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Guifeng Zongmi

Guifeng Zongmi (780–1 February 841) was a Tang dynasty Chinese Buddhist monk and scholar who is considered a patriarch of both the Huayan school and Chan Buddhism. Zongmi wrote a number of works on several Mahayana Sutras, Chan and Huayan, and he also discussed Taoism and Confucianism. His works are a major source for studying the various Chan schools of the Tang.

Zongmi was deeply interested in both the practical and doctrinal aspects of Mahayana Buddhism, especially the teachings of the Sutra of Perfect Enlightenment and the Mahayana Awakening of Faith. Zongmi's work is concerned with harmonizing the various Chan teachings (especially the doctrines of sudden awakening and original enlightenment) with other Chinese Buddhist traditions, especially Huayan, though he also drew on the work of Tiantai Zhiyi in his ritual works. His philosophy attempts to create a comprehensive worldview that includes and sublimates all Buddhist and non-buddhist teachings of his time into a single harmonious spiritual vision.

Early years (780–810)
Zongmi was born in 780 into the powerful and influential He family in what is now central Sichuan. In his early years, he studied the Chinese classics, hoping for a career in the provincial government. When he was seventeen or eighteen, Zongmi lost his father and took up Buddhist studies. In an 811 letter to Chengguan, he wrote that for three years he "gave up eating meat, examined [Buddhist] scriptures and treatises, became familiar with the virtues of meditation and sought out the acquaintance of noted monks."

At the age of twenty-two, he returned to the Confucian classics and deepened his understanding, studying at the Yixueyuan Confucian Academy in Suizhou. His later writings reveal a detailed familiarity with the Analects, the Classic of Filial Piety, and the Book of Rites, as well as historical texts and Daoist classics such as the works of Laozi. He eventually converted to Buddhism, but Zongmi's Confucian moral values never left him and he spent much of his career attempting to integrate Confucian ethics with Buddhism.

Chan (804–810)
At the age of twenty-four, Zongmi met the Chan master Suizhou Daoyuan and trained in Chan for two or three years. He received Daoyuan's seal in 807, the year he was fully ordained as a Buddhist monk. Daoyuan was part of a southern Chan tradition called the Jingzhong school, which was based in Chengdu, Sichuan. This lineage goes back to the Korean prince and Chan master Kim Hwasang (c. 684–762, also known as Jingzhong Wuxiang, 浄衆無相 ), and to his student Shenhui (not to be confused with the earlier Heze Shenhui). According to Broughton, Daoyuan's teacher, Yizhou Nanyin, the abbot of Shengshou Monastery, likely trained with Heze Shenhui and this other Shenhui of Sichuan.

In his autobiographical summary, Zongmi recounts how after a sutra chanting service, he encountered a copy of the Sutra of Perfect Enlightenment. According to Gregory, "after only reading two or three pages, he had an awakening, an experience whose intensity so overwhelmed him that he found himself spontaneously dancing for joy." Zongmi would later write: "at one word [from Daoyuan] my mind-ground opened thoroughly, and with one scroll [of the Scripture of Perfect Enlightenment] its meaning was as clear and bright as the heavens."

Zongmi's sudden awakening during this period had a profound impact upon his subsequent scholarly career. He spent the next several years studying the Sutra of Perfect Enlightenment and its commentaries extensively. He propounded the necessity of scriptural studies in Chan, and was highly critical of what he saw as the antinomianism of the Hongzhou lineage derived from Mazu Daoyi (709–788), which practiced "entrusting oneself to act freely according to the nature of one's feelings".

Huayan (810–816)
In 810, at the age of thirty, Zongmi met Lingfeng, a disciple of the preeminent Buddhist scholar and Huayan exegete Chengguan (, 738–839). Lingfeng gave Zongmi a copy of Chengguan's commentary and sub-commentary on the Avatamsaka Sutra. The two texts were to have a profound impact on Zongmi. He studied these texts and the sūtra with great intensity, declaring later that due to his assiduous efforts, finally "all remaining doubts were completely washed away." In 812 Zongmi travelled to the western capital, Chang'an, where he spent two years studying with Chengguan, who was not only the undisputed authority on Huayan, but was also highly knowledgeable in Chan, Tiantai, the vinaya and East Asian Mādhyamaka.

Mount Zhongnan and scholarly work (816–828)

In 816, Zongmi withdrew to the Zhongnan Mountains southwest of Chang'an and began his writing career, composing an annotated outline of the Sutra of Perfect Enlightenment, and a compilation of passages from four commentaries on the sūtra. For the next three years Zongmi continued his scholarly research, reading through the Buddhist canon, the Tripiṭaka, and traveling to various temples on Zhongnan. He returned Chang'an in 819 and continued his studies utilizing the extensive libraries of various monasteries in the capital city. In late 819 he completed a commentary and subcommentary  on the Diamond Sutra. In early 821 he returned to Cottage Temple beneath Gui Peak and hence became known as "Guifeng Zongmi". In mid-823, he finally finished his own commentary on the text that had led to his first awakening, Sutra of Perfect Enlightenment, and the culmination of a vow he had made some fifteen years earlier. For the next five years, Zongmi continued writing and studying in the Zhongnan Mountains as his fame grew.

Capital city (828–835)
Zongmi was summoned to the capital in 828 by Emperor Wenzong (r. 826–840) and awarded the purple robe and the honorific title "Great Worthy" (bhadanta). The two years he spent in the capital were significant for Zongmi. He was now a nationally honored Buddhist master with extensive contacts among the literati of the day, such as the devout Buddhist layman and scholar-official Pei Xiu and the famed poet Bai Juyi (772–846).

During this time, Zongmi turned his considerable knowledge and intellect towards writing for a broader audience rather than the technical exegetical works he had produced for a limited readership of Buddhist specialists. His scholarly efforts became directed towards the intellectual issues of the day and much of his subsequent work was produced at the appeals of assorted literati of the day. One such works directed at the lay literati was Inquiry into the Origin of Humanity.

During this period, Zongmi also began collecting every extant Chan text in circulation with the goal of producing a Chan canon to create a new section of the Buddhist canon. This work is lost but the title, Collected Writings on the Source of Chan remains.

Last years (835–841)
It was Zongmi's association with the great and the powerful that led to his downfall in 835 in an event known as the Sweet Dew Incident. A high official and friend of Zongmi, Li Xun, in connivance with Emperor Wenzong of Tang and his general Zheng Zhu, attempted to curb the power of the court eunuchs by killing them all. The plot failed and Li Xun fled to the Zhongnan Mountains, seeking refuge with Zongmi. Li Xun was quickly captured and executed and Zongmi was arrested and tried for treason. Impressed with Zongmi's bravery and honesty in the face of execution, the eunuch generals spared Zongmi.

Little is known about Zongmi's activities after this event, though he certainly would not have been welcome at court and evidence from his late commentary on the Yulanpen sutra indicates he returned home to Sichuan. Zongmi died in the zazen posture on 1 February 841 in Chang'an. According to his wishes, his body was left as food for scavengers, and was then cremated. Twelve years later he was awarded the posthumous title "Samādhi-Prajñā Dhyāna Master" and his remains were interred in a stupa called Blue Lotus.

Influence
Zongmi's work continued to be studied by later Chinese Buddhists. His ecumenical vision of Chan as a single family with many branches and his account of Tang Chan lineages were influential sources for later Song dynasty Chan works. His work was also very influential on the "Huayan-Chan" traditions of the Khitan Liao Empire (916-1125) and the Tangut kingdom (1038-1227) of the Western Xia.

Zongmi's views about sudden awakening, gradual cultivation, and the importance of scripture to Chan practice was also influential on the later Chan figure Yongming Yanshou, whose Mind Mirror (Zongjinglu) echoes Zongmi's views on these topics. The Mind Mirror was widely printed during the Song dynasty and widely circulated. According to Broughton, the Mind Mirror "conveyed to Song Chan the most fundamental elements of Zongmi's Chan Prolegomenon, sometimes in Zongmi's wording or close paraphrases". Zongmi's ideas also remained influential during the later Song and Ming dynasties. The paradigm of "sudden awakening, gradual cultivation" was promoted and defended by later Chan figures like as Dahui Zonggao (1089-1163) and Hanshan Deqing (1546-1623).

According to Albert Welter, during the Song era, there were to main contrasting styles of Chan, the "moderate" sutra based Chan (wenzi chan) of Zongmi and Yanshou, and the "rhetorical" Chan of the Linji school which is based on recorded sayings literature (like the Linji yulu). The rhetorical Chan held that Chan was "separate transmission outside the teachings" (jiaowai biechuan) and thus saw sutra based Chan as a confused and futile method and the sutras as "old toilet paper that wipes away filth" (Extensive Record of Yunmen).

Furthermore, Zongmi's thought also influenced later Neo-Confucianism. For example, Zhu Xi's (1130-1200) critique of Chan Buddhism is a but recapitulation, in Confucian terms, of Zongmi's critique of the Hongzhou school.

The Chan of Zongmi and Yanshou was also influential on Korean Buddhism, due to its impact on Jinul's (1158-1210) system, who also argued for the unity of Chan and the teachings. According to Broughton, Jinul's influential Excerpts from the Separately Circulated Record of the Dharma Collection with Inserted Personal Notes "is an expression of sutra-based sudden awakening gradual practice Guifeng Chan". The gradual practice is centered around the Kanhua Chan of Dahui, making Jinul's system a mix of Zongmi Chan and Dahui's Linji Chan. Zongmi's Chan Preface remained one of the most widely printed texts throughout Korean Buddhist history, with a large number of woodblock print editions. Zongmi's thought continues to be influential on modern Korean Seon (where the Preface remains part of the study curriculum of the Chogye order), and his ideas remain a major topic of discussion today.

Through the influence of the Mind Mirror, Zongmi-Yanshou Chan also reached Japan, were it was influential on the Daruma school of Zen and on the thought of early Rinzai figures like Myoan Yosai (1141-1215) and Enni Ben'en (1202-1280). The works of Zongmi and Yanshou were also printed by the monasteries of the Five Mountains of Rinzai (around Kyoto and Kamakura). This tradition valued textual study, and the influence of Zongmi can be seen in the works of some of its important literari figures, like Kiyō Hōshū (1361-1424), who promoted Zongmi's teachings with the aphorism: "No Zen separate from the teachings; no teachings separate from Zen."

Philosophy

Zongmi's lifelong work was the attempt to incorporate differing and sometimes conflicting systems into an integrated framework that harmonized various Buddhist teachings and traditions, especially those of Chan practice and Huayan doctrine. More specifically, according to Peter Gregory, one of Zongmi's main philosophical projects was to reformulate Huayan thought so as to "provide an ontological basis and philosophical rationale for Ch'an practice". This philosophical basis also provided a basic ethical theory for Chan which could stand up against Confucian critiques. It also provided the moral foundations needed to critique the more antinomian forms of Chan Buddhism that had developed in the Tang.

Metaphysically speaking, Zongmi moves beyond the thought of earlier Huayan thinkers like Fazang, who emphasized the doctrine of perfect interfusion or unobstructed interpenetration. Instead, Zongmi emphasizes the teachings of the Mahayana Awakening of Faith on the "one mind" (一心) and the teachings of the Sutra of Perfect Enlightenment on the teaching of sudden enlightenment (頓教) followed by gradual cultivation, which is based on the originally enlightened mind that all beings have.

Classification of teachings
As with many Chinese Buddhist scholars, doctrinal classification (Chinese: p’an chiao) was an integral part of Zongmi's work. Such taxonomical schemas (prefigured by Indian ideas like the three turnings of the wheel) were important in Chinese Buddhism, since they provided a sense of order and a hermeneutical structure to the mass of Buddhist teachings which contained numerous, seemingly conflicting ideas and which had been transmitted to China at different times.

Zongmi's classification system, influenced by previous Huayan schemas like those of Fazang and Chengguan, provided a schematic analysis of various teachings and practices which explained and harmonized their differences. Zongmi organized all teachings into a hierarchical schema that integrates and validates all traditions and teachings, seeing each lower category as partially true. According to Gregory, "the logic by which such a synthetic approach worked was dialectical. Each teaching overcame the particular shortcoming of the one that preceded it, and the highest teaching was accorded that vaunted position precisely because it succeeded in sublating all of the other teachings within itself."

Zongmi's mature panchiao (found in sources like Inquiry into the Origin of Humanity and Chan Preface) arranged the Buddhist teachings into five categories:

 * 1) The teaching of men and gods (人天教), teaches ethics, karmic retribution and rebirth;
 * 2) The teachings of the Hinayana ("Lesser Vehicle", non-Mahayana Śrāvakayāna Buddhism, Ch: 小乘教); including the doctrine of not-self and the classic teachings of Abhidharma Buddhism;
 * 3) The Mahayana teaching of the analysis of phenomenal appearances (大乘法相教), mainly referring to the Yogacara teachings which see all phenomena as constructed by mind;
 * 4) The Mahayana teaching of the negation of phenomenal appearances (大乘破相教), the teaching of emptiness and Madhyamaka which sees all phenomena, even mind, as equally unreal;
 * 5) The teaching of the one vehicle (ekayāna) that reveals the nature (一乘顯性教) also known as the "teaching of the nature" (xingzong, i.e. faxing zong, "Dharmata teaching"), which is associated with the teaching of intrinsically enlightened pure mind found in the Awakening of Faith and the Sutra of Perfect Enlightenment. It teaches that one's own mind is the true nature (Ch: zhenxing, Skt: tattva), which is the true mind (zhenxin), empty, calm, intrinsically pure. It is a clear, bright and constant knowing (Ch: zhi, Skt: jñana), also called buddha-nature, mind ground, etc.

In Zongmi's teaching, the "nature" of each person is identical with Buddha-nature, which is emphasised in Chan. He stated, "To designate it, initially there is only one true spiritual nature, that is not born, does not die, does not increase, does not decrease, does not become, and does not change." In giving this teaching the highest position, Zongmi's schema is quite different from the Huayan system of Fazang, who regarded the Huayan "perfect" teaching of the interpenetration of all dharmas (Ch: shih-shih wu-ai), as illustrated by the Indra's net metaphor, to be the supreme teaching.

Zongmi's thought instead focuses on the nature of the mind and on how all phenomena arise from that ultimate nature and are deeply interconnected with it. As such, Zongmi follows a doctrinal shift initiated by Chengguan, who also saw the idea of the "unobstructed interpenetration of the absolute and phenomenal" (li-shih wu-ai) and "nature origination" (how all phenomena arise from one ultimate nature) as a more foundational doctrine than the interpenetration of all phenomena (shih-shih wu-ai). Zongmi identifies this ultimate nature, the one true dharmadhatu, with the "one mind" taught in the Awakening of Faith and with buddha-nature. As such, for Zongmi, the highest teaching of the "One Vehicle" (Ekayāna), the culmination of Chan and Huayan, is grounded in this teaching of the ontological basis of all reality, which is the originally awakened mind, the ultimate "nature" or principle (li).

According to Gregory, Zongmi's classification scheme provides a dialectical progression, from a naive view, to increasingly apophatic positions (which culminate in the full apophasis of Madhyamaka that lets go of all views and words), to an advanced cataphasis which recovers the use of positive religious language as a method to directly reveal (hsien-shih) the true nature. Zongmi's defense of affirmative religious discourse (Ch: piao-ch'uan), against those who held that only a negative apophatic discourse (che-ch'uan, like Madhyamaka or some forms of Chan), is an important element of his Chan Preface. Zongmi held that the tradition which "takes the nature as its cardinal principle" made use of both types of discourse in order to reveal the true nature of reality. Zongmi thus affirms positive statements which describe ultimate reality as "empty tranquil awareness" (k'ung chi chih), "the enlightened illumination of awareness and vision", "the mirror-like radiance of the spirit", "brilliant refulgence", "clear tranquility," "clear and bright, unobscured, ever-present awareness" and so forth. According to Zongmi, without the positive elements, the negative descriptions of the ultimate cannot apply to anything but to a complete nothingness.

The intrinsically enlightened true mind
The central doctrine of the highest teaching which is at the core of Zongmi's thought is the idea that all beings are endowed with a "perfectly enlightened mind," which is none other than the buddha-nature (tathāgatagarbha), the Dharmadhātu (法界, absolute reality) and the "one mind" of the Awakening of Faith. According to Zongmi:"The teaching of the one vehicle that reveals the nature holds that all sentient beings without exception have the intrinsically enlightened true mind. From time without beginning it is permanently abiding and immaculate. It is shining, unobscured, clear and bright ever-present awareness. It is also called Buddha-nature, and it is also called tathagatagarbha. From time without beginning deluded thoughts cover it, and [sentient beings] by themselves are not aware of it. Because they only recognize their inferior qualities, they become indulgently attached, enmeshed in karma, and experience the suffering of birth-and-death. The Great Enlightened One took pity upon them and taught that everything without exception is empty. He further revealed that the purity of the numinous enlightened true mind is wholly identical with that of all Buddhas."

Zongmi saw the term "empty tranquil awareness" as expressing the positive and negative aspects of the ultimate reality:"'Empty' means empty of all phenomenal appearances and is still a negative term. 'Tranquil' just indicates the principle of the immutability of the true nature and is not the same as nothingness. 'Awareness' indicates the revelation of the very essence and is not the same as discrimination. It alone constitutes the intrinsic essence of the true mind."

Zongmi described the ultimate truth by the term "awareness" (chih), which he also sometimes calls the "numinous awareness" (ling-chih), "numinous awareness unobscured" (ling-chih pu-mei), "ever-present awareness" (ch'ang-chih), and "empty tranquil awareness" (k'ung chi [chih] chih). This awareness is not the defiled mind, mental discrimination, nor is it an object of the mind. According to Gregory, for Zongmi, this awareness is "the underlying ground of consciousness that is always present in all sentient life," it is "the noetic ground of both delusion and enlightenment, ignorance and wisdom, or, as he aptly terms it, the mind ground (hsin-li)." In the Chan Letter, Zongmi explains the mind ground through the simile of the wish fulfilling jewel (cintamani) which is like a crystal ball reflecting many colors. Because its substance is brightness, the jewel has the capacity to reflect. In terms of the analogy, the brightness stands for awareness while the color characteristics stand for objective supports. However, while it has the potential to reflect them in all their variety, the jewel itself has no differentiation at all in terms of colors. Zongmi says, "The variety is inherent in the color characteristics themselves; the bright jewel never changes."

According to Zongmi:

"Just the perfect brightness of jade-like purity [yingjing yuanming] is the jewel substance [zhuti]. [...] The black color, up to and including all the other colors, such as blue and yellow, etc., are unreal. [...] When one truly sees the color black, the black from the outset is not black. It is just the brightness. The blue from the outset is not blue. It is just the brightness, up to and including, all the [other colors], such as red, white, yellow, etc., are like this. They are just the brightness. If, at the locus of the color characteristics, one after the other you just see the perfect brightness of jade-like sparkling purity, then you are not confused about the jewel. [...] If you are just free of confusion about the jewel, then black is non-black; black is the bright jewel, and so on with all colors. This is freedom [from the two extremes of] existence and non-existence. The brightness and the blackness are in fusion. How could there be any further obstacle?"

Zongmi follows the Śrīmālādevī sutra and the Awakening of Faith in seeing this ultimate mind ground as being both empty and not-empty in the sense that it is empty of defilement and discrimination, but not empty of positive qualities like the four perfect qualities (Skt: guṇapāramitā, which are: permanence, purity, bliss, and self), as well as all the buddha qualities.

For Zongmi, any gradual practice of the Buddhist path must be grounded in faith in the originally enlightened mind. This is what makes it possible to attain a sudden awakening to our own originally enlightened state. This true awakened mind is what is revealed during the experience of sudden awakening. It is a recognition of what is already the case, one's innate originally awakened nature. This is like how gold is already present in ore.

Before awakening, the true mind has two principles: being in concord with and not being in concord with unreal thought. Insofar as it concords with unreal thought, it can contain both purity and impurity and is known as the storehouse consciousness. But in its principle of not being in concord with unreal thought its substance is immutable and it is known as thusness. Zongmi's metaphysics follows the "one mind two gates" model of the Awakening of Faith which sees the mind as having an unchanging absolute aspect and a phenomenonal appearing aspect. As such, Zongmi saw enlightenment and delusion as two parallel sides of the same non-dual reality.

Zongmi cites the Avatamsaka sutra which states that "there is not a single sentient being that is not fully endowed with the wisdom of the Tathagata" as a major source for this teaching. He also sees this teaching as exemplified by numerous sources, like the Sutra of Perfect Enlightenment, the Śūraṅgama Sūtra, Ghanavyuha, Śrīmālādevī, Tathagatagarbha sutra, Nirvana sutra, Awakening of Faith, Buddha-nature treatise (Foxing lun) and Ratnagotravibhāga.

Nature origination
This ultimate ground of reality, the "dharmadhatu of suchness" (chen-ju fa-chieh), is, according to Zongmi, "the pure mind that is the source of Buddhas and sentient beings." It is the nature or the essence of mind which gives rise to all dharmas (phenomena). Thus, "nature origination" (hsing-ch'i) refers to the manifestation of all phenomena in the universe from the ultimate nature.

Nature origination is also identified with the concept of the interpenetration of the absolute and the phenomenal realms (li-shih wu-ai). Like gold (i.e. the nature) and the various objects that can be produced from gold (dharmas), they are neither the same exact thing, nor are they different in essence.

According to Zongmi, this teaching of nature origination is one of the key differences between Huayan thought and Yogacara, since for Yogacara, the pure suchness is inert and unchanging, while in Huayan and in the Awakening of Faith, suchness also has a conditioned aspect that gives rise to all dharmas.

Furthermore, it is precisely due to the fact that all dharmas originate wholly from suchness that all dharmas also interpenetrate. As Zongmi writes, nature origination "is why Buddhas and sentient beings are inextricably interconnected and the pure and defiled lands harmoniously interpenetrate." As such, for Zongmi, the Huayan idea of the interfusion and interpenetration of all dharmas with each other (shih shih wu-ai) is subordinate to and derived from the interpenetration of the ultimate principle and phenomena that arise from it (li-shih wu-ai).

Zongmi also makes use of the East Asian discourse of essence and function (ti yong) to explain the ultimate nature. In this analysis, Buddha nature (i.e. suchness) is described as the essence (ti) which is the pure and absolute (pu-pien) aspect of the mind. Meanwhile, the innate capacities and manifestations of Buddha nature are its functioning, which includes an intrinsic unchanging awareness, and also the impermanent and conditioned (sui-yuan) aspects of mind which respond to various causes and conditions. According to Zongmi, all of these are aspects of a unitary reality which includes both dependent arising on the conventional level and also nature origination which is more fundamental and makes conventional causality possible. Ultimate reality thus includes all phenomena in the world which are all empty and interconnected with each other, as well as the ontological basis of reality, the dynamic mind ground of awareness (chih), which generates all phenomena.

The five stages of phenomenal evolution
Another key part of Zongmi's system is a theory of cosmogonic development which, as Gregory writes, "explains how the world of delusion and defilement, the world in terms of which unenlightened beings experience themselves, evolves out of a unitary ontological ground that is both intrinsically enlightened and pure." In this system, which is largely derived from the Awakening of Faith, the world of suffering, samsara, arises out of a primal dualistic delusion (i.e. the unenlightened aspect of the storehouse consciousness) which arises out of the primordial state of undifferentiated perfection. Zongmi compares the primal delusion to the act of a wealthy and well respected man falling asleep and forgetting who they are.

The five stages are:

 * 1) The One Mind, the ultimate source (Ch: pen-yuan) of all pure and impure phenomena. It is the underlying nature (hsing) of all reality, the buddha-nature in all beings that is the basis for samsara and nirvana and yet transcends all dualities. It is also termed the wondrous mind of perfect enlightenment (in the Sutra of Perfect Enlightenment) and the one true dharmadhatu (in Huayan). Enlightenment for Zongmi is thus a return to the original source, which is also said to be ineffable, inconceivable and beyond thought.
 * 2) The Two Aspects of the One Mind. As described in the Awakening of Faith, the one mind also has two aspects: the transcendent "mind as suchness", which is neither born nor dies, and the "mind subject to birth-and-death", which refers to the storehouse consciousness. As indicated by the Awakening of Faith, the second aspect (the storehouse) is an interfusion of buddha-nature and the conditioned deluded consciousness "in such a way that they are neither one nor different." This is compared to waves and water, though a wave is not identical to water itself, their nature is not different. As such, in this system, ignorance and delusion are adventitious manifestations of the one mind.
 * 3) The Two Modes of the storehouse consciousness. These two modes are the enlightened mode, which is free of thought and gives rise to pure dharmas (phenomena) through pure dependent arising, and the unenlightened mode, which gives rise to impure dharmas (i.e. the world of samsara). The enlightened mode refers to intrinsic awakening, which is none other than the undifferentiated Dharmakaya. The unenlightened mode is "primordial unenlightenment" which gives rise to impure dependent arising, and which is yet also based on the enlightened nature (i.e. they are of the same essence).
 * 4) The Three Subtle Phenomenal Appearances: 1. activation (yeh) or the activity of ignorance, 2. the perceiving subject, 3. the perceived object. The activity of ignorance is the initial "subtle movement of thought" which stirs the originally calm one mind, and this produces the duality of subject and object.
 * 5) The Six Coarse Phenomenal Appearances, which is a step by step process found in the Awakening of Faith, each one leading to the other, . The six are: 1. discrimination of likes and dislikes; 2. continuation: awareness and thoughts of pleasure and pain; 3. attachment to and objectification of perceptual objects; 4. conceptual elaboration; 5. karmic generation in which one's attachments lead to actions (karma); 6. suffering of karmic bondage - the experience of the consequences of one's past actions.

Zongmi also correlated these five stages to the various teachings in his doctrinal classification system, seeing different teachings as focusing on overcoming different stages of phenomenal development in a process that reverses the course of phenomenal evolution. As such, the first three stages of evolution are associated with the sudden and advanced teachings, while the fourth stage of evolution is correlated with the Yogacara teachings. The first two of the six coarse phenomenal appearances are then seen as covered by the Mahayana teachings on emptiness, the third and fourth appearances by Hinayana, and the last two phenomenal appearances are dealt with by the teaching of men and gods. Thus, this psycho-cosmogy provides another layer to Zongmi's soteriological map of Buddhist practice.

Sudden awakening, gradual cultivation
Zongmi tried to harmonize the different views on the nature of awakening (bodhi). A key topic of debate in Tang Chan circles was the nature of awakening, and how it could occur suddenly. Zongmi advocated the view of sudden awakening (Ch: tun-wu), accompanied by gradual cultivation. In the Chan Preface, Zongmi writes that all ancient Chan teachings "first show the original nature [kaishi benxing] and then require reliance on this nature to practice dhyana [yixing xiuchan]."

Indeed, sudden awakening and gradual cultivation were ultimately one and the same according to Zongmi who writes:

"It is only because of variations in the style of the World Honoured One’s exposition of the teachings that there are sudden expositions in accordance with the truth and gradual expositions in accordance with the capacities [of beings]…this does not mean that there is a separate sudden and gradual [teaching]."Zongmi thus understands "sudden" and "gradual" teachings as different ways or methods that express the same truth, not to two separate teachings or truths. Furthermore, although the sudden teaching reveals the truth directly, and results in a "sudden" understanding that all beings are Buddhas, this does not mean that one attained full Buddhahood right away. This is because the deeply rooted defiled seeds are only removed by extensive training which reduces these residual effects of past life conditionings.

As such, Zongmi differentiates between two types of awakening / enlightenment: the awakening of initial insight (chieh-wu) which is the basis of gradual cultivation and the awakening of full realization (cheng wu) which is the final awakening. This means Zongmi's analysis of the path has three main components: initial insight, gradual cultivation, full realization. Thus, Zongmi advocated "sudden enlightenment, gradual cultivation" (Ch: tun-wu chien-hsiu). Sudden awakening provides an initial insight into the true nature and gradual cultivation was needed to eliminate all remaining traces of defilements that prevented the total integration of the originally enlightened mind into all actions.

As Gregory writes: "the realization that one was a Buddha was not sufficient to guarantee that one acted like a Buddha. The gradual practices thus played a necessary role in the post-enlightenment actualization of the insight afforded by the sudden teaching to beings of superior capacity." Furthermore, for Zongmi, gradual practices are only truly authentic after the experience of sudden awakening. He writes "if one engages in spiritual cultivation without having first experienced enlightenment, then it is not authentic practice."

Zongmi described this sudden awakening event as follows:"Sudden enlightenment means that although [beings] have been deluded [from time] without beginning, recognizing the four elements as their body and deluded thoughts as their mind and taking them both together as constituting their self, when they meet a good friend who explains to them the meaning of the absolute and conditioned [aspects of suchness], the nature and its phenomenal appearance, the essence and its functioning ..., then they at once realize that [their own] marvelous awareness and vision is their true mind, that the mind-which is from the beginning empty and tranquil, boundless and formless-is the dharmakaya, that the nonduality of body and mind is their true self, and that they are no different from all Buddhas by even a hair."

Zongmi drew on various similes to explain the process of sudden awakening - gradual cultivation:

"In terms of the elimination of hindrances, it is like when the sun immediately comes out, yet the frost melts gradually. With respect to the perfection of virtue, it is like a child which, when born, immediately possesses four limbs and six senses. As it grows, it gradually develops control over its actions. Therefore, the Hua Yen [ Avatamsaka sutra] says that when the bodhicitta is first aroused, this is already the accomplishment of perfect enlightenment."

Zongmi also used the metaphor of water and waves found in the Awakening of Faith to explain this teaching. The essential tranquil nature of water which reflects all things (intrinsic enlightenment) is disturbed by the winds of ignorance (un-enlightenment, delusion). Although the wind may stop suddenly (sudden enlightenment), the disturbing waves subside only gradually (gradual cultivation) until all motion ceases and the water once again reflects its intrinsic nature (Buddhahood). However, whether disturbed by ignorance or not, the fundamental nature of the water (i.e., the pure luminous mind) never changes.

Stages of spiritual cultivation
Zongmi also outlined a ten stage process of spiritual development. Each stage overturns a corresponding stage of the development of samsara, moving from coarser aspects to more subtle ones. The stages are as follows:

 * 1) Sudden enlightenment: the initial insight into the true nature of the mind, the original enlightenment (benjue), which gives rise to faith.
 * 2) Resolving to attain Buddhahood: one generates compassion, wisdom, and bodhisattva vow fueled by bodhicitta.
 * 3) Cultivating the five practices of giving, morality, patience, striving and meditative insight (samatha-vipasyana).
 * 4) Spiritual development - one further develops compassion, wisdom, and vows, and the three minds: straight mind, deep mind, and mind of great compassion.
 * 5) Realizing the emptiness of self
 * 6) Realizing the emptiness of dharmas
 * 7) Mastery of form - one realizes that perceptual objects are manifestations of the mind and gains mastery over them.
 * 8) Mastery of mind - one sees the perceiving subject as also illusory and gains mastery of it
 * 9) Freedom from thought - one is fully aware of the origin of deluded thoughts and sees the true nature of mind.
 * 10) Attainment of Buddhahood - returning to the ultimate source, realizing the non-duality of all things, even delusion and enlightenment, samsara and nirvana.

Scripture and doctrinal study
Aside from promoting the importance of "gradual" classic Buddhist practices such as taking of precepts and taking part in rituals, Zongmi also promoted the importance of doctrinal study as an indispensable element of spiritual cultivation. Indeed, for Zongmi, the Buddhist scriptures were a key element of validating one's spiritual experiences, even in Chan, which relies on a direct "mind to mind" transmission of the essential meaning. According to Zongmi, even though Bodhidharma taught in this direct way, he did not reject Buddhist scripture or see enlightenment as a separate teaching from what is taught in the sutras. Zongmi held that those who believed that Chan was separate from doctrinal study were deeply confused.

As such, Zongmi promoted the "correspondence of the teachings and Chan" (chiao-ch'an i-chih), which sees the meaning of Chan as the same as the meaning of the scriptures. Because of this, Zongmi's Chan Preface states that: "the scriptures are like a marking line to be used as a standard to determine true and false....those who transmit Ch'an must use the scriptures and treatises as a standard." According to Zongmi, Chan and Buddhism rely on three sources of knowledge (pramana): inference, direct perception and the word of the Buddha. Those who only rely on one of these are unbalanced, and they may go astray by merely relying on their own unguided experiences, which may be faulty. Zongmi thus promotes an approach to spiritual cultivation that relies on a harmonious development of both meditation and doctrinal study.

That being said, for Zongmi, the scriptures themselves are not Chan, which relies on an intuitive mysterious or dark understanding based on "getting the idea and forgetting the words". As such, the meaning of Chan and the sutras "are spontaneously understood in a mysterious way", not in a conceptual intellectual way. The reading and study of the sutras therefore, are a useful guide to the true meaning, but they are not the true meaning itself, which appears in the source of one's mind suddenly without calculation, and without becoming caught up in the words or clinging to the text. This is the meaning of the phrase Zongmi attributes to Bodhidharma "no involvement with the written word." It is not a rejection of textual study, but an indication that Chan realization goes beyond words even while also skillfully making use of the scriptures. This is the way in which the following simile by Zongmi on Chan (which is beyond words) and the scriptures is to be understood:"The sutras are like an inked marking string [shengmo], serving as a model by which to establish the false and the correct. The inked marking string is not the skill itself; a skillful craftsman must use the string as a standard [wei ping]. The sutras and treatises are not Chan; one who transmits Chan must use the sutras and treatises as a norm [wei zhun]."Broughton argues that the sutra based "sudden enlightenment, gradual cultivation" Chan was the normative type of Chan during the Song and Ming dynasties, as well as in Goryeo Korea. Zongmi's sutra-based Chan view contrasts with another popular Zen narrative which sees Chan / Zen as being separate from textual learning altogether, and sees scriptural study as pointless for the Zen practitioner. According to Broughton, modern Japanese Rinzai Zen generally promotes this other popular narrative which is "an image of Zen framed by the Record of Linji—wherein the real teacher boldly discards the teachings of the Buddhist canon."

Account of Chan Buddhism
Zongmi's analysis of Chan (i.e. Zen Buddhism) also attempted to harmonize the various splits and debates which existed among the various Chan schools of his time, which often engaged in debate and disparaging attacks on each other. Citing the parable of the blind men and the elephant, Zongmi thinks that the various competing Chan schools have different parts of the whole truth, and his doctrinal framework seeks to provide a holistic account of all of the various schools of Chan, writing: "If taken in isolation, each of them is wrong. But if taken together, each of them is valid." This shows Zongmi's basic doctrinal tendency which as Gregory explains "is always to articulate a comprehensive framework in which such discrepant perspectives can be harmoniously subsumed. Such a comprehensive framework not only provides a larger context in which the divergent perspectives can be validated as parts of a whole, it also provides a new and higher perspective that is superior to the others because it succeeds in sublating them within itself."

Critiques of Chan schools
Zongmi gave critiques on seven Chan schools in his Prolegomenon to the Collection of Expressions of the Zen Source and although he promoted his own Heze school as exemplifying the highest practice, his accounts of the other schools are mostly balanced. His works on Chan remain an invaluable source for the development of Buddhism in China. Zongmi's view of the various Chan schools is closely tied to his doctrinal classification system and his critiques of the various Tang Chan schools draw on his broader philosophical position.

Zongmi categorized all Chan lineages or houses as belonging to one of three main types of doctrinal systems (Skt: siddhanta, Ch: zong):

 * the doctrine of " stopping thought of the unreal and cultivating mind (only)", which is associated with the consciousness-only school. According to Zongmi, Chan schools which focus on this doctrine include: Jingzhong, Northern school, Baotang, and the South Mountain Buddha-Recitation Gate school.
 * the doctrine of "cutting off and not leaning on anything", which is associated with the teaching of emptiness and Madhyamaka. The Shitou and Oxhead schools are associated with this doctrine.
 * the doctrine of "directly revealing the mind nature", which is associated with the buddha-nature teachings of the Dharma nature school (Zongmi's own view). This is considered to contain the clear and explicit (Skt: nitartha) meaning of the true nature. The Heze and Hongzhou lineages are associated with this doctrine.

Northern school
Zongmi's critique of Northern Chan was based on its practice of removing impurities in the mind to reach enlightenment. He associated this teaching with the Chinese Yogacara school and saw the practice as a gradual method that was not authentic Chan. Zongmi criticized this on the basis that the Northern school was under the misconception that impurities were inherently different or separate from the pure mind, making this a dualistic view that fails to understand the natural working of intrinsic enlightenment. Zongmi held that the impurities of the mind were merely adventitious, empty and ultimatelly non-dual with the pure mind, since they are nothing but the empty manifestations (yung) of the ultimate nature. Thus, the impurities are not inherently impure, they merely appear that way due to our dualistic misperception. In attempting to remove them one merely feeds that sense of duality. Instead, for Zongmi, we need to first have an initial awakening to the true nature of things, which includes the principle of the non-duality of the pure nature and the impure manifestations, of the identity of samsara and nirvana. It is only based on this awakening which validates the mundane world of appearances as a manifestation of the true nature that one can practice the path of Chan.

Oxhead school
Zongmi's criticism of another prominent Chan lineage, the Oxhead school, is based on their understanding of emptiness, which Zongmi saw as one sided. He claimed that the Oxhead School taught "no mind" (wuxin, 無心) but did not recognize the nonempty aspect of mind, assuming that the intrinsically enlightened nature is only "empty" and "that there is nothing to be cognized". Drawing on the buddha-nature sutras like the Nirvana sutra, Zongmi argues that emptiness does not mean a pure negation, since according to the Nirvana sutra, "when there is nothing in a jar, the jar is said to be empty-it does not mean that there is no jar." As such, the Chan teaching of "no mind" means that the true mind is empty of defilements and concepts. As such, the teaching of the Oxhead school is one sided since it only focuses on emptiness and negation, but does not know the intrinsically enlightened mind and the ultimate reality. In critiquing the purely negative dialectic of this school, Zongmi writes "if there are no real things whatsoever, then on the basis of what are illusions made to appear? Moreover, there has never been a case of the illusory things in the world before us being able to arise without being based on something real....therefore we know that this teaching merely destroys our attachment to feelings but does not yet reveal the nature that is true and numinous."

Hongzhou school
Zongmi was also critical of certain Chan sects that seemed to ignore the moral order of traditional Buddhism and Confucianism and that failed to understand the conventional truths of Buddhism in general. As such, Zongmi claimed the Hongzhou school, derived from Mazu Daoyi (709–788), believed "everything as altogether true". Zongmi does not critique Mazu himself (whom he sees as a great master), rather that "lineage's junior trainees" (bi zong houxue) who misundertood his teachings. As such, the target of his critique is not Mazu, but Zongmi's ninth century Hongzhou contemporaries.

According to Zongmi, the Hongzhou teaching led some foolish people to a radical nondualism that believed that all actions, good or bad, and all experiences (anger, greed, pleasure, pain) were totally equal expressions of the Buddha-nature. As such, it collapses the ultimate essence into its myriad functions, holding that there is no essence apart from its functioning. Zongmi saw this as ethically dangerous, and as denying the need for ethics and spiritual cultivation. Zongmi also described their teaching as "entrusting oneself to act freely according to the nature of one's feelings". For Zongmi, this was a dangerously antinomian view as it eliminated all moral distinctions and validated any actions as expressions of the essence of Buddha-nature.

Furthermore, Zongmi also held that this view failed to understand that while the essence (buddha-nature, the mind ground) and its functions (all phenomena) are two aspects of a non-dual reality, they are also still different in an important way, because the essence is more fundamental, being the basis for enlightenment. Indeed, for Zongmi, they are ultimately "neither one nor different". As such, from the perspective of a sentient being, there is still a distinction between awakening and delusion. As Gregory writes, "their inseparability is what makes religious cultivation possible, and their difference is what makes religious cultivation necessary."

Because of this, Zongmi sees the Hongzhou practice of "simply allowing the mind to act spontaneously" as an inadequate form of spiritual cultivation. As such, Zongmi writes:

"Hung-chou constantly says: 'Since greed, anger, compassion, and good deeds are all the Buddha-nature, how could there be any difference between them?' This is like someone seeing that there is never any difference in the wet nature [of the water] and not realizing that there is an enormous difference between the success of a boat that crosses over it and the failure of a boat that capsizes in it. Therefore, as far as this line's approach toward sudden enlightenment is concerned, even though it comes close, it still does not hit the mark, and, as far as its approach toward gradual cultivation is concerned, it is mistaken and completely backward."

Thus, while Zongmi acknowledged that the essence of Buddha-nature and its expressions in the world are non-dual, he insisted that there is still a difference between good and evil, and affirmed the need for spiritual cultivation. Gregory writes that to avoid the dualism he saw in the Northern Line and the radical antinomianism of the Hongzhou school, Zongmi's paradigm preserved "an ethically critical duality within a larger ontological unity".

As part of his critique, Zongmi introduces a key distinction between two types of the functioning of the ultimate nature:
 * "intrinsic functioning of the self-nature" (tzu-hsing pen-yung), which is "ever-present awareness". Zongmi compares this to the reflectivity of a mirror, its capacity to reflect.
 * "responsive functioning in accord with conditions" (sui-yuan ying-yung), which he compares to the images reflected in a mirror and corresponds to psycho-physical functions like speech, discrimination, bodily movement, etc.

According to Zongmi, the Hongzhou school emphasizes the responsive functioning of buddha-nature, but fails to understand the intrinsic functioning of the self-nature. Indeed, Zongmi thinks that the Hongzhou school mistakes the mirror's reflections for its capacity to reflect. That is to say, it mistakes the various impermanent appearances for the true nature itself. This mistake means that they are unable to make ethical and practical distinctions.

Furthermore, Zongmi argues that the true nature can only be truly seen in the state of no-thought (wu-nien), which is like a reflective jewel that is not reflecting any colors and so can be seen just as it is. He cites the Awakening of Faith and Chengguan in support of this view. For Zongmi, this direct perception of the true nature, an empty tranquil awareness in which there are no thoughts, is what is at the core of the experience of sudden awakening. Zongmi argues the Hongzhou school lacks this knowledge, and that their teaching of spontaneous action can even become a rationalization for deluded activity. Thus, they lack knowledge of what sudden enlightenment is, and also of gradual cultivation.

Baotang school
Zongmi also criticized the Baotang school in similar ways. According to Zongmi, this school had misinterpreted Shenhui's teaching on "no-thought" as entailing "the rejection of all forms of traditional Buddhist ethical practice and ritual observance." According to Zongmi, this school rejected conventional Buddhist practices like "worship, repentance, reciting scriptures, painting Buddhist images, and copying sutras" because they sought to "extinguish discriminative consciousness" (mieh-shih).

On the three teachings
Zongmi was also concerned with providing an inclusive view of the three main religions of China: Confucianism, Daoism and Buddhism. He saw all three as expedient means (upaya), functioning within a particular historical context. Although he saw Buddhism as revealing the highest truth, this had nothing to do with the level of understanding of the three sages (Confucius, Laozi and Buddha). Zongmi saw them as "consummate sages, who, in accord with the times and in response to beings, made different paths in setting up their teachings". As such, the three teachings differed in their goals and in the particular circumstances in which they arose, not in the truth realized by their founding figures.

As Zongmi writes: "Since encouraging the myriad practices, admonishing against evil, and promoting good contribute in common to order, the three teachings should all be followed and practiced. [However] if it be a matter of investigating the myriad phenomena, exhausting principle, realizing the nature, and reaching the original source, then Buddhism alone is the ultimate judgment."

Zongmi's early training in Confucianism led him to attempt to develop a syncretic framework where Chinese non-Buddhist principles could be integrated within Buddhist teachings. For example, Zongmi matches the five precepts with the five Confucian virtues (benevolence, righteousness, propriety, trustworthiness, wisdom), and he links Buddhist cosmological processes with Daoist cosmogony. Zongmi valued Confucian teachings greatly and did not reject its moral vision. He merely held that only Buddhism could provide the metaphysical foundation for it. He added Confucianism (along with Daoism) into his doctrinal classification scheme in an inclusive manner that validated them, something that previous Huayan scholars had not done.

Zongmi's worldview thus attempts to subtlate (Ch: shiyo) non-buddhist traditions (along with buddhist ones), which Zongmi saw as partially true, into what he considered to be a higher more comprehensive worldview. Since this supreme view was inclusive of other, supposedly more more limited views, it was seen as a more impartial view that was called "round" or "having no sides" (yuan). Zongmi considered this to be the harmonious consumation of other views.

That being said, Zongmi also presented several criticisms of Confucianism as well as Daoism, such as in the first part of Inquiry into the Origin of Humanity, which critiques such concepts as the Way (dao), spontaneity (ziran), primal qi (yuanqi), and the mandate of heaven. One of Zongmi's main critiques of the view that the way of heaven determines all things is that this provides no grounding for ethics and no way to distinguish between good and evil. He critiques Confucianism's narrow focus on this life, and their failure to understand karma and rebirth. Zongmi also mounts a theodicy critique of the concept of the mandate of heaven, the idea that heaven (tian) "monitors the sociopolitical world of human endeavor to ensure that it resonates with the larger rhythms of a universe functioning in natural harmony with Confucian moral principles". According to Zongmi, heaven cannot be seen as providential moral force due to the injustice and suffering found in the world.

Writings
Zongmi's writings were extensive and infuential. He wrote commentaries, ritual manuals, and popular essays for literati audiences. There is no certainty about the quantity of Zongmi's writings. Zongmi's epitaph, written by his student Pei Xiu, (787?–860) listed over ninety fascicles. Tsan-ning's (919–1001) biography claimed over two hundred.

For modern scholars, Zongmi provides the "most valuable sources on Tang dynasty Zen. There is no other extant source even remotely as informative". Unfortunately, many of Zongmi's works are lost, including his Collected Writings on the Source of Ch’an which would provide modern scholars with an invaluable source to reconstruct Tang dynasty Chan.

Work on the Sutra of Perfect Enlightenment
Zongmi's first major work was his commentary and sub-commentary on Sūtra of Perfect Enlightenment, completed in 823–824. The sub-commentary contains extensive data on the teachings, ideas and practices on the seven houses of Chan. These data are derived from personal experience and observations.

Zongmi also wrote a major work in eighteen fascicles called A Manual of Procedures for the Cultivation and Realization of Ritual Practice according to the Scripture of Perfect Enlightenment. In this work, Zongmi discusses the theoretical basis of Buddhist practice as well as the specific details of Buddhist practice. According to Gregory, the work is influenced by the writings of Zhiyi and explains a "complex program of ritual and meditation practice in terms of the Scripture of Perfect Enlightenment."

The Chart of Chan Succession
The Chart of the Master-Disciple Succession of the Chan Gate That Has Transmitted the Mind-Ground in China (Chung-hua ch’uan-hsin-ti ch’an-men shih-tzu ch’eng-his t’u), was written at the request of Pei Xiu sometime between 830 and 833. The work clarifies the major Chan traditions of the Tang era. It contains detailed critiques of the Northern School, the Ox-head School and the two branches of Southern Chan, the Hongzhou and his own Heze lineage.

The Chan Preface
The Prolegomenon to the Collection of Expressions of the Chan Source, also known as the Chan Preface, was written around 833. It provides a theoretical basis for Zongmi's vision of the correlation between Chan and the Buddhist scriptures. It gives accounts of the several lineages extant at the time, many of which had died out by the time of the Song dynasty (960–1279). This work was the preface to Collection of Expressions of the Chan Source, (Chan-yuan zhuquanji) which was a large collection of Chan texts that Zongmi compiled as a kind of Chan scriptural canon, i.e. a Chan pitaka (basket of scriptures). Unfortunately, this collection itself is lost, and only the Chan Preface exists.

On the Original Nature of Man
Zongmi's Inquiry into the Origin of Humanity, (or On the Original Nature of Man, or The Debate on an Original Person, Chinese: 原人論 Yüanren lun) was written sometime between 828 and 835. This essay became one of his best-known works and was popular with the literati of the time. The writing style is simple and straightforward, and the content not overly technical, making the work accessible to non-Buddhist intellectuals of the day. The essay surveys the current major Buddhist teachings of the day, as well as Confucian and Taoist teachings, defending his Buddhist view as the most comprehensive of all. It develops a holistic and inclusive perspective of all the Buddhist and non-buddhist teachings, while also discussing the nature of existence and the human condition. De Bary writes that the essay may have been written as a kind of rejoinder to the On the original Nature of Man (Yuan jen) and On the Tao (Yuan tao) by the contemporary Confucian Han Yu (768–824).

Other commentaries
Zongmi also wrote the following works:

 * Commentary on the Diamond Sutra
 * Commentary on the Nirvana Sutra, probably written between 823 and 828.
 * Commentary on the Chengweshilun
 * Commentary on the Bianzong lun of Xie Lingyun
 * Commentary on the Yulanpen Sutra

title: Guifeng Zongmi


text:
Vidago

Vidago is a village and a former-civil parish in the municipality of Chaves, in the Portuguese district of Vila Real. Until its merger into the parish of Vidago, Arcossó, Selhariz e Vilarinho das Paranheiras, the civil parish administered an area of 6.40 km2 and included a population of 1186 inhabitants.

History
Known as a spa town, Vidago was a centre of mineral water treatments dating back to the late 19th century and early 20th century.

For a long time distance was an impediment to the development of the town, due to the difficulty of transport. Guests would arrive by coach from Porto, using roads that were often little more than dirt path. Finally in 1907 the railroad between Régua and Vila Real was extended to Pedras Salgadas, 12 km to the south of Vidago. Three years later the first train arrived at the Vidago station, resulting in renewed popularity of the spa.

During the Gilded Age, the popularity of the waters in Vidago resulted in the construction of the Hotel Palace of Vidago, which became the retreat of some Portuguese monarchs. The spa town saw its period of splendor from 1875 to 1877 when King Luís I made consecutive visits. The fame of the curative powers of the waters spread so far that from 1876 to 1889 they were awarded prizes in Madrid, Paris, Vienna and Rio de Janeiro. For this, Vidago was once one of the most visited Portuguese spas; the water is apparently effective in the treatment of digestive problems, as it contains sodium bicarbonate and, according to publicity leaflets, radioactive elements.

Bottled mineral waters Salus and Campilho are extracted in Vidago.

When the Palace Hotel was built in 1910, it was one of the most luxurious on the Iberian Peninsula, that included concerts, tennis and croquet courts, and, on the small island within the lake, there was a spacious skating rink. Following a period of remodelling, refurbishing and renovation, the 5-star hotel reopened in 2010, which included an expanded golf course and park.

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

title: Vidago


text:
Horse gait

Horses can use various gaits (patterns of leg movement) during locomotion across solid ground, either naturally or as a result of specialized training by humans.

Classification
Gaits are typically categorized into two groups: the "natural" gaits that most horses will use without special training, and the "ambling" gaits that are various smooth-riding, four-beat footfall patterns that may appear naturally in some individuals. Special training is often required before a horse will perform an ambling gait in response to a rider's command.

Another system of classification that applies to quadrupeds uses three categories: walking and ambling gaits, running or trotting gaits, and leaping gaits. The British Horse Society dressage rules require competitors to perform four variations of the walk, six forms of the trot, five leaping gaits (all forms of the canter), halt, and rein back, but not the gallop. The British Horse Society equitation examinations also require proficiency in the gallop as distinct from the canter.

The so-called "natural" gaits, in increasing order of speed, are the walk, trot, canter, and gallop. Some consider these as three gaits, with the canter a variation of the gallop. All four gaits are seen in wild horse populations. While other intermediate speed gaits may occur naturally to some horses, these four basic gaits occur in nature across almost all horse breeds. In some animals the trot is replaced by the pace or an ambling gait. Horses who possess an ambling gait are usually also able to trot.

Walk

The walk is a four-beat gait that averages about 7 km/h. When walking, a horse's legs follow this sequence: left hind leg, left front leg, right hind leg, right front leg, in a regular 1-2-3-4 beat. At the walk, the horse will alternate between having three or two feet on the ground. A horse moves its head and neck in a slight up and down motion that helps maintain balance.

In detail, a horse starts a walk by lifting its left front leg (the other three feet are touching the ground). It then lifts its right hind leg (while being supported by the diagonal pair right front and left hind). Next, the left front foot touches the ground (the horse is now supported by all but the right hind leg); then the horse lifts its right front leg (it is now supported laterally on both left legs), and shortly afterwards it sets down the right hind leg (only the right front leg is now lifted). Then it lifts its left hind leg (diagonal support), puts down the right front (lateral support), lifts the left front, puts down the left hind, and the pattern repeats.

Ideally, the advancing rear hoof oversteps the spot where the previously advancing front hoof touched the ground. The more the rear hoof oversteps, the smoother and more comfortable the walk becomes. Individual horses and different breeds vary in the smoothness of their walk. However, a rider will almost always feel some degree of gentle side-to-side motion in the horse's hips as each hind leg reaches forward.

The fastest "walks" with a four-beat footfall pattern are actually the lateral forms of ambling gaits such as the running walk, singlefoot, and similar rapid but smooth intermediate speed gaits. If a horse begins to speed up and lose a regular four-beat cadence to its gait, the horse is no longer walking but is beginning to either trot or pace.

Trot
The trot is a two-beat gait that has a wide variation in possible speeds and averages about 13 km/h. A very slow trot is sometimes referred to as a jog. An extremely fast trot has no special name, but in harness racing, the trot of a Standardbred is faster than the gallop of the average non-racehorse. The North American speed record for a racing trot under saddle was measured at 48.68 km/h

In this gait, the horse moves its legs in unison in diagonal pairs. From the standpoint of the balance of the horse, this is a very stable gait, and the horse need not make major balancing motions with its head and neck. The trot is the working gait for a horse. Horses can only canter and gallop for short periods at a time, after which they need time to rest and recover. Horses in good condition can maintain a working trot for hours. The trot is the main way horses travel quickly from one place to the next.

Depending on the horse and its speed, a trot can be difficult for a rider to sit because the body of the horse drops a bit between beats and bounces up again when the next set of legs strike the ground. Each time another diagonal pair of legs hits the ground, the rider can be jolted upwards out of the saddle and meet the horse with some force on the way back down. Therefore, at most speeds above a jog, especially in English riding disciplines, most riders post to the trot, rising up and down in rhythm with the horse to avoid being jolted. Posting is easy on the horse's back and once mastered is also easy on the rider.

To not be jostled out of the saddle and to not harm the horse by bouncing on its back, riders must learn specific skills in order to "sit" the trot. Most riders can easily learn to sit a slow jog trot without bouncing. A skilled rider can ride even a powerfully extended trot without bouncing, but to do so requires well-conditioned back and abdominal muscles, and to do so for long periods is tiring for even experienced riders. A fast, uncollected, racing trot, such as that of the harness racing horse, is virtually impossible to sit.

Because the trot is such a safe and efficient gait for a horse, learning to ride the trot correctly is an important component in almost all equestrian disciplines. Nonetheless, "gaited" or "ambling" horses that possess smooth four-beat intermediate gaits that replace or supplement the trot (see "ambling gaits" below) are popular with riders who prefer for various reasons not to have to ride at a trot.

Two variations of the trot are specially trained in advanced dressage horses: the piaffe and the passage. The piaffe is essentially created by asking the horse to trot in place, with very little forward motion. The passage is an exaggerated slow motion trot. Both require tremendous collection, careful training and considerable physical conditioning for a horse to perform.

Canter (Lope)
The canter, or Lope as it is known in Western circles of riding, is a controlled three-beat gait that is usually a bit faster than the average trot but slower than the gallop. The average speed of a canter is 16–27 km/h (10–17 mph), depending on the length of the stride of the horse. Listening to a horse canter, one can usually hear the three beats as though a drum had been struck three times in succession. Then there is a rest, and immediately afterwards the three-beat occurs again. The faster the horse is moving, the longer the suspension time between the three beats. The word is thought to be short for "Canterbury gallop".

In the canter, one of the horse's rear legs – the right hind leg, for example – propels the horse forward. During this beat, the horse is supported only on that single leg while the remaining three legs are moving forward. On the next beat the horse catches itself on the left hind and right front legs while the other hind leg is still momentarily on the ground. On the third beat, the horse catches itself on the left front leg while the diagonal pair is momentarily still in contact with the ground.

The more extended foreleg is matched by a slightly more extended hind leg on the same side. This is referred to as a "lead". Except in special cases, such as the counter-canter, it is desirable for a horse to lead with its inside legs when on a circle. Therefore, a horse that begins cantering with the right hind leg as described above will have the left front and hind legs each land farther forward. This would be referred to as being on the "left lead".

When a rider is added to the horse's natural balance, the question of the lead becomes more important. When riding in an enclosed area such as an arena, the correct lead provides the horse with better balance. The rider typically signals the horse which lead to take when moving from a slower gait into the canter. In addition, when jumping over fences, the rider typically signals the horse to land on the correct lead to approach the next fence or turn. The rider can also request the horse to deliberately take up the wrong lead (counter-canter), a move required in some dressage competitions and routine in polo, which requires a degree of collection and balance in the horse. The switch from one lead to another without breaking gait is called the "flying lead change" or "flying change". This switch is also a feature of dressage and reining schooling and competition.

If a horse is leading with one front foot but the opposite hind foot, it produces an awkward rolling movement, called a cross-canter, disunited canter or "cross-firing".

Gallop

The gallop is very much like the canter, except that it is faster, more ground-covering, and the three-beat canter changes to a four-beat gait. It is the fastest gait of the horse, averaging about 40 to 48 km/h, and in the wild is used when the animal needs to flee from predators or simply cover short distances quickly. Horses seldom will gallop more than 1.5 to 3 km before they need to rest, though horses can sustain a moderately paced gallop for longer distances before they become winded and have to slow down.

The gallop is the gait of the classic race horse. Modern Thoroughbred horse races are seldom longer than 1.5 mi, though in some countries Arabian horses are sometimes raced as far as 2.5 mi. The fastest galloping speed is achieved by the American Quarter Horse, which in a short sprint of a quarter mile (0.25 mi) or less has been clocked at speeds approaching 55 mph. The Guinness Book of World Records lists a Thoroughbred as having averaged 43.97 mph over a two-furlong (0.25 mi) distance in 2008.

Like a canter, the horse will strike off with its non-leading hind foot; but the second stage of the canter becomes, in the gallop, the second and third stages because the inside hind foot hits the ground a split second before the outside front foot. Then both gaits end with the striking off of the leading leg, followed by a moment of suspension when all four feet are off the ground. A careful listener or observer can tell an extended canter from a gallop by the presence of the fourth beat.

Contrary to the old "classic" paintings of running horses, which showed all four legs stretched out in the suspension phase, when the legs are stretched out, at least one foot is still in contact with the ground. When all four feet are off the ground in the suspension phase of the gallop, the legs are bent rather than extended. In 1877, Leland Stanford settled an argument about whether racehorses were ever fully airborne: he paid photographer Eadweard Muybridge to prove it photographically. The resulting photos, known as The Horse in Motion, are the first documented example of high-speed photography and they clearly show the horse airborne. According to Equix, who analyzed the biometrics of racing Thoroughbreds, the average racing colt has a stride length of 24.6 ft; that of Secretariat, for instance, was 24.8 ft, which was probably part of his success.

A controlled gallop used to show a horse's ground-covering stride in horse show competition is called a "gallop in hand" or a hand gallop.

In complete contrast to the suspended phase of a gallop, when a horse jumps over a fence, the legs are stretched out while in the air, and the front legs hit the ground before the hind legs. Essentially, the horse takes the first two steps of a galloping stride on the take-off side of the fence, and the other two steps on the landing side. A horse has to collect its hindquarters after a jump to strike off into the next stride.

Pace
The pace is a lateral two-beat gait. In the pace, the two legs on the same side of the horse move forward together, unlike the trot, where the two legs diagonally opposite from each other move forward together. In both the pace and the trot, two feet are always off the ground. The trot is much more common, but some horses, particularly in breeds bred for harness racing, naturally prefer to pace. Pacers are also faster than trotters on average, though horses are raced at both gaits. Among Standardbred horses, pacers breed truer than trotters – that is, trotting sires have a higher proportion of pacers among their get than pacing sires do of trotters.

A slow pace can be relatively comfortable, as the rider is lightly rocked from side to side. A slightly uneven pace that is somewhat between a pace and an amble is the sobreandando of the Peruvian Paso. On the other hand, a slow pace is considered undesirable in an Icelandic horse, where it is called a lull or a "piggy-pace".

With one exception, a fast pace is uncomfortable for riding and very difficult to sit, because the rider is moved rapidly from side to side. The motion feels somewhat as if the rider is on a camel, another animal that naturally paces. However, a camel is much taller than a horse and so even at relatively fast speeds, a rider can follow the rocking motion of a camel. A pacing horse, being smaller and taking quicker steps, moves from side to side at a rate that becomes difficult for a rider to follow at speed, so though the gait is faster and useful for harness racing, it becomes impractical as a gait for riding at speed over long distances. However, in the case of the Icelandic horse, where the pace is known as the skeið, "flying pace" or flugskeið, it is a smooth and highly valued gait, ridden in short bursts at great speed.

A horse that paces and is not used in harness is often taught to perform some form of amble, obtained by lightly unbalancing the horse so the footfalls of the pace break up into a four beat lateral gait that is smoother to ride. A rider cannot properly post to a pacing horse because there is no diagonal gait pattern to follow, though some riders attempt to avoid jostling by rhythmically rising and sitting.

Based on studies of the Icelandic horse, it is possible that the pace may be heritable and linked to a single genetic mutation on DMRT3 in the same manner as the lateral ambling gaits.

Ambling
There are a significant number of names for various four-beat intermediate gaits. Though these names derive from differences in footfall patterns and speed, historically they were once grouped together and collectively referred to as the "amble". In the United States, horses that are able to amble are referred to as "gaited". In almost all cases, the primary feature of the ambling gaits is that one of the feet is bearing full weight at any one time, reflected in the colloquial term, "singlefoot".

All ambling gaits are faster than a walk but usually slower than a canter. They are smoother for a rider than either a trot or a pace, and most can be sustained for relatively long periods, making them particularly desirable for trail riding and other tasks where a rider must spend long periods of time in the saddle. There are two basic types: lateral, wherein the front and hind feet on the same side move in sequence, and diagonal, where the front and hind feet on opposite sides move in sequence. Ambling gaits are further distinguished by whether the footfall rhythm is isochronous (four equal beats in a 1–2–3–4 rhythm) or non-isochronous (1–2, 3–4 rhythm) created by a slight pause between the ground strike of the forefoot of one side to the hind foot of the other.

Not all horses can perform an ambling gait. However, many breeds can be trained to produce them. In most "gaited" breeds, an ambling gait is a hereditary trait. A 2012 DNA study of movement in Icelandic horses and mice have determined that a mutation on the gene DMRT3, which is related to limb movement and motion, causes a premature "stop codon" in horses with lateral ambling gaits.

The major ambling gaits include:

 * The fox trot is most often associated with the Missouri Fox Trotter breed, but it is also seen under different names in other gaited breeds. The fox trot is a four-beat diagonal gait in which the front foot of the diagonal pair lands before the hind. The same footfall pattern is characteristic of the trocha, pasitrote and marcha batida seen in various South American breeds.
 * Many South American horse breeds have a range of smooth intermediate lateral ambling gaits. The Paso Fino's speed variations are called (from slowest to fastest) the paso fino, paso corto, and paso largo. The Peruvian Paso's lateral gaits are known as the paso llano and sobreandando. The lateral gait of the Mangalarga Marchador is called the marcha picada.
 * The rack or racking is a lateral gait most commonly associated with the five-gaited American Saddlebred. In the rack, the speed is increased to be approximately that of the pace, but it is a four-beat gait with equal intervals between each beat.
 * The running walk, a four-beat lateral gait with footfalls in the same sequence as the regular walk but characterized by greater speed and smoothness. It is a distinctive natural gait of the Tennessee Walking Horse.
 * The slow gait is a general term for various lateral gaits that follow the same general lateral footfall pattern, but the rhythm and collection of the movements are different. Terms for various slow gaits include the stepping pace and singlefoot.
 * The tölt is a gait that is often described as being unique to the Icelandic horse. The footfall pattern is the same as for the rack, but the tölt is characterized by more freedom and liquidity of movement. Some breeds of horses that are related to the Icelandic horse, living in the Faroe Islands and Norway, also tölt.
 * The revaal or ravaal is a four-beat lateral gait associated with Marwari, Kathiawari or Sindhi horse breeds of India.

title: Horse gait


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Pedras Salgadas

Pedras Salgadas is a small spa town in the district of Vila Real, in Northern Portugal, located approximately 37 km north of the district capital of Vila Real. It is famous for its mineral waters. There are several small hotels in the area, which cater to tourists who want to relax in the green countryside and benefit from the spa facilities located there. In 2010 the Portuguese beer and water company Unicer opened a brand new spa complex in the town. Pedras Salgadas is located in the municipality of Vila Pouca de Aguiar. Both are located on N2, the national road linking Chaves with Vila real. The new four-lane A24 highway passes a few kilometers west of the town and is connected by a feeder road. The railway was closed in the nineteen eighties and the abandoned railway bed has now been tarmacked for use as a  cycling and walking path. This path extends to Vila Pouca de Aguiar, at a distance of about 10 kilometers.

title: Pedras Salgadas


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Thutmose (sculptor)

Thutmose, also known as "The King's Favourite and Master of Works, the Sculptor Thutmose" (also spelled Djhutmose, Thutmosis, and Thutmes), was an Ancient Egyptian sculptor in the Amarna style. He flourished around 1350 BC, and is thought to have been the official court sculptor of the Egyptian pharaoh Akhenaten in the latter part of his reign. A German archaeological expedition digging in Akhenaten's deserted city of Akhetaten, known today as Amarna, found a ruined house and studio complex (labeled P47.1-3) in early December 1912; the building was identified as that of Thutmose based on an ivory horse blinker found in a rubbish pit in the courtyard inscribed with his name and job title. Since it gave his occupation as "sculptor" and the building was clearly a sculpture workshop, the determination seemed logical and has proven to be accurate.

Recovered works
Among many other sculptural items recovered at the same time was the polychrome bust of Nefertiti, apparently a master study for others to copy, which was found on the floor of a storeroom. In addition to this now-famous bust, twenty-two plaster casts of faces—some of which are full heads, others just the face—were found in Rooms 18 and 19 of the studio, with an additional one found in Room 14. Eight of these have been identified as various members of the royal family, including Akhenaten, his other wife Kiya, his late father Amenhotep III, and his eventual successor Ay. The rest represent unknown individuals, presumably contemporary residents of Amarna.

A couple of the pieces found in the workshop depict realistic images of older noblewomen, something rare in Ancient Egyptian art, which more often portrayed women in an idealized manner as always young, slender, and beautiful. One of the plaster faces depicts an older woman, with wrinkles at the corner of her eyes, bags under them, and a deeply lined forehead. This piece has been described as showing "a greater variety of wrinkles than any other depiction of an elite woman from ancient Egypt" It is thought to represent the image of a wise, older woman. A small statue of an aging Nefertiti also was found in the workshop, depicting her with a rounded, drooping belly, thick thighs, and a curved line at the base of her abdomen showing that she had borne several children, perhaps intended to project an image of fertility.

Examples of his work recovered from his abandoned studio may be viewed at the Egyptian Museum of Berlin, the Egyptian Museum in Cairo, and the Metropolitan Museum of Art, New York City. Upon the death of Akhenaten, the seat of government was returned from Amarna to Thebes and the associated bureaucratic and professional industries followed.

Tomb
In 1996 the French Egyptologist Alain Zivie discovered at Saqqara the decorated rock cut tomb of the "head of the painters in the place of truth", Thutmose. The tomb dates to the time shortly after the Amarna Period. Although the title of the Thutmose in Saqqara is slightly different from the title of the Thutmose known from Amarna, it seems likely that they refer to the same person and that the different titles represent different stages in his career.

An extensive article by Zivie in the July–August 2018 edition of Biblical Archaeology Review provides great detail and many images of artifacts recovered in an adjacent tomb, discussion of many aspects of several topics regarding Ancient Egyptian research and identification, as well as information about the sculptor, Thutmose.

title: Thutmose (sculptor)


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Talk:Indian family name

Plural
The plurals in "adopting their patronyms as family names" were dropped. Does it mean all Tamils share one patronym or some sort of collective system? --Menchi 19:10, Aug 10, 2003 (UTC)

 * Umm, no, I dropped it so that it is gramatically consistent with the rest of the sentence. -- Arvindn 04:08, 11 Aug 2003 (UTC)

 * I've added another -s after "them...acquire surname". Does it sound grammatical now? Feel free to improve it further. --Menchi 04:12, Aug 11, 2003 (UTC)

Cast surnames
Could we have some example caste-derived surnames, and some subcaste-derived ones? It'd be helpful to also point out the differences between caste- and subcaste-derived surnames, if there's any. --Menchi 19:30, Aug 10, 2003 (UTC)

Hindu only?
I don't mean to offend, but this article covers only Hindu names, or caste-based family names--there are enough Muslims with Gandhi and Raja in their name to make this issue be not only about Hindus as a religious community. Or am I confused? —iFaqeer (Talk to me!) 00:17, Mar 19, 2005 (UTC)

PS
I don't believe it is only "a number" of Sikhs that are named "Singh"...I am tempted to put an notice on this article.  —iFaqeer (Talk to me!) 00:17, Mar 19, 2005 (UTC)

Merge with Indian name
All the names by category culture category articles have the format {region name[s]} and none specifically mention 'family' in them specifically. To retain that consistency, it might be worth considering merging this into Indian name rather than vice versa. Shyamal 11:58, 5 August 2005 (UTC)
 * Fair point. I had a look around and didn't see a concensus - but I didn't check the categories.  I suggested merging Indian name into this one because it has less information so that's an easier merge.  But we can always move the merged article to Indian name if that makes more sense.  Actually I think Indian names is a better title - what do you think? -- Muntfish 12:29, 5 August 2005 (UTC)

 * I fully agree. Shyamal 03:21, 8 August 2005 (UTC)

 * second.. uh third that. --n00b

Creating Indian names
I plan to create a new page with the title Indian names and merge the contents from Indian name and Indian family name into the new page, and when the page is eveolved, we can keep redirects at the Indian family name and Indian name pages. Doctor Bruno 07:52, 26 November 2005 (UTC)

title: Talk:Indian family name


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Involution (mathematics)

In mathematics, an involution, involutory function, or self-inverse function is a function $f$ that is its own inverse,

for all $x$ in the domain of $f : X → X$. Equivalently, applying $f$ twice produces the original value.

General properties
Any involution is a bijection.

The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation ($f(f(x)) = x$), reciprocation ($f$), and complex conjugation ($x ↦ −x$) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.

The composition $x ↦ 1/x$ of two involutions $z ↦ \overline{z}$ and $g ∘ f$ is an involution if and only if they commute: $f$.

Involutions on finite sets
The number of involutions, including the identity involution, on a set with $g$ elements is given by a recurrence relation found by Heinrich August Rothe in 1800:
 * $$a_0 = a_1 = 1$$ and $$a_n = a_{n - 1} + (n - 1)a_{n-2}$$ for $$n > 1.$$

The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 ; these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The number $g ∘ f = f ∘ g$ can also be expressed by non-recursive formulas, such as the sum $$a_n = \sum_{m=0}^{\lfloor \frac{n}{2} \rfloor} \frac{n!}{2^m m! (n-2m)!} .$$

The number of fixed points of an involution on a finite set and its number of elements have the same parity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem.

Real-valued functions
The graph of an involution (on the real numbers) is symmetric across the line $n = 0, 1, 2, ...$. This is due to the fact that the inverse of any general function will be its reflection over the line $an$. This can be seen by "swapping" $y = x$ with $y = x$. If, in particular, the function is an involution, then its graph is its own reflection. Some basic examples of involutions include the functions $$ \begin{alignat}{4} f_1(x) &= -x, \\ f_2(x) &= \frac{1}{x}, \\ f_3(x) &= \frac{x}{x - 1}, \\ \end{alignat}$$ the composition $$f_4(x) := (f_1 \circ f_2)(x) = (f_2 \circ f_1)(x) = -\frac {1}{x},$$ and more generally the function $$g(x) = \frac{b - x}{1 + c x}$$ is an involution for constants $x$ and $y$ that satisfy $b$.

Other nonlinear examples include (note the possibility of interpreting these as compositions): $$f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right),$$ $$f(x) = \exp\left(\frac {1}{\ln(x)}\right),$$ $$f(x) = \frac{x}{\sqrt{x^2-1}}.$$

Other elementary involutions are useful in solving functional equations.

Euclidean geometry
A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates.

Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example.

These transformations are examples of affine involutions.

Projective geometry
An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points.
 * Any projectivity that interchanges two points is an involution.
 * The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution. This theorem has been called Desargues's Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus of Alexandria.
 * If an involution has one fixed point, it has another, and consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called double points.

Another type of involution occurring in projective geometry is a polarity that is a correlation of period 2.

Linear algebra
In linear algebra, an involution is a linear operator $c$ on a vector space, such that $bc ≠ −1$. Except for in characteristic 2, such operators are diagonalizable for a given basis with just $T$s and $2 = I$s on the diagonal of the corresponding matrix. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

For example, suppose that a basis for a vector space $1$ is chosen, and that $−1$ and $V$ are basis elements. There exists a linear transformation $e_{1}$ that sends $e_{2}$ to $f$, and sends $e_{1}$ to $e_{2}$, and that is the identity on all other basis vectors. It can be checked that $e_{2}$ for all $e_{1}$ in $f(f(x)) = x$. That is, $x$ is an involution of $V$.

For a specific basis, any linear operator can be represented by a matrix $f$. Every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution.

The definition of involution extends readily to modules. Given a module $V$ over a ring $T$, an $M$ endomorphism $R$ of $R$ is called an involution if $f$ is the identity homomorphism on $M$.

Involutions are related to idempotents; if $^{2}$ is invertible then they correspond in a one-to-one manner.

In functional analysis, Banach *-algebras and C*-algebras are special types of Banach algebras with involutions.

Quaternion algebra, groups, semigroups
In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation $$x \mapsto f(x)$$ then it is an involution if
 * $$ f(f(x))=x $$ (it is its own inverse)
 * $$ f(x_1+x_2)=f(x_1)+f(x_2) $$ and $$ f(\lambda x)=\lambda f(x) $$ (it is linear)
 * $$ f(x_1 x_2)=f(x_1) f(x_2) $$

An anti-involution does not obey the last axiom but instead
 * $$ f(x_1 x_2)=f(x_2) f(x_1) $$

This former law is sometimes called antidistributive. It also appears in groups as $M$. Taken as an axiom, it leads to the notion of semigroup with involution, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the full linear monoid) with transpose as the involution.

Ring theory
In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples of involutions in common rings:
 * complex conjugation on the complex plane, and its equivalent in the split-complex numbers
 * taking the transpose in a matrix ring.

Group theory
In group theory, an element of a group is an involution if it has order 2; that is, an involution is an element $2$ such that $(xy)−1 = (y)−1(x)−1$ and $a$, where $a ≠ e$ is the identity element. Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is, group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution.

A permutation is an involution if and only if it can be written as a finite product of disjoint transpositions.

The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups.

An element $a^{2} = e$ of a group $e$ is called strongly real if there is an involution $x$ with $G$ (where $t$).

Coxeter groups are groups generated by a set $xt = x−1$ of involutions subject only to relations involving powers of pairs of elements of $xt = x−1 = t−1 ⋅ x ⋅ t$. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

Mathematical logic
The operation of complement in Boolean algebras is an involution. Accordingly, negation in classical logic satisfies the law of double negation: $S$ is equivalent to $S$.

Generally in non-classical logics, negation that satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. Examples of logics that have involutive negation are Kleene and Bochvar three-valued logics, Łukasiewicz many-valued logic, fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in t-norm fuzzy logics.

The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras. For instance, involutive negation characterizes Boolean algebras among Heyting algebras. Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic. The same relationship holds also between MV-algebras and BL-algebras (and so correspondingly between Łukasiewicz logic and fuzzy logic BL), IMTL and MTL, and other pairs of important varieties of algebras (resp. corresponding logics).

In the study of binary relations, every relation has a converse relation. Since the converse of the converse is the original relation, the conversion operation is an involution on the category of relations. Binary relations are ordered through inclusion. While this ordering is reversed with the complementation involution, it is preserved under conversion.

Computer science
The XOR bitwise operation with a given value for one parameter is an involution on the other parameter. XOR masks in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state. The NOT bitwise operation is also an involution, and is a special case of the XOR operation where one parameter has all bits set to 1.

Another example is a bit mask-and-shift function operating on colour values stored as integers, say in the form $¬¬A$, that swaps $A$ and $(R, G, B)$, resulting in the form $R$: $B$.

The RC4 cryptographic cipher is an involution, as encryption and decryption operations use the same function.

Practically all mechanical cipher machines implement a reciprocal cipher, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.

title: Involution (mathematics)


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Talk:G2 (mathematics)

Should we move G2 to G2(math)? Someone just claimed G2 is also the name of an Apple release. Phys

To ease uniformization, I'll move this to G2 (mathematics) Phys

G2 is the name of a Peer to Peer Network... --Nate3000 08:28, 1 Jun 2005 (UTC)

Ambiguity for new learners
This page refers to G2 in both plural and singular mode:

Here the plural is used: G2 is the name of some Lie groups and also their Lie algebras. They are the smallest of the five exceptional simple Lie groups.

The very next sentence uses singular.

Is there only one G2 group (up to isomorphism) or are there many?

67.190.17.110 13:55, 15 February 2007 (UTC) BL

 * Well the language could certainly be cleaned up. There are many things called G2 but usually only one thing (up to isomorphism) of any given type:
 * There is one root system called G2.
 * There is one complex Lie algebra called G2.
 * There is one complex Lie group called G2.
 * There are two real Lie algebras called G2, although usually the compact form is meant.
 * There are two real Lie groups called G2, but again one almost always means the compact one.
 * -- Fropuff 18:48, 15 February 2007 (UTC)

Possible Typo in the Section 'Algebra'?
The arrow in the Dynkin diagram should be facing left, since in the text below root 1 is short and root 2 is long — Preceding unsigned comment added by 144.32.240.72 (talk) 10:59, 17 February 2017 (UTC)

Possible Typo in the Section 'Real Forms'?
Lie algebra aren't my specialism but I know enough to know that

Its maximal compact subgroup is SU(2) × SU(2)/(−1,−1)

looks a bit suspect. Should it perhaps be "SU(2) × SU(2)/(1,−1)", as in the quotient group taken with Z2? PanDTV (talk) 04:21, 6 June 2015 (UTC)

title: Talk:G2 (mathematics)