio. and by 1939, it owned or controlled 29 local transportation companies in 27 different cities in 10 states.

American City Lines, which had been organized to acquire local transportation systems in the larger metropolitan areas in various parts of the country in 1943 merged with NCL in 1946. By 1947 the company owned or controlled 46 systems in 45 cities in 16 states.

In 1947 National City Lines, with others was indicted in the Federal District Court of Southern California on two counts: 'conspiring to acquire control of a number of transit companies, forming a transportation monopolize' and 'Conspiring to monopolize sales of buses and supplies to companies owned by National City Lines' in what became known as the Great American streetcar scandal (or 'General Motors streetcar conspiracy', 'National City Lines conspiracy').

In 1948, the United States Supreme Court (in United States v. National City Lines Inc.) permitted a change in venue to the Federal District Court in Northern Illinois. National City Lines merged with Pacific City Lines the same year.

In 1949, General Motors, Standard Oil of California, Firestone and others were convicted of conspiring to monopolize the sale of buses and related products to local transit companies controlled by NCL and other companies; they were acquitted of conspiring to monopolize the ownership of these companies. The verdicts were upheld on appeal in 1951. The corporations involved were fined $5000, their executives $1 apiece.

Operating areas and companies
There is considerable uncertainty and variability amongst sources as to where National City Lines operated.

The 1948 ruling stated that:

"Forty-four cities in sixteen states are included. The states are as widely scattered as California, Florida, Maryland, Michigan, Nebraska, Texas and Washington. The larger local transportation systems include those of Baltimore, St. Louis, Salt Lake City, Los Angeles and Oakland. The largest concentrations of smaller systems are in Illinois, with eleven cities; California with nine (excluding Los Angeles); and Michigan with four. The local operating companies were not named as parties defendant."

This table attempts to bring together the many sources detailing the cities in which, at one time or another, National City Lines owned or controlled transit companies. A star (*) indicates that NCL is understood to have had significant control but not ownership:

 * Alabama: Mobile, Montgomery
 * California: Burbank, Eureka, Fresno, Glendale, Los Angeles*, Oakland*, Sacramento, Inglewood, Long Beach, San Jose, Pasadena, Stockton
 * Florida: Tampa, Jacksonville*
 * Illinois: Aurora/Elgin, Bloomington, Champaign, Danville, East St. Louis, Decatur, Galesburg, Joliet, Kewanee, LaSalle/Peru, Peoria, Quincy, Rock Island
 * Indiana: South Bend, Terre Haute
 * Iowa: Burlington, Davenport, Cedar Rapids, Ottumwa, Sioux City
 * Maryland: Baltimore*
 * Michigan: Jackson, Lansing, Kalamazoo, Saginaw, Pontiac
 * Mississippi: Jackson
 * Missouri: St. Louis*
 * Montana: Butte, Great Falls
 * Nebraska: Lincoln
 * Ohio: Canton, Portsmouth
 * Oklahoma: Tulsa
 * Pennsylvania: Philadelphia*
 * Texas: Beaumont, El Paso, Houston, Port Arthur, Wichita Falls
 * Utah: Salt Lake City
 * Washington: Bellingham, Everett, Spokane
 * Wisconsin: Oshkosh

Additional information: In Los Angeles the Los Angeles Railway (Yellow Cars) was controlled by NCL but not Pacific Electric Railway (Red Cars)

Montgomery City Lines and the Montgomery bus boycott
Montgomery City Lines was the National City Lines subsidiary that operated the municipal transit system for Montgomery, Alabama.

On 1 December 1955, Rosa Parks was arrested for refusing to move to the back of a Montgomery City Lines bus. This led to the Montgomery bus boycott. Montgomery City Lines was placed in the middle of a dispute between Montgomery's black citizenry and Montgomery city laws. In a letter published in the Montgomery Advertiser on December 3, 1955, Montgomery's Transportation Superintendent J. H. Bagley wrote: "The Montgomery City Lines is sorry if anyone expects us to be exempt from any state or city law ... [w]e are sorry that the colored people blame us for any state or city ordinance which we didn't have passed.'"

After Martin Luther King Jr. and the Montgomery Improvement Association wired a letter to National City Lines on 8 December 1955, the company's vice president, Kenneth E. Totten, traveled to Montgomery the following week.

The boycott lasted for just over a year, and cost the company $750,000 (equivalent to $ million in ). The boycott ended only after the United States Supreme Court affirmed Browder v. Gayle, a ruling that black bus passengers had a right to sit in any publicly available seat.

Later history
National City Lines acquired the trucking company Los Angeles-Seattle Motor Express (LASME) in 1959. In 1968, LASME merged with DC International and T.I.M.E. to form T.I.M.E.-DC. National City Lines sold its transportation management division in 1978.

National City Lines was later acquired by Harold C. Simmons early in 1981. T.I.M.E.-DC ceased operations in 1988. The company continued as a fully controlled subsidiary of Simmon' Contran operation until December 31, 2007, when it was dissolved.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: National City Lines


text:
Wallpaper group

A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations, tiles and physical wallpaper.

What this page calls pattern
Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the boundaries between these tiles. Conversely, from every wallpaper we can construct such a tiling by identical tiles edge‑to‑edge, which bear each identical ornaments, the identical outlines of these tiles being not necessarily visible on the original wallpaper. Such repeated boundaries delineate a repetitive surface added here in dashed lines.

Such pseudo‑tilings connected to a given wallpaper are in infinite number. For example image 1 shows two models of repetitive squares in two different positions, which have $$\text{equal areas of }\,a.$$ Another repetitive square has an $$\text{area of 5 times}\,~a.~$$ We could indefinitely conceive such repetitive squares larger and larger. An infinity of shapes of repetitive zones are possible for this Pythagorean tiling, in an infinity of positions on this wallpaper. For example in red on the bottom right‑hand corner of image 1, we could glide its repetitive parallelogram in one or another position. In common on the first two images: a repetitive square concentric with each small square tile, their common center being a point symmetry of the wallpaper.

Between identical tiles edge‑to‑edge, an edge is not necessarily a segment of a straight line. On the top left‑hand corner of image 3, point C  is a vertex of a repetitive pseudo‑rhombus with thick stripes on its whole surface, called pseudo‑rhombus because of a concentric repetitive rhombus $$\text{of same area}\,~a,~$$ constructed from it by taking out a bit of surface somewhere to append it elsewhere, and keep the area unchanged. By the same process on image 4, a repetitive regular hexagon filled with vertical stripes is constructed from a rhombic repetitive zone $$\text{of area }\,~a.~$$ Conversely, from elementary geometric tiles edge‑to‑edge, an artist like M. C. Escher created attractive surfaces many times repeated. On image 2,  $$a~\text{ represents}$$ the minimum area of a repetitive surface by disregarding colors, each repetitive zone in dashed lines consisting of five pieces in a certain arrangement, to be either a square or a hexagon, like in a proof of the Pythagorean theorem.

In the present article, a pattern is a repetitive parallelogram of minimal area in a determined position on the wallpaper. Image 1 shows two parallelogram‑shaped patterns — a square is a particular parallelogram —. Image 3 shows rhombic patterns — a rhombus is a particular parallelogram —.

On this page, all repetitive patterns (of minimal area) are constructed from two translations that generate the group of all translations under which the wallpaper is invariant. With the circle shaped symbol ⵔ of function composition, a pair like $$\{T, U\}$$ or $$\{\,U,\; T \circ U^{\,-1}\}$$ generates the group of all translations that transform the Pythagorean tiling into itself.

Possible groups linked to a pattern
A wallpaper remains on the whole unchanged under certain isometries, starting with certain translations that confer on the wallpaper a repetitive nature. One of the reasons to be unchanged under certain translations is that it covers the whole plane. No mathematical object in our minds is stuck onto a motionless wall! On the contrary an observer or his eye is motionless in front of a transformation, which glides or rotates or flips a wallpaper, eventually could distort it, but that would be out of our subject.

If an isometry leaves unchanged a given wallpaper, then the inverse isometry keeps it also unchanged, like translation $$\mathit{T}\text{  or }\mathit{T}^{\,-1}~$$ on image 1, 3 or 4, or a ± 120° rotation around a point like S  on image 3 or 4. If they have both this property to leave unchanged a wallpaper, two isometries composed in one or the other order have then this same property to leave unchanged the wallpaper. To be exhaustive about the concepts of group and subgroups under the function composition, represented by the circle shaped symbol ⵔ, here is a traditional truism in mathematics: everything remains itself under the identity transformation. This identity function can be called translation of zero vector or rotation of 360°.

A glide can be represented by one or several arrows if parallel and of same length and same sense, in same way a wallpaper can be represented either by a few patterns or by only one pattern, considered as a pseudo‑tile imagined repeated edge‑to‑edge with an infinite number of replicas. Image 3 shows two patterns with two different contents, and the one in dark dashed lines or one of its images under $$\,\text{ rotation }\,\mathit{R}\,\text{ or }\,\mathit{R}\,^{-1}\;$$ represents the same wallpaper on the following image 4, by disregarding the colors. Certainly a color is perceived subjectively whereas a wallpaper is an ideal object, however any color can be seen as a label that characterizes certain surfaces, we might think of a hexadecimal code of color as a label specific to certain zones. It may be added that a well‑known theorem deals with colors.

Groups are registered in the catalog by examining properties of a parallelogram, edge‑to‑edge with its replicas. For example its diagonals intersect at their common midpoints, center and symmetry point of any parallelogram, not necessarily symmetry point of its content. Other example, the midpoint of a full side shared by two patterns is the center of a new repetitive parallelogram formed by the two together, center which is not necessarily symmetry point of the content of this double parallelogram. Other possible symmetry point, two patterns symmetric one to the other with respect to their common vertex form together a new repetitive surface, the center of which is not necessarily symmetry point of its content.

Certain rotational symmetries are possible only for certain shapes of pattern. For example on image 2, a Pythagorean tiling is sometimes called pinwheel tilings because of its rotational symmetry of 90 degrees about the center of a tile, either small or large, or about the center of any replica of tile, of course. Also when two equilateral triangles form edge‑to‑edge a rhombic pattern, like on image 4 or 5 (future image 5), a rotational symmetry of 120 degrees about a vertex of a 120° angle, formed by two sides of pattern, is not always a symmetry point of the content of the regular hexagon formed by three patterns together sharing a vertex, because it does not always contain the same motif.

First examples of groups
The simplest wallpaper group, Group p1, applies when there is no symmetry other than the fact that a pattern repeats over regular intervals in two dimensions, as shown in the section on p1 below.

The following examples are patterns with more forms of symmetry:

Examples A and B have the same wallpaper group; it is called p4m in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

A proof that there are only 17 distinct groups of such planar symmetries was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Pólya in 1924. The proof that the list of wallpaper groups is complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in.

Symmetries of patterns
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (in other words, shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

The types of transformations that are relevant here are called Euclidean plane isometries. For example:

 * If one shifts example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the starting pattern. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
 * If one turns example B clockwise by 90°, around the centre of one of the squares, again one obtains exactly the same pattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.
 * One can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.

However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If one flips across a diagonal line, one does not get the same pattern back, but the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.

Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection.

Formal definition and discussion
Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations.

Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glide reflections, and rotational symmetry is at most of order 2).

Unlike in the three-dimensional case, one can equivalently restrict the affine transformations to those that preserve orientation.

It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed to e.g. frieze groups, of which two are isomorphic with Z).

2D patterns with double translational symmetry can be categorized according to their symmetry group type.

Isometries of the Euclidean plane
Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).
 * Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.
 * Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.
 * Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror.
 * Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d.

The independent translations condition
The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.

The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups, which only repeat along a single axis.

(It is possible to generalise this situation. One could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.)

The discreteness condition
The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector, but the independent translations condition prevents this, since any set that contains the zero vector is linearly dependent by definition and thus disallowed).

The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, one might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.

One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180°, 120°, 90°, or 60°. This fact is known as the crystallographic restriction theorem, and can be generalised to higher-dimensional cases.

Crystallographic notation
Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus one can use a similar notation for both kinds of groups, that of Carl Hermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style (also called IUCr notation) is p31m, with four letters or digits; more usual is a shortened name like cmm or pg.

For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis that is the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.

A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.

 * Examples
 * p2 (p2): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
 * p4gm (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
 * c2mm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
 * p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.

Here are all the names that differ in short and full notation.

 * {| class="wikitable"

! Short ! pm ! pg ! cm ! pmm ! pmg ! pgg ! cmm ! p4m ! p4g ! p6m ! Full || p1m1 || p1g1 || c1m1 || p2mm || p2mg || p2gg || c2mm || p4mm || p4gm || p6mm
 * + Crystallographic short and full names
 * - style="background:white"
 * }

The remaining names are p1, p2, p3, p3m1, p31m, p4, and p6.

Orbifold notation
Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. One can fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.

 * A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
 * An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows:
 * Digits before * denote centres of pure rotation (cyclic).
 * Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (dihedral).
 * A cross, ×, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so need no notation.
 * The "no symmetry" symbol, o, stands alone, and indicates there are only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.

The group denoted in crystallographic notation by cmm will, in Conway's notation, be 2*22. The 2 before the * says there is a 2-fold rotation centre with no mirror through it. The * itself says there is a mirror. The first 2 after the * says there is a 2-fold rotation centre on a mirror. The final 2 says there is an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries.

The group denoted by pgg will be 22×. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.

Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations.

Why there are exactly seventeen groups
An orbifold can be viewed as a polygon with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.

When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces, which must be consistent with the Euler characteristic. Reversing the process, one can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.

The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:
 * A digit n without or before a * counts as $n − 1⁄n$.
 * A digit n after a * counts as $n − 1⁄2n$.
 * Both * and × count as 1.
 * The "no symmetry" o counts as 2.

For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.

 * Examples
 * 632: $5⁄6$ + $2⁄3$ + $1⁄2$ = 2
 * 3*3: $2⁄3$ + 1 + $2⁄6$ = 2
 * 4*2: $3⁄4$ + 1 + $1⁄4$ = 2
 * 22×: $1⁄2$ + $1⁄2$ + 1 = 2

Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.

Feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).

Guide to recognizing wallpaper groups
To work out which wallpaper group corresponds to a given design, one may use the following table.

See also this overview with diagrams.

The seventeen groups
Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows (it is the shape that is significant, not the colour): On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.

The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern that is repeated.

The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the left sometimes show a larger area.

Group p1 (o)

 * Orbifold signature: o
 * Coxeter notation (rectangular): [∞+,2,∞+] or [∞]+×[∞]+
 * Lattice: oblique
 * Point group: C1
 * The group p1 contains only translations; there are no rotations, reflections, or glide reflections.
 * Examples of group p1

The two translations (cell sides) can each have different lengths, and can form any angle.

Group p2 (2222)

 * Orbifold signature: 2222
 * Coxeter notation (rectangular): [∞,2,∞]+
 * Lattice: oblique
 * Point group: C2
 * The group p2 contains four rotation centres of order two (180°), but no reflections or glide reflections.

 * Examples of group p2

Group pm (**)

 * Orbifold signature: **
 * Coxeter notation: [∞,2,∞+] or [∞+,2,∞]
 * Lattice: rectangular
 * Point group: D1
 * The group pm has no rotations. It has reflection axes, they are all parallel.

 * Examples of group pm

(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)

Group pg (××)

 * Orbifold signature: ××
 * Coxeter notation: [(∞,2)+,∞+] or [∞+,(2,∞)+]
 * Lattice: rectangular
 * Point group: D1
 * The group pg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.

 * Examples of group pg

Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction between brown and black it is pgg.

Ignoring the wavy borders of the tiles, the pavement is pgg.

Group cm (*×)

 * Orbifold signature: *×
 * Coxeter notation: [∞+,2+,∞] or [∞,2+,∞+]
 * Lattice: rhombic
 * Point group: D1
 * The group cm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes.
 * This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.

 * Examples of group cm

Group pmm (*2222)

 * Orbifold signature: *2222
 * Coxeter notation (rectangular): [∞,2,∞] or [∞]×[∞]
 * Coxeter notation (square): [4,1+,4] or [1+,4,4,1+]
 * Lattice: rectangular
 * Point group: D2
 * The group pmm has reflections in two perpendicular directions, and four rotation centres of order two (180°) located at the intersections of the reflection axes.

 * Examples of group pmm

Group pmg (22*)

 * Orbifold signature: 22 *
 * Coxeter notation: [(∞,2)+,∞] or [∞,(2,∞)+]
 * Lattice: rectangular
 * Point group: D2
 * The group pmg has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.

 * Examples of group pmg

Group pgg (22×)

 * Orbifold signature: 22 ×
 * Coxeter notation (rectangular): [((∞,2)+,(∞,2)+)]
 * Coxeter notation (square): [4+,4+]
 * Lattice: rectangular
 * Point group: D2
 * The group pgg contains two rotation centres of order two (180°), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.

 * Examples of group pgg

Group cmm (2*22)

 * Orbifold signature: 2 *22
 * Coxeter notation (rhombic): [∞,2+,∞]
 * Coxeter notation (square): [(4,4,2+)]
 * Lattice: rhombic
 * Point group: D2
 * The group cmm has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is not on a reflection axis. It also has two rotations whose centres are on a reflection axis.
 * This group is frequently seen in everyday life, since the most common arrangement of bricks in a brick building (running bond) utilises this group (see example below).

The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties.

The pattern corresponds to each of the following:
 * symmetrically staggered rows of identical doubly symmetric objects
 * a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric
 * a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image

 * Examples of group cmm

Group p4 (442)

 * Orbifold signature: 442
 * Coxeter notation: [4,4]+
 * Lattice: square
 * Point group: C4
 * The group p4 has two rotation centres of order four (90°), and one rotation centre of order two (180°). It has no reflections or glide reflections.

 * Examples of group p4

A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor √2 smaller and rotated 45°.

Group p4m (*442)

 * Orbifold signature: *442
 * Coxeter notation: [4,4]
 * Lattice: square
 * Point group: D4
 * The group p4m has two rotation centres of order four (90°), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180°) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.

This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a checkerboard pattern of two of such squares.

 * Examples of group p4m

Examples displayed with the smallest translations horizontal and vertical (like in the diagram):

Group p4g (4*2)
A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).
 * Orbifold signature: 4 *2
 * Coxeter notation: [4+,4]
 * Lattice: square
 * Point group: D4
 * The group p4g has two centres of rotation of order four (90°), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180°) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45° with these.

 * Examples of group p4g

Group p3 (333)

 * Orbifold signature: 333
 * Coxeter notation: [(3,3,3)]+ or [3[3]]+
 * Lattice: hexagonal
 * Point group: C3
 * The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.

Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal it is p6, if they are each other's mirror image it is p31m, if they are both symmetric it is p3m1; if two of the three apply then the third also, and it is p6m). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles.

Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by √3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six it is p6, if they are symmetric with respect to the main diagonals it is p31m, if they are symmetric with respect to lines perpendicular to the sides it is p3m1; if two of the three apply then the third also, it is p6m). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.

 * Examples of group p3

Group p3m1 (*333)

 * Orbifold signature: *333
 * Coxeter notation: [(3,3,3)] or [3[3]]
 * Lattice: hexagonal
 * Point group: D3
 * The group p3m1 has three different rotation centres of order three (120°). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the blue or the green triangles.

 * Examples of group p3m1

Group p31m (3*3)

 * Orbifold signature: 3 *3
 * Coxeter notation: [6,3+]
 * Lattice: hexagonal
 * Point group: D3
 * The group p31m has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices must be the red triangles, not the blue triangles.

 * Examples of group p31m

Group p6 (632)

 * Orbifold signature: 632
 * Coxeter notation: [6,3]+
 * Lattice: hexagonal
 * Point group: C6
 * The group p6 has one rotation centre of order six (60°); two rotation centres of order three (120°), which are each other's images under a rotation of 60°; and three rotation centres of order two (180°) which are also each other's images under a rotation of 60°. It has no reflections or glide reflections.

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C6 symmetry (with the edges of the tiles not necessarily part of the pattern).

 * Examples of group p6

Group p6m (*632)

 * Orbifold signature: *632
 * Coxeter notation: [6,3]
 * Lattice: hexagonal
 * Point group: D6
 * The group p6m has one rotation centre of order six (60°); it has two rotation centres of order three, which only differ by a rotation of 60° (or, equivalently, 180°), and three of order two, which only differ by a rotation of 60°. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.

 * Examples of group p6m

Lattice types
There are five lattice types or Bravais lattices, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
 * In the 5 cases of rotational symmetry of order 3 or 6, the unit cell consists of two equilateral triangles (hexagonal lattice, itself p6m). They form a rhombus with angles 60° and 120°.
 * In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).
 * In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm). It may also be interpreted as a centered rhombic lattice. Special cases: square.
 * In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm). It may also be interpreted as a centered rectangular lattice. Special cases: square, hexagonal unit cell.
 * In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic or oblique lattice, itself p2). Special cases: rectangle, square, rhombus, hexagonal unit cell.

Symmetry groups
The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.

The numbers of degrees of freedom are:
 * 6 for p2
 * 5 for pmm, pmg, pgg, and cmm
 * 4 for the rest.

However, within each wallpaper group, all symmetry groups are algebraically isomorphic.

Some symmetry group isomorphisms:
 * p1: Z2
 * pm: Z × D∞
 * pmm: D∞ × D∞.

Dependence of wallpaper groups on transformations

 * The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).
 * Translational symmetry is preserved under arbitrary bijective affine transformations.
 * Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.
 * Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.

Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.

Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.

If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.

Web demo and software
Several software graphic tools will let you create 2D patterns using wallpaper symmetry groups. Usually you can edit the original tile and its copies in the entire pattern are updated automatically.
 * MadPattern, a free set of Adobe Illustrator templates that support the 17 wallpaper groups
 * Tess, a shareware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
 * Wallpaper Symmetry is a free online JavaScript drawing tool supporting the 17 groups. The main page has an explanation of the wallpaper groups, as well as drawing tools and explanations for the other planar symmetry groups as well.
 * TALES GAME, a free software designed for educational purposes which includes the tessellation function.
 * Kali, online graphical symmetry editor Java applet (not supported by default in browsers).
 * Kali, free downloadable Kali for Windows and Mac Classic.
 * Inkscape, a free vector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See )
 * SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.
 * EscherSketch is a free online JavaScript drawing tool supporting the 17 groups.
 * Repper is a commercial online drawing tool supporting the 17 groups plus a number of non-periodic tilings

title: Wallpaper group


text:
Mesolimbic pathway

The mesolimbic pathway, sometimes referred to as the reward pathway, is a dopaminergic pathway in the brain. The pathway connects the ventral tegmental area in the midbrain to the ventral striatum of the basal ganglia in the forebrain. The ventral striatum includes the nucleus accumbens and the olfactory tubercle.

The release of dopamine from the mesolimbic pathway into the nucleus accumbens regulates incentive salience (e.g. motivation and desire for rewarding stimuli) and facilitates reinforcement and reward-related motor function learning; it may also play a role in the subjective perception of pleasure. The dysregulation of the mesolimbic pathway and its output neurons in the nucleus accumbens plays a significant role in the development and maintenance of an addiction.

Anatomy
The mesolimbic pathway is a collection of dopaminergic (i.e., dopamine-releasing) neurons that project from the ventral tegmental area (VTA) to the ventral striatum, which includes the nucleus accumbens (NAcc) and olfactory tubercle. It is one of the component pathways of the medial forebrain bundle, which is a set of neural pathways that mediate brain stimulation reward.

The VTA is located in the midbrain and consists of dopaminergic, GABAergic, and glutamatergic neurons. The dopaminergic neurons in this region receive stimuli from both cholinergic neurons in the pedunculopontine nucleus and the laterodorsal tegmental nucleus as well as glutamatergic neurons in other regions such as the prefrontal cortex. The nucleus accumbens and olfactory tubercle are located in the ventral striatum and are primarily composed of medium spiny neurons. The nucleus accumbens is subdivided into limbic and motor subregions known as the NAcc shell and NAcc core. The medium spiny neurons in the nucleus accumbens receive input from both the dopaminergic neurons of the VTA and the glutamatergic neurons of the hippocampus, amygdala, and medial prefrontal cortex. When they are activated by these inputs, the medium spiny neurons' projections release GABA onto the ventral pallidum.

Function
The mesolimbic pathway regulates incentive salience, motivation, reinforcement learning, and fear, among other cognitive processes.

The mesolimbic pathway is involved in motivational cognition. Depletion of dopamine in this pathway, or lesions at its site of origin, decrease the extent to which an animal is willing to go to obtain a reward (e.g. the number of lever presses for intravenous nicotine delivery in rats or time spent searching for food). Dopaminergic drugs are also able to increase the extent an animal is willing to go to obtain a reward. Moreover, the firing rate of neurons in the mesolimbic pathway increases during anticipation of reward, which may explain craving. Mesolimbic dopamine release was once thought to be the primary mediator of pleasure, but is now believed to have only a minor or secondary role in pleasure perception.

Mechanisms of addiction
The mesolimbic pathway and a specific set of the pathway's output neurons (e.g. D1-type medium spiny neurons within the nucleus accumbens) play a central role in the neurobiology of addiction.  Drug addiction is an illness caused by habitual substance use that induces chemical changes in the brain's circuitry. An addictive drug is defined as a substance that affects the mesolimbic system directly or indirectly by increasing extracellular levels of dopamine.

Common addictive substances such as cocaine, alcohol, and nicotine have been shown to increase extracellular levels of dopamine within the mesolimbic pathway, preferentially within the nucleus accumbens. The mechanisms by which these drugs do so vary depending on the drug prototype. For example, cocaine precludes the re-uptake of synaptic dopamine through blocking the presynaptic dopamine transporter. Another stimulant, amphetamine, reverses the dopamine transporter and induces the release of dopamine from synaptic vesicles. Non-stimulant drugs typically bind with ligand-gated channels or G protein-coupled receptors. Such drugs include alcohol, nicotine, and tetrahydrocannabinol (THC). These dopaminergic activations of the mesolimbic pathway are accompanied by the perception of reward. This stimulus-reward association shows a resistance to extinction and creates an increased motivation to repeat that same behavior that caused it. Additionally, drug intake changes synaptic plasticity in the ventral tegmental area and the nucleus accumbens. Repeated exposure to the drug can lead to lasting changes in the brain that gives rise to addictive behavior.

Relation to other neurological and psychological disorders
The mesolimbic pathway is implicated in schizophrenia, depression,  and Parkinson's disease. It is also theorized to be implicated in overuse of digital media, although it could simply be a consequence of a sedentary lifestyle. Each involves distinct structural changes within the mesolimbic pathway.

Other dopamine pathways

 * Mesocortical pathway
 * Nigrostriatal pathway
 * Tuberoinfundibular pathway

title: Mesolimbic pathway