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Mike Foster (American politician)

Murphy James Foster Jr. (July 11, 1930 – October 4, 2020) was an American businessman and politician who served as the 53rd governor of Louisiana from 1996 to 2004.

Early life and career
Murphy James Foster Jr. was born in Franklin, the seat of government of St. Mary Parish. His father, also named Murphy James Foster, was an area sugar planter and owner of oil and natural gas lands, whose own father was Murphy J. Foster Sr., who was Louisiana governor from 1892 to 1900 and a U.S. Senator from 1901 to 1913. As governor, Murphy Foster led the campaign to disenfranchise Black citizens through the Louisiana Constitution of 1898, calling for an end to "the leprous virus of negro suffrage, thereby creating a running sore which has ever since tainted our government, both federal and state, with foul corruption and loathsome disease."

Foster attended public high school in Franklin, graduated from Louisiana State University in Baton Rouge in 1952 with a Bachelor of Science in chemistry, and Southern University Law Center with a Juris Doctor in 2004, the year he left the governorship. He became an Eagle Scout in the Boy Scouts of America in 1946 and was a recipient of the Distinguished Eagle Scout Award. He was a member of Delta Kappa Epsilon fraternity (Zeta Zeta chapter) and The Friars. He joined the Air Force and served in the Korean War. By the time Mike Foster entered politics, he had already become a wealthy sugar planter and owner of a construction firm. He resided at Oaklawn Manor, an antebellum plantation mansion in Franklin.

Election as governor, 1995
Foster entered the 1995 gubernatorial race as a minor candidate whom most local political observers discounted. Then in September 1995, Foster announced he would qualify for the race as a Republican. The Republicans had not coalesced on a candidate, and Foster's announcement that he was switching parties vaulted him from single digits in the polls to serious contention. Foster rode a wave of popular dissatisfaction with the more unsavory aspects of the casino gambling that had been legalized under outgoing Governor Edwin W. Edwards. Foster came out strongly against gambling and pledged to run Louisiana "like a business." His conservative platform included attacks on welfare abuse, gun control, affirmative action, racial quotas, and political corruption.

He carried the endorsement of the columnist and former Republican presidential candidate Patrick J. Buchanan.

Foster edged out two more well-known candidates for a seat in the runoff with then-United States Representative Cleo Fields from Louisiana's 4th congressional district, a prominent black Democratic politician. Future U.S. Senator Mary Landrieu ran third and missed the general election berth by just 8,983 votes (0.6 percent of the total votes cast). Former Governor Buddy Roemer, seeking a gubernatorial comeback, came in fourth place. Foster's embrace of the Republican label and his conservative platform undercut Roemer, another Democrat-turned-Republican.

Reminiscent of his grandfather's inauguration virtually a century earlier, Mike Foster's inauguration ceremony on January 8, 1996, occurred at the Old State Capitol. Always a man of few words, Foster remarked briefly about the historicity of the occasion and made cordial statements about outgoing four-term Governor Edwin Edwards, who was present.

Foster defeated black Democratic candidates in both of his campaigns for governor—Cleo Fields in 1995 and Congressman William Jefferson of Louisiana's 2nd congressional district in 1999. He defeated Jefferson in a landslide, avoiding a runoff with 64 percent of the vote. His second inauguration took place on January 10, 2000.

Foster as governor
Foster was widely seen as having favored business to a greater degree than had previous governors. Even though his tax corporate America to death policy, caused all oil companies to move their corporate headquarters from New Orleans, Louisiana, to Houston Texas, causing Louisiana to lose out on billions of dollars in tax revenue. He retained the secretary of economic development, former legislator Kevin P. Reilly Sr. of Baton Rouge, the former CEO of Lamar Advertising Company of Baton Rouge. He ended state affirmative action and set-aside programs, which earned him the support of the business community but prompted protests from civil rights groups. Foster also targeted tort reform and ended the practice by which trial lawyers could seek punitive damages from businesses. Foster had close relations with the statewide pro-business lobbying group Louisiana Association of Business and Industry (LABI) for most of his tenure, though there were short-lived tensions in 2000 over Foster's attempt to raise business taxes in an effort to secure funding for higher education. By the end of his second term, Foster was receiving criticism for his reluctance to take business trips in order to attract businesses and jobs to Louisiana, and for enrolling in part-time law school classes while still in office. He also appointed then 24-year-old Bobby Jindal, later a two-term Republican governor, as head of the Louisiana Department of Health and Hospitals.

As his executive counsel, Foster appointed the Democrat Cheney Joseph Jr. (1942–2015), a member of the LSU Law School faculty and a former district attorney for East Baton Rouge Parish.

Foster worked to re-organize the state's community college system by creating the Louisiana Community and Technical College System, and expanded the Tuition Opportunity Program for Students (TOPS). Foster instituted mandatory standardized testing for grade advancement in a move described by his administration as an effort to make public schools more accountable. He made increasing teacher salaries a major priority, at one point promising to stop cashing his paychecks until teachers' salaries reached the Southern average. Andy Kopplin served as Governor Foster's chief of staff.

In 1997, Foster named former state budget director Ralph Perlman as secretary of the Louisiana Gaming Control Board, a position that Perlman held for five years while in his eighties.

Despite having run on an anti-gambling platform, in office Foster became a quiet supporter of the gambling industry. His advocacy of a bailout bill for the Harrah's casino in New Orleans helped ensure the passage of the measure.

Atchafalaya Basin Program
In November 1996 the U.S. Army Corps of Engineers requested that Foster appoint a lead agency to coordinate state participation in the Atchafalaya Basin Project. Foster chose the Louisiana Department of Natural Resources as the lead agency. In December 1996, the Atchafalaya Basin Advisory Committee was created, members appointed, and planning initiated that resulted in the Atchafalaya Basin Master Plan, as authorized by the U.S. Congress. A result of this plan was the creation of the Sherburne Complex Wildlife Management Area (Section 4.41-B) that includes the partnership of the U.S. Fish and Wildlife Service, the Louisiana Department of Wildlife and Fisheries (LDWF), and the U.S. Army Corps of Engineers. The area consists of 44000 acre, and is managed by the Louisiana Department of Wildlife and Fisheries.

Foster and David Duke
In his 1995 campaign, Foster paid more than $150,000 for former Ku Klux Klansman David Duke's mailing list of supporters. After failing to report the purchase as a campaign expenditure, Foster became the first Louisiana governor to admit and pay a fine for a violation of the state's ethics code. Foster insisted that he did not need to report the expenditure because he paid Duke with his personal funds and did not utilize the list in his campaign. Duke also endorsed Foster in the 1995 campaign.

Post governorship
In retirement, Foster lived with his wife Alice C. Foster (born 1940), to whom he was married for over 50 years, on the family estate near Franklin.

In 2003, Foster was inducted into the Louisiana Political Museum and Hall of Fame in Winnfield. In 2013, the state agreed to fund $2 million to renovate part of Franklin City Hall to provide housing for Foster's papers.

On September 28, 2020, news reports confirmed Foster entered hospice care.

After spending a week in hospice care, Mike Foster died on October 4, 2020, at the age of 90.

Electoral history
State Senator, 21st Senatorial District, 1987

Threshold > 50%

First Ballot, October 24, 1987

State Senator, 21st Senatorial District, 1991

Threshold > 50%

First Ballot, October 19, 1991

Governor of Louisiana, 1995

Threshold > 50%

First Ballot, October 21, 1995

Second Ballot, November 18, 1995

Governor of Louisiana, 1999

Threshold > 50%

First Ballot, October 23, 1999

Videos
(1) Foster's Inauguration as Louisiana's 53rd Governor on January 8, 1996, at the Old State Capitol Grounds

(2) Second Inauguration on the State Capitol Grounds on January 10, 2000

(3) State of the State Address on April 29, 1996

(4) Joint Session of the Louisiana State Legislature from May 30, 1996

(5) Opening Address to the Louisiana State Legislature on March 31, 1997

(6) Special Session of the Louisiana State Legislature from March 23, 1998

(7) Opening Address to Fiscal Session of the Louisiana State Legislature from April 27, 1998

(8) Opening Address to the Louisiana State Legislature from March 29, 1999

(9) Gubernatorial Debate Forum from October 8, 1999

(10) Opening Address to the Louisiana State Legislature from March 19, 2000

(11) Fiscal Session from April 24, 2000

(12) Special Session from March 11, 2001

(13) Opening Address to the Louisiana State Legislature from March 26, 2001

(14) Press conference on the September 11th terrorist attacks on September 12, 2001

(15) Opening Address to the Louisiana State Legislature from April 29, 2002

(16) Final State of the State Address from March 31, 2003

title: Mike Foster (American politician)


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Ram Khamhaeng

Ram Khamhaeng (รามคำแหง, ) or Pho Khun Ram Khamhaeng Maharat (พ่อขุนรามคำแหงมหาราช, ), also spelled Ramkhamhaeng, was the third king of the Phra Ruang Dynasty, ruling the Sukhothai Kingdom (a historical kingdom of Thailand) from 1279 to 1298, during its most prosperous era.

He is credited for the creation of the Thai alphabet and the firm establishment of Theravada Buddhism as the state religion of the kingdom.

Birth and name
Ram Khamhaeng was a son of Pho Khun Bang Klang Hao, who ruled as Pho Khun Si Inthrathit, and his queen, Sueang, though folk legend claims his real parents were an ogress named Kangli and a fisherman. He had two brothers and two sisters. The eldest brother died while very young. The second, Ban Mueang, became king following their father's death and was succeeded by Ram Khamhaeng on his death.

At age 19, he participated in his father's successful invasion of the city of Sukhothai, formerly a vassal of the Khmer, establishing the independent Sukhothai Kingdom. Due to his courage in the war, he allegedly was given the title "Phra Ram Khamhaeng” or “Rama the Bold”. After his father's death, his brother Ban Mueang ruled the kingdom, assigning Ram Khamhaeng control of the city of Si Satchanalai.

The Royal Institute of Thailand speculates that Ram Khamhaeng's birth name was "Ram" (derived from Rama, the name of the hero of the Hindu epic Ramayana), as his name following his coronation was "Pho Khun Ramarat" (พ่อขุนรามราช). Furthermore, the tradition at the time was to give the name of a grandfather to a grandson; according to both the 11th Stone Inscription and the Ayutthaya Chronicles by Prasoet Aksoranit, Ram Khamhaeng had a grandson named "Phraya Ram", and two grandsons of Phraya Ram were named "Phraya Ban Mueang" and "Phraya Ram".

In English, an alternate spelling of his name is Ramkhamhaeng. The title Maharat (มหาราช) is the Thai translation of “the Great King”.

Accession
Tri Amattayakun (ตรี อมาตยกุล), a Thai historian, suggests that Ram Khamhaeng should have acceded to the throne in 1279, the year he planted a sugar palm tree in Sukhothai. Prasert na Nagara of the Royal Institute speculates that this was a tradition of Tai Ahom monarchs, who planted banyan or sugar palm trees on their coronation day in the hope that their reign would achieve the same stature as the tree.

The most significant event at the beginning of his reign was the elopement of one of his daughters, Thai: แม่นางสร้อยดาว, RTGS: Mae-nang Soidao, "Lady Soidao" May Hnin Thwe-Da, with the captain of the palace guards, a commoner. The commoner would found the Burmese Hanthawaddy Kingdom and commission compilation of the Code of Wareru, which would provide a basis for the law of Thailand used in Siam until 1908, and in Burma to the present.

Reign
Ram Khamhaeng sent embassies to Yuan China from 1282 to 1323 and imported the techniques to make the ceramics now known as Sangkhalok ceramic ware. He had close relationships with the rulers of nearby city-states, especially Ngam Muang, the ruler of neighboring Phayao (whose wife, according to legend, he seduced), and King Mangrai of Chiang Mai. His campaign against Cambodia left the Khmer country "utterly devastated".

According to Thai history, Ram Khamhaeng is credited with creating the Thai alphabet (Lai Nangsue Thai) from a combination of the Khmer, Sanskrit, Pali, and Grantha alphabets.

It is speculated that Ram Khamhaeng expanded his kingdom as far as Lampang, Phrae, and Nan in the north, Phitsanulok and Vientiane in the east, the Nakhon Si Thammarat Kingdom in the south, the Mon kingdoms of what is now Myanmar in the west, and the Bay of Bengal in the northwest. However, in the mandala political model, kingdoms such as Sukhothai lacked distinct borders, instead being centered on the strength of the capital itself. Claims of Ram Khamhaeng's large kingdom were intended to assert Siamese dominance over mainland Southeast Asia.

Death
According to the Chinese History of Yuan, King Ram Khamhaeng died in 1298 and was succeeded by his son, Loe Thai, though George Cœdès says it is "more probable" it was "shortly before 1318". Legend holds that Khamhaeng perished in the rapids of the rivers of Sawankhalok. Another source says he was slain by a Malay warrior princess named Adruja Wijayamala Singha during a battle between Thai and Malay armies, in a campaign to conquer Malay lands that make up a third of modern Thailand today.

Ram Khamhaeng Inscription
Much of the traditional biographical information comes from the inscription on the Ram Khamhaeng stele, composed in 1292, and contains vague facts about the king. It is now found in the Bangkok National Museum. The formal name of the stele is the "King Ram Khamhaeng Inscription". It was added to the Memory of the World Register in 2003 by UNESCO.

Sangkhalok ceramic ware
Ram Khamhaeng is credited with bringing the skills of ceramic making from China and laying the foundation of a strong ceramic ware industry in the Sukhothai Kingdom. Sukhothai for centuries was the major exporter of the ceramics known as "Sangkhalok ware" (เครื่องสังคโลก) to countries such as Japan, the Philippines, Indonesia, and even to China. The industry was one of the main revenue generators during his reign and long afterward.

Banknote
The reverse of the 20 Baht note (series 16), issued in 2013, depicts images of the royal statue of Ram Khamhaeng seated on the Manangkhasila Asana Throne and commemorates the invention of the Thai script by the king.

Honour
Ramkhamhaeng University, the first Thai university with an open-door policy and with campuses throughout the country was named after Ram Khamhaeng.

Video games
King Ramkhamhaeng is a playable ruler for the Siamese in Sid Meier's Civilization V.

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

title: Ram Khamhaeng


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title: File:Triumph Thunder Seven Cover.jpg


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Leafcutter ant

Leafcutter ants, a non-generic name, are any of 47 species of leaf-chewing ants belonging to the two genera Atta and Acromyrmex, within the tribe Attini. These species of tropical, fungus-growing ants are all endemic to South and Central America, Mexico, and parts of the southern United States. Leafcutter ants can carry twenty times their body weight and cut and process fresh vegetation (leaves, flowers, and grasses) to serve as the nutritional substrate for their fungal cultivates.

Acromyrmex and Atta ants have much in common anatomically; however, the two can be identified by their external differences. Atta ants have three pairs of spines and a smooth exoskeleton on the upper surface of the thorax, while Acromyrmex ants have four pairs and a rough exoskeleton. The exoskeleton itself is covered in a thin layer of mineral coating, composed of rhombohedral crystals that are generated by the ants.

Next to humans, leafcutter ants form some of the largest and most complex animal societies on Earth. In a few years, the central mound of their underground nests can grow to more than 30 m across, with smaller radiating mounds extending out to a radius of 80 m, taking up 30 to 600 m2 and containing eight million individuals.

Reproduction and colony founding
Winged females and males leave their respective nests en masse and engage in a nuptial flight known as the revoada (Portuguese) or vuelo nupcial (Spanish). Each female mates with multiple males to collect the 300 million sperm she needs to set up a colony.

Once on the ground, the female loses her wings and searches for a suitable underground lair in which to found her colony. The success rate of these young queens is very low, and only 2.5% will go on to establish a long-lived colony. To start her own fungus garden, the queen stores bits of the parental fungus garden mycelium in her infrabuccal pocket, which is located within her oral cavity. Colonies are generally founded by individual queens haplometrosis. Because colonies with multiple queens over the lifespan of the colony have been found by a large number of investigators by Weber 1937, Jonkman 1977, Huber 1907, Moser & Lewis 1981, Mariconi & Zamith 1963, Moser 1963, and Walter et al 1938  it is believable that some colonies have multiple foundresses  termed pleometrosis. Pleometrosis is confirmed only for Atta texana by Vinson 1985.

Colony hierarchy
In leafcutter colonies, ants are divided into castes, based mostly on size, that perform different functions. Acromyrmex and Atta exhibit a high degree of polymorphism, four castes being present in established colonies—minims, minors, mediae, and majors. Majors are also known as soldiers or dinergates. Atta ants are more polymorphic than Acromyrmex, meaning comparatively less difference occurs in size from the smallest to largest types of Acromyrmex.

 * Minims are the smallest workers, and tend to the growing brood or care for the fungus gardens. Head width is less than 1 mm.
 * Minors are slightly larger than minima workers, and are present in large numbers in and around foraging columns. These ants are the first line of defense and continuously patrol the surrounding terrain and vigorously attack any enemies that threaten the foraging lines. Head width is around 1.8–2.2 mm.
 * Mediae are the generalized foragers, which cut leaves and bring the leaf fragments back to the nest.
 * Majors, the largest worker ants, act as soldiers, defending the nest from intruders, although recent evidence indicates majors participate in other activities, such as clearing the main foraging trails of large debris and carrying bulky items back to the nest. The largest soldiers (Atta laevigata) may have total body lengths up to 16 mm and head widths of 7 mm.

Ant–fungus mutualism
Their societies are based on an ant–fungus mutualism, and different species of ants use different species of fungus, but all of the fungi the ants use are members of the family Lepiotaceae. The ants actively cultivate their fungus, feeding it with freshly cut plant material and keeping it free from pests and molds. This mutualistic relationship is further augmented by another symbiotic partner, a bacterium that grows on the ants and secretes chemicals; essentially, the ants use portable antimicrobials. Leaf cutter ants are sensitive enough to adapt to the fungi's reaction to different plant material, apparently detecting chemical signals from the fungus. If a particular type of leaf is toxic to the fungus, the colony will no longer collect it. The only two other groups of insects to use fungus-based agriculture are ambrosia beetles and termites. The fungus cultivated by the adults is used to feed the ant larvae, and the adult ants feed on leaf sap. The fungus needs the ants to stay alive, and the larvae need the fungus to stay alive, so mutualism is obligatory.

The fungi used by the higher attine ants no longer produce spores. These ants fully domesticated their fungal partner 15 million years ago, a process that took 30 million years to complete. Their fungi produce nutritious and swollen hyphal tips (gongylidia) that grow in bundles called staphylae, to specifically feed the ants. Leucoagaricus gongylophorus is the most commonly documented fungi farmed by higher attine ant species.

Waste management
Leafcutter ants have very specific roles in taking care of the fungal garden and dumping the refuse. Waste management is a key role for each colony's longevity. The necrotrophic parasitic fungus Escovopsis threatens the ants' food source and thus is a constant danger to the ants. The waste transporters and waste-heap workers are the older, more dispensable leafcutter ants, ensuring the healthier and younger ants can work on the fungal garden. The Atta colombica species, unusually for the Attine tribe, have an external waste heap. Waste transporters take the waste, which consists of used substrate and discarded fungus, to the waste heap. Once dropped off at the refuse dump, the heap workers organise the waste and constantly shuffle it around to aid decomposition. A compelling observation of A. colombica was the dead ants placed around the perimeter of the waste heap.

In addition to feeding the fungal garden with foraged food, mainly consisting of leaves, it is protected from Escovopsis by the antibiotic secretions of Actinomycetota (genus Pseudonocardia). This mutualistic micro-organism lives in the metapleural glands of the ant. Actinomycetota are responsible for producing the majority of the world's antibiotics today.

Parasitism
When the ants are out collecting leaves, they are at risk of attack by some species of phorid flies, parasitoids that lay eggs into the crevices of the worker ants' heads. Often, a minim will sit on a worker ant and ward off any attack.

Also, the wrong type of fungus can grow during cultivation. Escovopsis, a highly virulent fungus, has the potential to devastate an ant garden, as it is horizontally transmitted. Escovopsis was cultured, during colony foundation, in 6.6% of colonies. However, in one- to two-year-old colonies, almost 60% had Escovopsis growing in the fungal garden.

Nevertheless, leafcutter ants have many adaptive mechanisms to recognize and control infections by Escovopsis and other micro-organisms. The most common known behaviors rely on workers reducing the number of fungal spores by grooming, or removing an infected piece of the fungus garden and throwing it away at the waste dump (described as weeding).

Communication
Leafcutter ants use chemical communication and stridulation (substrate-borne vibrations) to communicate with each other.

Prey plants
Leafcutter ants prefer disturbed habitats, likely due to higher concentrations of pioneer plant species. These are more attractive food sources because pioneer plants have lower levels of secondary metabolites and higher nutrient concentrations than the shade-tolerant species that will come later.

Interactions with humans
In some parts of their range, leafcutter ants can be a serious agricultural pest, defoliating crops and damaging roads and farmland with their nest-making activities. For example, some Atta species are capable of defoliating an entire citrus tree in less than 24 hours. A promising approach to deterring attacks of the leafcutter ant Acromyrmex lobicornis on crops has been demonstrated. Collecting the refuse from the nest and placing it over seedlings or around crops resulted in a deterrent effect over a period of 30 days.

title: Leafcutter ant


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User talk:Kn1kda

Hello, welcome to Wikipedia. Here are some useful links in case you haven't already found them;

 * How to edit a page
 * How to write a great article
 * Naming conventions
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If you have any questions, see the help pages or add a question to the village pump. I hope you enjoy editing here and being a Wikipedian!

Kosebamse 21:43, 16 Nov 2003 (UTC)

You can sign your name using three tildes, like Kosebamse. If you use four, you can add a datestamp too.

Outstanding soap bubbles article! We seldom see anything that comprehensive on a first edit. :) It'll probably need to be moved to soap bubble though to fit with the naming convention. Great work, thanks! - Hephaestos 23:31, 17 Nov 2003 (UTC)
 * Came here to say just the same thing. Its a joy to read! Muriel 14:12, 19 Nov 2003 (UTC)

3 months is too long
According to your user contribution list, you were gone for three months. That is much too long a time for such a valuable contributor. I hope you decide to spend more time here writing and editing! :) Kingturtle 01:50, 9 Mar 2004 (UTC)

Soap bubble pic
Hi! Where did you get Image:Soapbubble.merging.250px.png from? - Ta bu shi da yu 11:36, 28 Sep 2004 (UTC)

Image license
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title: User talk:Kn1kda


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27 (number)

27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28.

Mathematics
Twenty-seven is the cube of 3, or three tetrated $$^{2}3 = 3^{3} = 3\times 3\times 3$$, divisible by the number of prime numbers below it (9|nine).

The first non-trivial decagonal number is 27.

27 has an aliquot sum of 13 (the sixth prime number) in the aliquot sequence $$(27, 13, 1, 0)$$ of only one composite number, rooted in the 13-aliquot tree.

The sum of the first four composite numbers is $$4 + 6 + 8 + 9 = 27$$, while the sum of the first four prime numbers is $$2 + 3 + 5 + 7 = 17$$, with 7 the fourth indexed prime.

In the Collatz conjecture (i.e. the $$3n+1$$ problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number.

27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27.

A prime reciprocal magic square based on multiples of $$\tfrac {1}{7}$$ in a $$6 \times 6$$ square has a magic constant of 27.

Including the null-motif, there are 27 distinct hypergraph motifs.

There are exactly twenty-seven straight lines on a smooth cubic surface, which give a basis of the fundamental representation of Lie algebra $$\mathrm {E_{6}}$$.

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra $$\mathrm {F_{4}}.$$

There are twenty-seven sporadic groups, if the non-strict group of Lie type $$\mathrm {T}$$ (with an irreducible representation that is twice that of $$\mathrm {F_{4}}$$ in 104 dimensions) is included.

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold $$\sigma(n) < e^\gamma n \log \log n $$ for values $$n \leq 5040,$$ where $$\gamma$$ is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger $$n.$$

Base-specific
In decimal, 27 is the first composite number not divisible by any of its digits, as well as:
 * the third Smith number and sixteenth Harshad number,
 * the only positive integer that is three times the sum of its digits,
 * equal to the sum of the numbers between and including its digits: $$2+3+4+5+6+7=27$$.

Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27.
 * In similar fashion, any multiple of 27 can be mirrored and spaced with a zero each for another multiple of 27 (i.e. 27 and 702, 54 and 405, and 378 and 80703 are all multiples of 27).
 * Any multiple of 27 with "000" or "999" inserted yields another multiple of 27 (20007, 29997, 50004, and 59994 are all multiples of 27).

In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513.

In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in '$\pi::

$$3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color{red}27}9\ldots$$

If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known.

In science

 * The Moon revolves around the Earth the same direction as Earth spins but 27 (27.3) times slower.
 * 27km is the circumference of the LHC located at CERN in Meyrin Switzerland
 * The atomic number of cobalt.
 * Dark matter is thought to make up 27% of the universe.
 * 27 is the number of bones in the human hand.

Astronomy

 * The Messier object M27, a magnitude 7.5 planetary nebula in the constellation Vulpecula, also known as the Dumbbell Nebula.
 * The New General Catalogue object NGC 27, a spiral galaxy in the Andromeda constellation.
 * The Saros number of the solar eclipse series, which began on March 9, 1993, BCE and ended on April 16, 713 BCE. The duration of Saros series 27 was 1,280.1 years, and it contained 72 solar eclipses. Further, the Saros number of the lunar eclipse series, which began on July 28, 1926, BCE and ended on January 23, 411 BCE. The duration of Saros series 27 was 1532.5 years, and it contained 86 lunar eclipses.

Electronics

 * The type 27 vacuum tube (valve), a triode introduced in 1927, was the first tube mass-produced for commercial use to incorporate an indirectly heated cathode. This made it the first vacuum tube that could function as a detector in AC-powered radios. Prior to the introduction of the 27, home radios were powered by a set of three or more storage batteries with voltages of 3 volts to 135 volts.

In language and literature

 * The number of letters in the Spanish alphabet.
 * The number of books in the New Testament.
 * The number of generations from David to the Christ according to the Gospel of Matthew. (Mt 1,1-17)
 * The number of Grievances of the United States Declaration of Independence.
 * The number of ratified Amendments to the United States Constitution.
 * Until 1835, the English Alphabet consisted of 27 letters: after "Z" the 27th letter of the alphabet was Ampersand (&)
 * The total number of letters in the Hebrew alphabet (22 regular letters and 5 final consonants).
 * Alternate name for The Hunt, a book by William Diehl.
 * Abbé Faria's prisoner number in the book The Count of Monte Cristo.
 * In Stephen King's novel It, It returns every 27 years to Derry.
 * The war led by the Antichrist will begin in the year 1999 and will last 27 years according to Nostradamus.

In astrology

 * 27 Nakṣatra or lunar mansions in Hindu astrology.

In sports

 * The value of all the colors in snooker add up to 27.
 * The number of outs in a regulation baseball game for each team at all adult levels, including professional play, is 27.
 * The New York Yankees have won 27 World Series championships, the most of any team in the MLB.

In other fields
Twenty-seven is also:
 * A-27, American attack aircraft.
 * The code for international direct-dial phone calls to South Africa.
 * The name of a cigarette, Marlboro Blend No. 27.
 * The number of the French department Eure.
 * The current number of countries in the European Union, as of 2024.

title: 27 (number)


text:
Hereford cattle

The Hereford is a British breed of beef cattle originally from Herefordshire in the West Midlands of England. It was the result of selective breeding from the mid-eighteenth century by a few families in Herefordshire, beginning some decades before the noted work of Robert Bakewell.

It has spread to many countries; in 2023 the populations reported by 62 countries totalled over seven million head; populations of over $100,000$ were reported by Uruguay, Brazil and Chile. The breed reached Ireland in 1775, and a few went to Kentucky in the United States in 1817; the modern American Hereford derives from a herd established in 1840 in Albany, New York. It was present in Australia before 1850, and in Argentina from 1858. In the twenty-first century there are breed societies in those countries and in the Czech Republic, Denmark, Estonia, France, Hungary, the Netherlands, Norway, Portugal, Spain and Sweden in Europe; in Brazil, Chile, Paraguay and Uruguay in South America; in New Zealand; and in South Africa.

History
Until the 18th century, the cattle of Herefordshire resembled other cattle of southern England, being wholly red with a white switch, similar to the modern North Devon and Sussex breeds. In the 18th and early 19th centuries, other cattle (mainly Shorthorns) were used to create a new type of draught and beef cattle which at first varied in colour, with herds ranging from yellow to grey and light brown, and with varying amounts of white. By the end of the 18th century the white face characteristic of the modern breed was well established, as was the modern colour during the 19th century.

The Hereford is still seen in the Herefordshire countryside today and featured strongly at agricultural shows. The first imports of Herefords to the United States were made about 1817 by the politician Henry Clay, with larger importation beginning in the 1840s.

Polled Hereford
The Polled Hereford is an American hornless variant of Hereford with a polled gene, a natural genetic mutation selected into a separate breed from 1889.

Iowa cattle rancher Warren Gammon capitalised on the idea of breeding Polled Herefords and started the registry with 11 naturally polled cattle. The American Polled Hereford Association (APHA) was formed in 1910. The American Polled Hereford and American Hereford breeds have been combined since 1995 under the same American Hereford Association name.

In Australia the breed is known as the Poll Hereford.

Traditional Hereford

Many strains of Hereford have used other cattle breeds to import desired characteristics, which has led to changes in the breed as a whole. However, some strains have been kept separate and retained characteristics of the earlier breed, such as hardiness and thriftiness. The Traditional Hereford is now treated as a minority breed of value for genetic conservation.

Health
Eye cancer (ocular squamous cell carcinoma) occurs in Herefords, notably in countries with continued bright sunlight and among those that prefer traits of low levels of red pigmentation round the eye. Studies of eye cancer in Hereford cattle in the US and Canada showed lid and corneoscleral pigment to be heritable and likely to decrease the risk of cancer. Vaginal prolapse is considered a heritable problem, but may also be influenced by nutrition. Another problem is exposed skin on the udder being of light pigmentation and so vulnerable to sunburn.

Dwarfism is known to occur in Hereford cattle, caused by an autosomal recessive gene. Equal occurrence in heifers and bulls means that dwarfism is not considered a sex-linked characteristic.

title: Hereford cattle


text:
User talk:Chris Jones

Wow, your first post is an immediate jump to another user's talk page. Why do I suspect you're DW in yet again another disguise? RickK 21:49, 16 Nov 2003 (UTC)

I'm afraid I do not understand your remark? I was about to edit but decided not to because of your change. Could you explain, please. Chris Jones 21:52, 16 Nov 2003 (UTC)

title: User talk:Chris Jones


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Talk:Dot product

Regarding bases
In 2D and 3D Euclidean space, the scalar product a.b between vectors a and b was defined for me as |a||b|cos@. If you use an orthonormal basis then it can be shown that a.b = (a1, a2, a3).(b1, b2, b3) = a1b1 + a2b2 + a3b3, where (x1, x2, x3) means x1e1 + x2e2 + x3e3 where {ei} is an orthonormal basis (and the xi are scalars).

However, for any other basis, the scalar product will NOT turn out to be the sum of the products of the components with respect to the basis, i.e. will not equal a1b1 + a2b2 + a3b3. However this is how the scalar product is defined by the article.

What is the resolution of this apparent contradiction? --81.152.176.225 (talk) 17:46, 30 March 2011 (UTC)

 * Your confusion is between the notions of inner product an dot product (at least as they are used here; their meanings are not always well distinguished). A Euclidean space comes equipped with an inner product. It is related to the norms of vectors and the cosine of the angle as you indicated, although I would say this defines the angle rather than the inner product. As defined here, a dot product is just an algebraic expression in terms of entries, like the determinant of a matrix (but a different expression of course). It happens that Euclidean spaces always have (many) orthonormal bases, and that in terms of the coordinates on a given orthonormal bases the inner product is given by the dot product of the respective coordinates. In terms of coordinates on other bases this is not the case, but that is no contradiction, it is actually more of a surprise that it is given by the very same expression in terms of coordinates of any orthonormal bases (in general changing coordinates also changes the expressions for functions of the coordinates). As Euclidean spaces are often identified with Rn (in which case the dot product of the vector entries is an inner product), the notions are sometimes confounded, but this should not really be done (and I'm not sure to which one of the two "scalar product" should refer; I think inner product would be the most appropriate). The dot product is defined here in such a manner that the phrase "a coefficient of a matrix product is given by the dot product of a row of the left matrix and a column of the right matrix" makes sense; in this case there is no Euclidean space or inner product involved at all. Hope this helps. Marc van Leeuwen (talk) 17:29, 31 March 2011 (UTC)

 * The deeper issue here is that a vector space alone does not have any concept of orthogonality or length. For a finite dimensional vector space over the reals, if we take any basis, then we can define the scalar product relative to that basis. The basis will be orthonormal relative to the scalar product defined from that basis; the lengths and angles between other vectors are again only relative to the choice of basis, and they can be found using the usual formulas and the scalar product for that basis. — Carl (CBM · talk) 23:24, 30 June 2013 (UTC)

 * To be as general as possible maybe it makes sense to define the dot product by using the metric tensor. In a general non-orthonormal basis the dot product is given as: $$u \cdot v = \sum_{i,j} u_i v_j (\vec{e}_i \cdot \vec{e}_j) = \sum_{i,j} u_i v_j g_{i,j}$$. Then one may argue that for an orthonormal basis the metric tensor is the identity so the simpler formula is retrieved. Length and angles are invariant to change of bases. As it is currently written, the formula is correct only for orthonormal bases. Vassillen (talk) 11:18, 22 June 2020 (UTC)
 * You are talking about the inner product, of which the dot product is a specific case. D.Lazard (talk) 11:24, 22 June 2020 (UTC)

 * The dot product is the inner product in Euclidean space as noted in the article on the inner product. You can pick any basis for the N-dimensional Euclidean space. And in the case of a non-orthonormal basis (an Euclidean space, but not the standard Euclidean space) the current definition does not work. The main issue here is that the way length and angles are measured is not invariant to change of basis anymore. I believe that it should be mentioned somewhere at the very least, before one introduces the assumptions about the orthonormality of the basis. Vassillen (talk) 11:39, 22 June 2020 (UTC)

 * This seems to be mainly a contention about the meaning of the dot/scalar product. A definition can be found in: "Аналитическая геометрия, Том 1", Б. Н. ДЕЛОНЕ и Д. А. РАЙКОВ, 1948 (Boris Delaunay, see Books). The definition can be found in 1.2.9, specifically the definition is introduced through $$\vec{a}\vec{b} = \|a\|\|b\|\cos\theta$$. Later on the algebraic definition is derived from linearity, and the explicit general form using the metric tensor can be found in: 1.2.11.2. Vassillen (talk) 13:21, 22 June 2020 (UTC)

Remove section Dot product, Generalization to tensors...
...since this section doesn't say much at all. All the info is in the main articles tensor, tensor contraction. It would be better to briefly include various other generalizations such as weight functions, inner products on functions, dot products in matrices and dyadics... which are not mentioned, in additon to complex vectors and tensors.Maschen (talk) 20:24, 25 August 2012 (UTC)

Recent edits
Following my major revision of the article, I thought I should summarize here the most important points of my edit, and highlight what I think still needs to be done.

 * My chief concern in the sequence of edits was the demands of WP:NPOV, that we should give equal weight to the two different definitions of the dot product: the geometric and algebraic one. The earlier revision of the article took for granted that the dot product should be defined algebraically in terms of the components of a vector.  As a result, the article had to worry about things like coordinate dependence and orthonormal bases.  This is generally a bad way to define the dot product for that reason.  That is why in physics/tensor analysis, the dot product is defined in a coordinate-independent manner at the outset, and its algebraic properties are then deduced.  Of course, in much of applied mathematics, the dot product is just defined componentwise.  That's fine, but it seems like we should present both points of view in parallel.  It simplifies much of the discussion I think.
 * I have removed the section on Rotations. This was so badly written that it completely failed to convey the relevant point: that the dot product is invariant under rotations.  That is something that should be mentioned in the "Properties" section (which also needs to be taken in hand and organized a bit better).  It is a trivial consequence of the geometric definition of the dot product (!) that doesn't require matrices to explain.
 * I have added a more intuitive proof sketch of the equivalence of the algebraic and geometric definitions using the scalar projection. In doing so, I have unfortunately removed the connection with the law of cosines.  This should probably be mentioned somewhere in the article, but it was not particularly enlightening where it was anyway, situated in the middle of a very dull proof that no one was going to read.
 * I removed the paragraph from the lead that stated: "In Euclidean space, the inner product of two vectors (expressed in terms of coordinate vectors on an orthonormal basis) is the same as their dot product. However, in more general contexts (e.g., vector spaces that use complex numbers as scalars) the inner and the dot products are typically defined in ways that do not coincide." This is misleading because many many authors define the dot product geometrically in Euclidean space rather than in terms of components in an orthonormal basis.
 * I shortened the discussion of the scalar product and geometric definition. This section managed to drag on without saying anything substantive.

-- Sławomir Biały (talk) 15:52, 25 November 2012 (UTC)

 * Thanks, a massive improvement. Maschen (talk) 16:12, 25 November 2012 (UTC)

 * If it's ok I added the cosine law to the end of the properties section. Maschen (talk) 17:02, 25 November 2012 (UTC)
 * Good idea.  Sławomir Biały  (talk) 18:09, 25 November 2012 (UTC)

More recent edits
I edited the properties section without realizing Quondum's edit before:

 * "The cross product can also produce a vector (when the inputs are a vector and a pseudovector); is this worth getting into in this article?)"

I would say no since this article is about the dot product, not the cross...

Hope my changes to the properties section are ok - it is harder or clearer to read? M∧Ŝc2ħεИτlk 18:20, 14 December 2012 (UTC)

IMHO, A simple 2D example of the algebraic formula would make this page useful to a wider audience
IMHO, this page assumes a high level of comfort with mathematical terminology.

As a software writer, using computer graphics, I was looking for confirmation of the simple case of dot product for a vector in (x,y): "(x1,y1) dot (x2,y2) = (x1*x2 + y1*y2)"

I presume the Algebraic section confirms that. But it would take me too long to think through what it is saying. (So I looked elsewhere.)

I suggest this example to anyone who might be thinking "this page could be made more accessible"... ToolmakerSteve (talk) 06:37, 5 February 2013 (UTC)

 * +1. With a diagram where the dot product is identified. Jordan Brown (talk) 21:30, 2 September 2017 (UTC)

"Sequences of numbers" revert
Hoo-boy. This [revert] somewhat surprises me. It is understood that the dot product may be defined on the Cartesian product of any finite number of copies of R. Indeed, it seems that this generalizes to any field. However, "sequences of numbers" does not adequately capture the requirement that the sequences must form a vector space before it can be considered for the dot product; also usually that the two vectors must belong to the same vector space. And damningly, this definition leads to explicit basis dependence (and even if one starts with a coordinate space, one can always find a nonstandard basis and thus another sequence of numbers, to which this definition would still apply, but with a basis-dependent result). Thus, however you word it, you need to either imply a standard basis, or restrict the definition to the original Cartesian product/power if you do not want to reference any basis. Perhaps, Sławomir, you are objecting to my reference to bases? These can be removed, but not the reference to a vector space. Every reference I've seen has clearly intended the definition in refer to a vector space, even if said vector space was defined in terms of the Cartesian product of n copies of R. One disclaimer: my access to references is limited, but certainly in Shaum's Outlines the intended vectorial nature of the "sequence" (and it is not called a sequence) is clear. The article also claims equivalence of the geometric and algebraic definitions, which does not hold as defined in the lead. What makes the idea of arbitrary "sequences of numbers" so notable? — Quondum 01:57, 30 June 2013 (UTC)

 * If you're referring to vectors, instead of "sequences of numbers", why not "ordered tuple" (of components)? M∧Ŝc2ħεИτlk 08:01, 30 June 2013 (UTC)

 * My point is that the properties of a vector space are needed, specifically scalar multiplication and addition. These are not implied in the definition of a tuple, which is the same thing as "a sequence".  Without these, half the listed properties of the dot product do not follow.  Let's get back to basic WP principles: what notable reference defines a dot product on an arbitrary tuple of numbers (even for the case of reals), without the assumption that the tuple belongs to a vector space?  And if you find such a reference, we will have to make it clear that two different dot products exist in the literature, one for which properties 2, 3, 4, 6, 7 and 8 cannot be derived.  — Quondum 12:13, 30 June 2013 (UTC)

 * I'm afraid I don't understand what specifically the issue is. Is it the wording "sequences"?  I'm not married to this wording, and would be happy with n-tuples or other equivalent expression.  My complaint with the original edit was that there is no need to mention bases (or even vector spaces) to define the dot product.  It is specifically defined on $$\mathbf R^n$$.  By definition, $$\mathbf R^n$$ is the space of n-tuples of real numbers, and the dot product of two n-tuples is $$(a_1,\dots,a_n)\cdot (b_1,\dots,b_n) = a_1b_1+\cdots+a_nb_n$$.  That's really all there is to it.  The dot product is not defined on a general vector space.  (That's a related notion, called an inner product.)  In a general vector space V, if you have a basis $$\phi:\mathbf R^n\to V$$, then you can push forward the dot product along φ to define an inner product on V, but this is not the same thing as the dot product since V is not in general a space of tuples (or even if it is a space of tuples, then the inner product defined in this manner might be different from the dot product on that space).  Here you can discuss basis-dependence, because the basis is actually relevant, but that's not really a topic for the article.
 * I really also don't see what the issue is with the properties section. It's certainly well-known without bringing in special bases how to add tuples and multiply them by scalars.  Perhaps these notions could be described more clearly than they currently are in the properties section, but I don't see how User:Quondum could possibly be confused by what is intended.
 * I also don't understand the final charge that our article somehow diverges from the account in most published sources. Many thousands of sources exist that define the dot product the way this article does (some of them give both definitions; some deduce one as a consequence of the other).  See, for instance, the Harvard consortium textbook "Calculus" (the "reform calculus" textbook, which includes both definitions side by side).  Volume 1 of Tom Apostol's Calculus, a classic book which is well-regarded by many mathematicians, Peter Lax's "Linear algebra", Paul Halmos's "Finite dimensional vector spaces" all include the algebraic definition and deduce the geometric definition from it.  Josiah Willard Gibbs's classic textbook "Vector analysis" starts from the geometric definition and deduces the algebraic definition from it, as does Richard Courant's "Differential and integral calculus".  There are slight differences in exposition in these sources, but there is certainly no essential logical departure from the account given in this article.   Sławomir Biały  (talk) 23:54, 30 June 2013 (UTC)
 * The article starts, In mathematics, not In linear algebra. As such, the concept of tuple covers the Cartesian product of arbitrary sets, and does not carry with it the structure of addition or scalar multiplication of tuples.  The texts that you mention, from their very titles are in the context of linear algebra, where tuples (sequences) could well be taken to refer specifically to vectors, and to form a vector space with the associated properties.  — Quondum 00:09, 1 July 2013 (UTC)
 * I'm sorry, but I am still grasping to see what the point of this discussion actually is. The structure of addition and scalar multiplication is not actually needed to define the dot product.  You just need a tuple of numbers (let's say real numbers, for the sake of conversation).  Many of the aforementioned texts do not even mention vector spaces, which would seem to obviate your point that they all implicitly are relying on this more general perspective.  But, in any case, what additional sources would you have us consider that might elucidate your point of view on the subject?  Is there some particular text in the article that you would like to change and, if so, what would you change it to?   Sławomir Biały  (talk) 00:16, 1 July 2013 (UTC)
 * I am happy to have the dot product defined on tuples of pair-wise multiplication-compatible components. However, said dot product then does not in general have all the properties attributed to it in the article; it can only acquire these if the tuples are then additionally constrained to being in a vector space.  What I am really driving at is trying to distinguish whether the dot product for the purposes of this article should be this sum-of-products-over-tuples that you evidently prefer, or else it is defined specifically on a vector space.  If we assume the former, then half the guts of the article does not belong, and the claimed equivalence with the more abstract geometric definition on Euclidean spaces does not hold (because the latter is at best a subclass).  — Quondum 00:34, 1 July 2013 (UTC)
 * Well, I wouldn't say that it's necessarily my preference to define the dot product as a sum of products over tuples. But that's certainly the prevailing definition in most sources I have seen.  I've already given you a sample.  To say that the dot product is "defined specifically on a vector space" is as immaterial as saying that it is "defined specifically on a topological space" or "defined specifically on a measure space".  It's defined on $$\mathbf R^n$$.  This space of tuples happens to support operations of addition and scalar multiplication (among other things: it's also a measure space, a topological space, a topological group, and so on).  It also happens to be the Cartesian model for Euclidean space, and that's how the two definitions are related.  (Incidentally, even the geometric definition does not rely specifically on the vector space structure of Euclidean space, even though Euclidean space does support such a structure in addition to being a topological space, a measure space, and so on.)  Saying that "half the guts of the article does not belong, and the claimed equivalence with the more abstract geometric definition on Euclidean spaces does not hold" seems either to be WP:POINTy, or at least to show a disregard for what many high quality sources have to say on the matter.   Sławomir Biały  (talk) 01:32, 1 July 2013 (UTC)
 * I can imagine addition on Rn that does not correspond to component-wise addition, so unless we specifically define a tuple addition, to me it is not defined. I can imagine it being used to parameterize a map on a manifold, which would make predefined addition and scalar multiplication undesirable. As you suggest, this process is absorbing your and my energy – too much now relative to the potential benefit to WP. I'm not happy that what I consider to be at best non-obvious properties of Rn are not stated in the article (or anywhere else?). You obviously have the superior mathematical background and experience, but this leaves me concerned by what you feel the reader can be expected to assume. The dot product should be a simple topic, but as you can see, I am having serious difficulty pinning down a rigorous understanding of the starting point.  I suggest that I should take a (long) rest from it.  — Quondum 02:22, 1 July 2013 (UTC)

Constraint on basis transformation for invariance of the dot product
§Properties currently reads: Ignoring the dispute about the algebraic definition, the geometric definition is invariant under any change of basis, exactly as it should be. Any definition of the dot product that produces a value that could vary with a change in basis is simply insane. Why is there any restriction placed on the change of basis? — Quondum 22:32, 30 June 2013 (UTC)
 * Invariance under isometric changes of basis

 * I am new to this particular article, but it seems to me that if we divide all the components of a vector by 2 (which corresponds to a certain change of basis) then the dot product of that vector with itself will not usually remain the same, if the dot product is defined (as I assume it is) in terms of the components. Really the issue is that the dot product is only defined in that sense relative to a basis, and then "invariance" is just jargon for a theorem that two bases which are isometric with respect to each other (whatever that means...) will give the same values for all dot products. Actually, I am not certain exactly what an "isometric change of basis" is supposed to be. One possibility: if the new basis gives every vector the same length as the old basis, then by the Parallelogram law the two dot products have to be the same. — Carl (CBM · talk) 23:19, 30 June 2013 (UTC)

 * There used to be a paragraph in the lede that clarified what was going on, but it was removed in this edit: . As that paragraph used to explain, the dot product defined in terms of components in an arbitrary basis is not, in general, the same as the inner product of two vectors in $$\mathbb{R}^n$$, which is the special case of the dot product in the standard basis. — Carl (CBM · talk) 23:53, 30 June 2013 (UTC)
 * That removed paragraph maybe does not clarify much. However what you might be reading from it is that the dot product is defined on the coordinate vector representing an abstract vector in terms of a basis. In this picture, the original vector and the basis are not relevant, and we are dealing with a linear transformation of coordinate vectors.  The section would have to be updated to Invariance under isometric linear transformations, though this is a tautology: an isometry is defined as a transformation that preserves the dot product. — Quondum 00:24, 1 July 2013 (UTC)
 * Because any coordinate-free definition of the dot product is (as you said originally) preserved under every isomorphism, for that definition of an isometry to make sense, the dot product has to be defined relative to a basis in a way that makes essential use of the specific coordinates. The geometric definition is such a definition: it is just a special case of the coordinate-based dot product relative to a particular basis (the standard basis). — Carl (CBM · talk) 00:30, 1 July 2013 (UTC)
 * Okay, so we define the dot product as a sum of products of corresponding (real) coordinates of two tuples. Then you identify a map onto Euclidean space so that the original tuple happens to be equal to the coordinate vector of each Euclidean vector relative to some basis, as well as this basis being orthonormal.  Fine, you can do this.  The dot product then matches the Euclidean length (we'll ignore such inconvenient things such as units for now).  The problem comes in with the implicit pull-back (well, I think you call it that) of properties such as scalar multiplication and vector addition: you added these properties when you did the mapping.  You do not have these properties without the mapping, or defining them on the original space of tuples.  You also do not have the concept of a basis without these properties.  — Quondum 01:02, 1 July 2013 (UTC)
 * If I have $$\mathbb{R}^n$$ as an abstract vector space (i.e. I have a vector space over $$\mathbb{R}$$ of dimension n but I know nothing else about the space) then I already have scalar multiplication and vector addition, because I have a vector space. But I have no dot product or inner product, because I just have an abstract vector space. One way to define an inner product on my space is to choose any basis for the space (the definition of a basis is coordinate-free, and a basis exists for each vector space), so that I can write a coordinate tuple for each vector relative to the basis. Then I can define an inner product on the original space to be the dot product on the coordinate tuples of the two vectors. Now I can ask: for which pairs of bases B, B′ do I obtain the same values for all these dot products from B and from B′? One answer is that if B and B′ give every vector the same length (i.e the dot product of every vector with itself relative to B is the same as the dot product with itself relative to B′) then all the dot products will be equal under B and B′. That is what the original sentence says to me. The key point is that the dot product of two vectors is only defined relative to a choice of basis, when all I know about the vectors is that they belong to an abstract copy of $$\mathbb{R}^n$$. — Carl (CBM · talk) 01:31, 1 July 2013 (UTC)

The main issue here is that what I call elementary books (e.g. for Calculus 3, or elementary linear algebra) define $$\mathbb{R}^n$$ to be a particular space, the space of n-tuples of real numbers with pointwise operations. But from a categorical viewpoint $$\mathbb{R}^n$$ is any space that is a product $$\mathbb{R}\times\cdots\times\mathbb{R}$$ where there are n factors, and this product is not unique - like any categorical product it is only unique up to isomorphism. From the viewpoint of the elementary texts $$\mathbb{R}^n$$ already comes with a standard basis using the components of the tuples; from the categorical viewpoint the choice of basis is arbitrary, because $$\mathbb{R}^n$$ is only defined up to isomorphism. The original versions of this article were written from the categorical perspective, i.e., where the dot product can be interpreted geometrically when an appropriate basis is used. At some point someone edited the article to use more of the elementary viewpoint, where the dot product is the geometrical one - but then parts of the previous article that talk about the dot product in other bases no longer fit. — Carl (CBM · talk) 01:43, 1 July 2013 (UTC)

 * I don't think this distinction between "elementary" and "categorical" is right. As a categorical product $$\mathbf R^n$$ is equipped with n projection maps into $$\mathbf R$$.   This is all we need to define the dot product.  Sławomir Biały  (talk) 02:00, 1 July 2013 (UTC)

 * Each individual copy of $$\mathbb{R}^n$$ has projections, but there is no unique space that "is" $$\mathbb{R}^n$$ in the category of vector spaces over $$\mathbb{R}$$. So there is an element of arbitrariness in choosing one particular space to call "$$\mathbb{R}^n$$", which is the same as the arbitrariness of choosing a basis. But in any case my point is that the original article was written from a viewpoint where a vector space does not come equipped with a basis - including $$\mathbb{R}^n$$ - while the present version assumes that $$\mathbb{R}^n$$ is a specific space of tuples. Either approach is fine with me, but it helps if everyone realizes which approach is being taken, because it will affect the way that the material is presented. — Carl (CBM · talk) 02:15, 1 July 2013 (UTC)

 * I still think this is a false dichotomy. Surely defining a dot product requires that we work in $$\mathbf R^n$$, not in a vector space that is abstractly isomorphic to $$\mathbf R^n$$.  The latter would more properly be called an inner product (perhaps an inner product induced by the isomorphism), but not the dot product.  See, for instance, the text "Linear algebra" by Greub.  He uses the term "standard inner product" to refer to the dot product on $$\mathbf R^n$$ and then discusses the induced inner product in a basis as a completely different case referring to it as an "inner product".   Sławomir Biały  (talk) 12:04, 1 July 2013 (UTC)
 * We both agree that the "dot" product requires working with tuples of real numbers. The distinction is whether $$\mathbb{R}^n$$ is defined to be the set of such tuples, as is common in elementary settings, or is instead a more abstract structure. In the former case, because vectors are defined as tuples, we can define the "dot product of two vectors" by referring directly to the tuples, or by identifying the vectors with geometric objects and referring to their length. In the latter case, when a "vector" does not come pre-equipped with components, it is impossible to define the "dot product of two vectors", only "the dot product of two vectors relative to a basis". The latter is exactly the induced inner product on a basis. In this more abstract setting, for example, the "angle" between two vectors is only defined after we pick (arbitrarily) an inner product, the angle is not an inherent property of the vectors themselves. (The same distinction would come if we treated $$\mathbb{R}^n$$ as an affine space: if $$\mathbb{R}^n$$ is the set of tuples then we know which vector is "really" zero, but if $$\mathbb{R}^n$$ is presented as an abstract affine space we have no way to tell.) — Carl (CBM · talk) 13:02, 1 July 2013 (UTC)
 * Then I think the point is not that there is a problem with the meaning of $$\mathbf R^n$$, but rather what "Euclidean space" signifies. Then I would agree that there is an elementary notion of Euclidean space (one that satisfies the axioms); an elementary model of Euclidean space: the "Cartesian model" $$\mathbf R^n$$ equipped with the usual metric; and a less elementary model of Euclidean space: an inner product space.   Sławomir Biały  (talk) 19:44, 1 July 2013 (UTC)
 * Back on topic: I have removed the property under discussion. It seems to me that this is already mentioned in a more appropriate context at the end of the section on the equivalence of the two definitions.   Sławomir Biały  (talk) 12:25, 1 July 2013 (UTC)

Section on "Scalar projection and the equivalence of the definitions" is jumbled and missing steps
1. It is not clear how "Scalar projection" and "equivalence of the definitions" are related. 2. It is not clear where discussion of scalar projection ends and proof of equivalence begins. 3. It is not clear how :$$\mathbf A\cdot\mathbf B = \sum_i B_i(\mathbf A\cdot\mathbf e_i)$$ follows. Furthermore, the section does not start with the trig definition and end with algebraic definition, or visa versa. So where is the equivalence? I'm addressing this to whomever reverted my changes from 2013-07-27, which were an improvement. You know who you are. Terse mathematics is fine for research papers and dissertations where the reader is expected to be an expert and math is a common language. But "Dot product" is not advanced mathematics. This subject can and should be readily accessible by first year college students who have never had course in abstract algebra and probably never will. So please step down off your high horse and join us peasants laboring in the fields. You may not be a smart as you think. — Preceding unsigned comment added by ThomasMcLeod (talk • contribs) 02:01, 2 August 2013 (UTC)

Operation, or result of the operation, or both?
The opening of the lead says the dot product is an operation that returns a number, and the lead sticks with this definition throughout. Then the lengthy section "Definition" consistently refers to the dot product as the scalar result of the operation. Then the untitled beginning of the section "Properties" treats the term as referring to an operation. Then the next five sections or subsections treat it as a result of the operation. Then two sections treat as an operation, then two sections as a result.

If it is standard to call the operation as well as its result the dot product, then this should be made clear in the lead. If there is a different term that is conventionally used for the operation, then that should be substituted where appropriate. Loraof (talk) 21:29, 7 October 2014 (UTC)

 * Can you point out where exactly you are seeing this, i.e. quote an offending sentence. I've just read the definition section and can see no problems as you describe.-- JohnBlackburne wordsdeeds 23:29, 7 October 2014 (UTC)

 * You ask for an offending sentence. The offense is the absence of a mention that the term refers to two related but different things: an operation, and a scalar that results from the operation.

 * sent. 1: the dot product ... is an algebraic operation

 * section "Definition": the length of a vector is defined as the square root of the dot product of the vector by itself -- the square root of the dot product is the square root of a number, not of an operation.

 * next subsection: the dot product of vectors [1, 3, −5] and [4, −2, −1] is ... 3. -- 3 is not an operation, even though sentence 1 of the article says that the dot product is an operation.

 * For further examples, see the exact statement of their locations throughout the article in my original post above. Loraof (talk) 01:50, 8 October 2014 (UTC)

 * The dot product is an operation; the dot product of two vectors is a number, as being the operation applied to two vectors. What is wrong with this formulation? D.Lazard (talk) 08:41, 8 October 2014 (UTC)

 * As at Cross product, this presumably relates to language, where we use a form of abbreviation, e.g. "the dot product of two vectors a and b" to mean "the result of applying the dot product to two vectors a and b". Thus, the "dot product" does not mean the result, even though the language use would suggest this on first reading. Where it might be confusing, we can be a little more explicit and not use this form of abbreviation/elision. —Quondum 13:55, 8 October 2014 (UTC)
 * I don't see any need to be any more explicit, it just makes it less clear. The language is far more elementary than the maths here: one might say "the sum of three and five is eight" and it's perfectly clear. Changing it to e.g. "the result of applying the sum to two numbers three and five is eight" is more clumsy and so far less clear. And this is the same example but with "sum" for "dot product" and "three and five" for "two vectors a and b".
 * If anything the dot product is clearer and easier to interpret. To take one example from above: "the length of a vector is defined as the square root of the dot product of the vector by itself", this can only mean one thing as the dot product is a binary operation on vectors and so requires two of them. There's only one vector so the dot product is with itself, A ⋅ A as is given in the article (so anyone who finds the language unclear can look at the algebra).-- JohnBlackburne wordsdeeds 14:39, 8 October 2014 (UTC)

This article appears to contradict another article
See the discussion here. Jarble (talk) 02:57, 6 January 2015 (UTC)

Geometric definition
As far as I can see, the geometric definition given is circular. It is, at best, a definition of the angle $θ$ in terms of the dot product imo. Taking the POV that Euclidean space is endowed with a norm (but not an inner product) in terms of which the dot product (here a full-fledged inner product) is defined seems rather bizarre, especially since what the angle between two vectors in $ℝ^{77}$ is isn't automatically clear to me (before it is defined in terms of the inner product). YohanN7 (talk) 14:16, 24 November 2015 (UTC)

 * I'm not clear what the objection here is. If a Euclidean space is equipped with a norm, then the dot product is given by the law of cosines identity $$A\cdot B = \frac{1}{2}(\|A+B\|^2-\|A\|^2-\|B\|^2) = \|A\|\, \|B\|\cos\theta$$.  There's nothing circular about that.  The definition $$A\cdot B = \|A\|\,\|B\|\,\cos\theta$$ goes back at least to Gibbs and Wilson, and remains standard in engineering and applied mathematics (see, for example, Kreyszig "Advanced engineering mathematics").   S ławomir  Biały  15:49, 24 November 2015 (UTC)

 * I'll try to explain exactly what I mean, therefore the essay below. It is not a lecture, it is a question.

 * Of course, if norm and angle are defined, it is not circular. But this is not always the approach. Euclidean space (linked from the article in connection with the definition) begins with the dot product, defines the norm through it, and finally defines the angle using both the norm and the dot product. Doing it the other way around requires well-motivated definitions of norm and angle.

 * I think can visualize a way of defining length using only two-dimensional concepts, such as the Pythagorean theorem in the plane and a suitable decomposition of the vector into terms having only one non-zero component, then successively applying the Pythagorean theorem to appropriate two-dimensional subspaces, finally arriving at the standard norm. But this presupposes that coordinate axes are (pairwise) at right angles, angles yet so far undefined. Then for the angles. Two linearly independent vector span a two-dimensional plane. In this plane, the angle is well defined from plane geometry (law of cosines). In order to make all this explicit, I'd appreciate the presence of the orthogonal group, at least $O(2)$. I think I'd need it, at least its existence to motivate the choice of appropriate bases.

 * Another approach would be to skip and "derivation" or "motivation" in terms of two-dimensional concepts, and just straight away define the norm by generalizing the formula of the Pythagorean theorem to $n$ dimensions (implicit here is that coordinate axes are at right angles), and, similarly, use the law of cosines as is to define the angle.

 * Out of curiosity, how is this done in the literature? YohanN7 (talk) 12:31, 25 November 2015 (UTC)

 * Concepts of norm and angle don't require a dot product or special Cartesian coordinate systems to define them. The concept of magnitude (as a length) in geometry predates Descartes by at least two thousand years.  Rigorous notions of length of a line segment, and the algebra of lengths, go back at least to Plato.  In applications, it's often assumed that one already knows what magnitude means, because we all were supposed to have studied geometry as a child.  This is the approach taken by sources like Gibbs and Wilson, Arfken and Weber, Kreyszig, and most "applied mathematics" sources: "A vector has a magnitude and direction."  This definition wouldn't  be much good if we didn't already know what "magnitude" means.  Anyway, there are articles that do discuss inner products, and using those to assign magnitudes to vectors.  Let's remember that the target audience of this article is someone who doesn't necessarily know what a dot product is, so the subject should be the dot product, not the foundations of geometry.   S ławomir  Biały  13:13, 25 November 2015 (UTC)

The algebraic definition seems flawed as well. Why talk about vectors? This brings in exactly the basis-dependence Q is talking about above. Only $n$-tuples should be mentioned here, taking for granted only the structure each factor ($ℝ$) comes with in the Cartesian product $ℝ^{n}$. Later, scalar multiplication and addition of $n$-tuples could be defined in order to make sense of the properties in the properties section for $n$-tuples as well. (With the definitions, this is the prototype of all finite-dimensional vector spaces.) YohanN7 (talk) 15:02, 24 November 2015 (UTC)

 * The word "vector" does not refer to a unique thing. It's clear in the context here that we mean coordinate vectors.  It might be less ambiguous to refer to n-tuples, but most sources just say "vectors", and I think here accord with sources is more important than imagined sources of ambiguity.  Many readers no doubt will encounter this in a text on basic physics or calculus, where "vector" refers to an element of $$\mathbb R^3$$.  More refined concept of "vector" are definitely possible, but presumably readers familiar with things like covariance and contravariance of vectors and vector spaces are also capable of inferring the intended meaning of the word "vector" in the algebraic definition.  In any case, there already is a paragraph introducing the definitions that discusses the use of coordinates to model Euclidean space.  It currently lacks a proper discussion of vectors, though.   S ławomir  Biały  15:51, 24 November 2015 (UTC)

 * The geometric definition makes sense if you have a clear understanding of what an angle is, which you do in elementary geometry, in two or three dimensions. you can e.g. form a plane triangle with two of the sides being the vectors then the angle is the well-defined angle between them. I think is often taught that way, which helps tie it into the cross product which has a similar definition in 3D space. It can them be turned around to define the angle in other spaces. In higher Euclidian dimensions, although you can do so geometrically even in $ℝ^{77}$ as you can construct a triangle on a plane and then work in two dimensions, and in other ways.

 * As for vector it is a vector product in that it is independent of the basis chosen. This is not obvious from the algebraic definition but must be true for the geometric definition to hold, and they are shown to be equivalent later. It is also a product on n-tuples but it is only interesting as it is a product on vectors and so basis independent.-- JohnBlackburne wordsdeeds 15:53, 24 November 2015 (UTC)

 * The following post has been written before reading the two replies (edit conflict) Moreover the definition of a vector that is given here (and also in Vector (mathematics and physics)) is not really a definition, but only an explanation. The correct definition of a vector appears in Equipollence (geometry). Then, a geometrical definition of the dot product may be deduced from the notion of Euclidean distance as follows: define orthogonality of vectors  from Pythagorean theorem; then either use it for defining Cartesian coordinates and dot product, or define orthogonal projection; in the latter case the dot product is the product of the length of one vector by the length of the projection of the second vector on the first one. This is rather complicate and probably too technical for a definition, even this must appear as a property. Thus, I would suggest something like In geometry, the dot product of two vectors is the dot product of their coordinate vectors over an orthonormal frame, which is an affine frame in which all vectors are pairwise orthogonal and of unit length. Then one could continue as in the considered section, by saying that two vectors are orthogonal if and only if their dot product is zero, and that the dot product of a vector by itself is the square of its length.  (Note that orthonormal frame needs to be completely rewritten).D.Lazard (talk) 16:03, 24 November 2015 (UTC)
 * The proposed text is very non-elementary. I would not add it to the "Definition" section of the article.  Anyway, I don't think it's entirely correct either.  The (algebraic) dot product is defined for vectors in $$\mathbb R^n$$.  There one doesn't have a concept of orthonormal frames, without already knowing what the dot product is (see previous discussions).  Certainly, the relationship between the two definitions involves a choice of Cartesian coordinates, which is already mentioned.  But we don't need to make such a huge breakfast out of this small point.   S ławomir  Biały  16:22, 24 November 2015 (UTC)

Thank you JohnBlackburne, D.Lazard and S ławomir Biały. I wrote my first reply (above) to Sławomir before reading the subsequent posts. The subsequent posts clarify a bit, and are mostly (but not all together) in accord with what I am thinking. YohanN7 (talk) 12:55, 25 November 2015 (UTC)

Discussion of non-Cartesian Basis
I wrote the new section on non-Cartesian basis, which D.Lizard removed. I've moved the section to be under Dot Product for now. I disagree with the comment saying it should be moved to be under Inner Product - a much broader category. The whole idea was to show that if you want to keep using the geometric definition of the dot product, if you change the basis, the algebraic definition needs to be altered. The introduction of the metric tensor and associated notation from differential geometry only make sense in this context (where they wouldn't be relevant in talking about the general inner product). StoicLaughter (talk) 21:57, 14 January 2018 (UTC)

 * But this remains unsourced and smacks of WP:OR, so I have removed it again. If you want to include this material you will need to provide reliable secondary sources that back it up. --Bill Cherowitzo (talk) 00:20, 15 January 2018 (UTC)

 * Which part do you consider to be 'original research'? The definition of a basis means that it's possible to write any vectors with respect to them, and the algebraic manipulations follow the exact steps in the main section. The metric tensor is a well known object. I don't have a textbook handy, but | Wolram Alpha provides an equivalent definition at the bottom of the page, where the lower indices imply the metric tensor implicitly. I can understand putting a flag requesting further citation, don't see why you think the section should be completely deleted.-- StoicLaughter (talk) 01:17, 15 January 2018 (UTC)

 * Wikipedia's definition of "original research" is much broader than the term commonly denotes and this can sometimes be confusing. It appears to me that you have taken several ideas from various places and combined them to synthesize a logical consequent. This is something mathematicians do all the time, it is the way we do business. However, Wikipedia views this as WP:SYNTH, a form of original research, unless this has already been done and vetted by the mathematical community. Editors are only allowed to report on what is already available in the existing literature. I do flag passages for further citation if I feel that such citations actually exist, otherwise I revert. Of course, I could make a wrong call and that can be fixed if a reliable secondary source can be cited, but this has only rarely happened.--Bill Cherowitzo (talk) 05:36, 15 January 2018 (UTC)
 * Here's another reference from my reply in the "regarding bases" thread. A definition can be found in: "Аналитическая геометрия, Том 1", Б. Н. ДЕЛОНЕ и Д. А. РАЙКОВ, 1948 (Boris Delaunay, see Books). The definition can be found in 1.2.9, specifically the definition is introduced through $$\vec{a}\vec{b} = \|a\|\|b\|\cos\theta$$. Later on the algebraic definition is derived from linearity, and the explicit general form using the metric tensor can be found in: 1.2.11.2.

 * The explanation comes from this course by Prof. Pavel Grinfeld at Drexel - the relevant discussion starts here, and continues here with the introduction of the index juggling convention. I changed the notation to be consistent with that presented in the main section, as well as the more standard notation g for the metric tensor. I don't have a copy handy, but Grinfeld's textbook is here with reference .  It seems to me that the discussion connecting the geometric and algebraic dot product is necessarily incomplete without mentioning the metric tensor - though I would not have realized that without Grinfeld! I strongly believe that that idea should at least somewhere be mentioned, whether it's through my submission or another approach. StoicLaughter (talk) 10:20, 15 January 2018 (UTC)

 * YouTube videos are not considered as reliable sources. So, Grinfeld's book is the only reliable source.
 * However, your edit remains original research, as Grinfeld certainly did not change (did not "alter") the definition of dot product. This does not mean that it would not be useful to have a section about the expression of the dot product on non-orthonormal bases, but your edit is too confusing for being accepted. Moreover, such a section seems too technical for this article, which is very elementary and supposes that a given basis has been chosen once for all (this is explicit in the algebraic definition, and implicit in the geometric definition). IMO, it would be better placed in Inner product space (the dot product is a particular inner product), but this is a personal opinion. D.Lazard (talk) 17:42, 3 February 2019 (UTC)

 * "Grinfeld certainly did not change (did not "alter") the definition of dot product." - This could just be reworded to "the method used to calculate the dot product must be changed. "your edit is too confusing for being accepted...such a section seems too technical for this article" - If you look at the other sections under "Generalizations", they all go far beyond an elementary (by which I mean high school) discussion of the topic. The presentation could be improved, but it's tricky to use consistent notation and properties from the above sections, get the point across with an example. and not get too non-elementary (i.e. a deeper discussion of contra- and covariant vectors). I still find it puzzling that the current solution is not to just add tags requesting further citations and/or cleanup, but to delete information that people could find useful. Since this page is supposed to cover the geometric and algebraic dot product on roughly equal footing, and the former is intimately tied up with the metric tensor, I strongly feel there should be more information on this page to guide curious readers. StoicLaughter (talk) 19:17, 4 February 2019 (UTC)

Text in the figure
I had a question about the formula shown with the illustration. It shows theta = arccos(x DOT y / |x| |y|). Should it be theta = arccos(x DOT y / (|x| |y|)) ? — Preceding unsigned comment added by 2601:280:5B7F:F890:2948:8B9C:53B:6F2 (talk) 15:14, 23 September 2019 (UTC)
 * You are right, but it is difficult to edit figures. Moreover, as formatted, I doubt that anybody would interpreted wrongly the formula. Also, in the case of a doubt, it is easy to look at the test of the aricle. D.Lazard (talk) 16:05, 23 September 2019 (UTC)

Curvilinear case
I agree with 's removal of 's edits. No one calls applying the metric tensor to two vectors a "dot product" in all but the Euclidean case. It also must be noted that this does not have the usual properties of a dot product when the manifold isn't Riemannian. Note too that there is not just one choice of metric on a finite-dimensional real vector space, since scaling the Euclidean metric by any positive constant factor yields another positive-definite quadratic form. We need a reliable source that uses the term "dot product" in this manner. It is also confusing to implicitly use raising and lowering indices without explaining what it is, which would leave our readers scratching their heads unless they already know differential geometry. Finally, the text as-written contained numerous grammar errors (often missing definite articles). All of these must be addressed before this can be reinserted.--Jasper Deng (talk) 07:58, 19 February 2020 (UTC)

Equivalence of definitions
I don't see a valid proof. The equivalence depends on homogeneity and distributivity of the geometric definition, which are not proved or justified. Zaslav (talk) 07:41, 13 April 2023 (UTC)

color usage
This page is great and informative. Though, I've seldom seen such stark and oversaturated colors in the definition of a mathematical concept. The edit in the history justifies the colors to "improve readability of the algebraic definition", but I don't think that the straightforward coordinate definition of a dot product necessitates this. Moreover, I feel the minor inconsistencies and arbitrary choices of what is colored adds confusion to the definition. I don't mind how it's done in the Law of Cosines section as a corresponding colored diagram is present, but I think I'm going to strip the colors in the coordinate definition unless anyone has strong feelings or can convince me otherwise. ^_^ EuclideanSwag (talk) 23:22, 6 May 2023 (UTC)

 * if anyone cares I did this. There's a few sentences in the article I'd also like to clean up later. :3 EuclideanSwag (talk) 22:40, 29 May 2023 (UTC)

θ =cos^ -1 ( vec a . vec b /| vec a || vec b | )
It will come handy if someone is trying to understand the dot product. Yuthfghds (talk) 05:18, 9 September 2023 (UTC)

title: Talk:Dot product


text:
Jakob Hutter

Jakob Hutter (also spelled Jacob Hutter, Huter or Hueter; c. 1500 – 25 February 1536) was a Tyrolean Anabaptist leader and founder of the Hutterites.

Biography
Hutter was born in the small hamlet of Moos near St. Lorenzen in the Puster Valley, in the County of Tyrol (present-day South Tyrol, Italy). He learned hat making in nearby Prags and became an itinerant craftsman. Later he settled in Spittal, Carinthia.

He probably first encountered Anabaptists in Klagenfurt and soon thereafter was converted to their belief. He began preaching in the Puster Valley region, forming several small congregations. As soon as the Habsburg authorities in Tyrol learned of these activities in early 1529, they began to persecute the Anabaptists. In 1527, the Habsburg archduke Ferdinand I of Austria had declared that seductive doctrines and heretical sects "will not be tolerated". In turn, Hutter and a few others went to investigate Moravia, because they heard the persecution was not as severe there. They visited Auspitz, where they found the situation was indeed more tolerant and the Tyrolean Anabaptists decided to emigrate. As small groups moved to Moravia, Hutter first remained in Tyrol to pastor to those who remained. He escaped capture by the authorities because other captured Anabaptists would not reveal his whereabouts, even under severe torture.

Hutter arrived in Moravia in 1533, when the persecution of the Anabaptists in Tyrol was at its peak. Many Anabaptists from the Palatinate, Swabia and Silesia also went to Moravia. Hutter united the local Anabaptist congregations, enabling Anabaptism in Moravia to flourish. Under Hutter's leadership, several of the congregations adopted the early Christian practice of communal ownership of goods, in addition to their Anabaptist beliefs of nonviolence, and adult baptism.

In 1535, however, the Moravian Landtag diet had all Anabaptists expelled from Moravia and they scattered to surrounding countries. Hutter returned to Tyrol, where he and his wife were arrested on 30 November 1535 in Klausen and brought to the fortress of Branzoll (Bronzolo). On 9 December, Hutter was deported to the Tyrolean capital Innsbruck, where he was interrogated and pressured to recant. Even under severe torture he would not recant or reveal the names of other Anabaptists. Hutter was sentenced to death by fire and burned at the stake on 25 February 1536 in Innsbruck in front of the Golden Roof. According to the Hutterian Chronicle, a total of 360 Anabaptists were executed in Tyrol.

Hutter's words are recorded in eight letters, written under severe persecution.

Commemoration

In Innsbruck, Hutter is remembered by a plaque at the Golden Roof.

In 2006–2007, a working group with representatives from Protestant and Catholic churches, the peace movement Pax Christi and the Association of Evangelical churches in Tyrol worked toward reconciliation with the Hutterites. On 25 February 2007, the group, along with three couples invited to represent the Hutterites, held a memorial ceremony at the Golden Roof and a joint prayer service in the old city hall in Innsbruck.

title: Jakob Hutter