text:
Jack Williamson

John Stewart Williamson (April 29, 1908 – November 10, 2006), who wrote as Jack Williamson, was an American science fiction writer, one of several called the "Dean of Science Fiction". He is also credited with one of the first uses of the term genetic engineering. Early in his career he sometimes used the pseudonyms Will Stewart and Nils O. Sonderlund.

Early life
Williamson was born April 29, 1908, in Bisbee, Arizona Territory. According to his own account, the first three years of his life were spent on a ranch at the top of the Sierra Madre Mountains on the headwaters of the Yaqui River in Sonora, Mexico. He spent much of the rest of his early childhood in western Texas. In search of better pastures, his family migrated to rural New Mexico in a horse-drawn covered wagon in 1915. The farming was difficult there and the family turned to ranching, which they continue to this day near Pep. He served in the U.S. Army Air Corps in World War II as a weather forecaster.

Writing career

As a child Williamson enjoyed storytelling to his brother and two sisters. As a young man, he discovered the magazine Amazing Stories, established in 1926 by Hugo Gernsback, after answering an ad for one free issue. He strove to write his own fiction and sold his first story to Gernsback at age 20: "The Metal Man" was published in the December 1928 issue of Amazing. During the next year Gernsback published three more of his stories in the new pulp magazines Science Wonder Stories and Air Wonder Stories, and separately published "The Girl from Mars" by Miles J. Breuer and Williamson as Science Fiction Series #1. His work during this early period was heavily influenced by A. Merritt, author of The Metal Monster (1920) and other fantasy serials. Noting the Merritt influence, Algis Budrys described "The Metal Man" as "a story full of memorable images".

Early on, Williamson became impressed by the works of Miles J. Breuer and struck up a correspondence with him. A doctor who wrote science fiction in his spare time, Breuer had a strong talent and turned Williamson away from dreamlike fantasies towards more rigorous plotting and stronger narrative. Under Breuer's tutelage, Williamson would send outlines and drafts for review. Their first work together was the novel Birth of a New Republic in which Moon colonies were undergoing something like the American Revolution, a theme later taken up by many other SF writers, particularly in Robert A. Heinlein's The Moon Is a Harsh Mistress.

Wracked by emotional storms and believing many of his physical ailments to be psychosomatic, Williamson underwent psychiatric evaluation in 1933 at the Menninger Clinic in Topeka, Kansas, in which he began to learn to resolve the conflict between his reason and his emotion. From this period, his stories take on a grittier, more realistic tone.

By the 1930s, he was an established genre author, and the teenaged Isaac Asimov was thrilled to receive a postcard from Williamson, whom he had idolized, which congratulated him on his first published story and offered "welcome to the ranks". Williamson remained a regular contributor to the pulp magazines but did not achieve financial success as a writer until many years later.

An unfavorable review of one of his books, which compared his writing to that of a comic strip, brought Williamson to the attention of The New York Sunday News, which needed a science fiction writer for a new comic strip. Williamson wrote the strip Beyond Mars (1952–55), loosely based on his novel Seetee Ship, until the paper dropped all comics.

Beginning 1954 and continuing into the 1990s, Williamson and Frederik Pohl wrote more than a dozen science fiction novels together, including the series Jim Eden, Starchild, and Cuckoo. Williamson continued to write as a nonagenarian and won both the Hugo and Nebula Awards during the last decade of his life, by far the oldest writer to win those awards.

In his later years, he would also criticize attempts to write "serious" science fiction:

"Maybe because of my own background of writing commercial SF for so many years, I have a great deal of respect for good craftsmanship of the sort that commercial writers must develop. The labels you hear so much of—'commercial,' 'serious writer,' 'mainstream,' 'hack,' 'New Wave,' 'experimental'—are usually very misleading. In my own field, Ed Hamilton and Hank Kuttner and more recently Bob Silverberg are all writers who formed a fine command of the SF genre early in their careers and who later on used this to do work that is more consciously 'literary' and hence more admired by critics. But certainly the writing they did earlier was deservedly popular among SF fandom, who evidently found these works 'serious' enough to merit reading.

I am opposed, however, to literary tricks that tend towards obscurity or artificial difficulty, though I can see arguments for that kind of approach. My own experience as a teacher of writing confirms my sense that new authors with artistic ambitions may find themselves scorning too many of the old forms and patterns simply because they blindly associate them with hack work. The point is that these patterns and structures form the basic vocabulary through which all SF writers must speak. That's one reason I'm not completely sympathetic with contemporary writers like Silverberg and Chip Delany and Tom Disch, who are clearly aiming to get themselves recognized as 'serious' or mainstream authors."

Academic career
Williamson received his Bachelor of Arts and Master of Arts degrees in English in the 1950s from Eastern New Mexico University (ENMU) in Portales (near the Texas panhandle), joining the faculty of that university in 1960. He remained affiliated with the school for the rest of his life. In the late 1990s, he established a permanent trust to fund the publication of El Portal, ENMU's journal of literature and art. In the 1980s, he made a sizable donation of books and original manuscripts to ENMU's library, which resulted in the formation of a Special Collections department; the library now is home to the Jack Williamson Science Fiction Library, which ENMU's website describes as "one of the top science fiction collections in the world". In addition, Williamson hosted the Jack Williamson Lectureship Series, an annual lectureship where Guests of Honor and other noted authors give lectures, read from their works, and participate in lively panel discussions on a variety of topics. The lectureship is still celebrated at ENMU each year. The Jack Williamson Liberal Arts building houses the Languages & Literature, Mathematical Sciences, History, Religion & Social Sciences, and Psychology & Political Science Departments of the university, as well as the College of Liberal Arts and Sciences Dean's Office.

Williamson completed his PhD in English literature at the University of Colorado in Boulder, focused on H.G. Wells' earlier works, demonstrating that Wells was not the naive optimist that many believed him to be.

Terms coined
Williamson coined the word terraforming in a science-fiction story published in 1942 in Astounding Science Fiction.

He is also credited with one of the first uses of the term genetic engineering.

The word "psionics" first appeared in print in a novella by Williamson—The Greatest Invention —published in Astounding Science Fiction magazine in 1951.

Later years
The Science Fiction Writers of America named Williamson its second Grand Master of Science Fiction after Robert Heinlein, presented 1976.

After having been let go from ENMU during the university's financial crisis in 1977, Williamson spent some time concentrating on his writing, but after being named professor emeritus by ENMU, he was coaxed back to co-teach two evening classes, "Creative Writing" and "Fantasy and Science Fiction" (he pioneered the latter at ENMU during his full-time professorship days). Williamson continued to co-teach these two classes into the 21st century. After he made a large donation of original manuscripts and rare books from his personal collection to the ENMU library, a special collections area was created to house these and it was named the "Jack Williamson Special Collection".

In 1994 Williamson received a World Fantasy Award for Lifetime Achievement.

The Science Fiction and Fantasy Hall of Fame inducted Williamson in 1996, its inaugural class of two deceased and two living persons.

The Horror Writers Association conferred its Bram Stoker Award for Lifetime Achievement in 1998 and the World Horror Convention elected him Grand Master in 2004.

In November 2006, Williamson died at his home in Portales, New Mexico, at age 98. Despite his age, he had made an appearance at the Spring 2006 Jack Williamson Lectureship and published a 320-page novel, The Stonehenge Gate, in 2005.

Minor planets 5516 Jawilliamson and 235281 Jackwilliamson are named in his honor.

Legion of Space series
While attending a Great Books course, Williamson learned that Henryk Sienkiewicz had created one of his works by taking The Three Musketeers of Alexandre Dumas and pairing them with John Falstaff of William Shakespeare. Williamson took this idea into science fiction with The Legion of Space.

Desperate for money, he searched for a quick source of income. While most pulps of the time were slow to pay, the recently restarted Astounding was an exception. However, they did not accept novels, so Williamson submitted three short stories and a novelette. Learning that they were also accepting novels for serialization, he sent in The Legion of Space, which was published as an anthology in six parts. It quickly became a genre favorite, and was quickly collected into a hardcover.

The story takes place in an era when humans have colonized the Solar System but dare not go farther, as the first extra-solar expedition to Barnard's Star failed and the survivors came back as babbling, grotesque, diseased madmen. They spoke of a gigantic planet, populated by ferocious animals and the single city left of the evil "Medusae". The Medusae bear a vague resemblance to jellyfish, but are actually elephant-sized, four-eyed, flying beings with hundreds of tentacles. The Medusae cannot speak and communicate with one another via a microwave code.

The Falstaff character is named Giles Habibula. He was once a criminal, and can open any lock ever made. In his youth he was called Giles The Ghost. Jay Kalam (Commander of The Legion) and Hal Samdu are the names of the other two warriors. In this story, these warriors of the 30th Century battle the Medusae, the alien race from the lone planet of Barnard's Star. The Legion itself is the military and police force of the Solar System after the overthrow of an empire called the Purple Hall that once ruled all humans.

In this novel, renegade Purple pretenders ally themselves with the Medusae as a means to regain their empire. But the Medusae, who are totally unlike humans in all ways, turn on the Purples, seeking to destroy all humans and move to the Solar System, as their own world, far older than Earth, is finally spiraling back into Barnard's Star. One of the Purples, John Ulnar, supports the Legion from the start, and he is the fourth great warrior. His enemy is the Purple pretender Eric Ulnar, who sought the Medusae out in the first place, seeking to become the next Emperor of The Sun.

The Medusae conquered the Moon, set up their bases there, and went on to attempt conquest of the Solar System. The Medusae had for eons used a reddish, artificial greenhouse gas to keep their dying world from freezing. The Medusae learned from the first human expedition to their world that the gas rots human flesh, and the Medusae use it as a potent chemical weapon, attempting ecological destruction by means of projectiles fired from the Moon. Their vast spaceships also have very effective plasma weapons, very similar to those the Romulans had in a Star Trek episode called Balance of Terror.

The Legion works also featured a force field called AKKA which can erase from the Universe any matter, of any size, anywhere, even a star or a planet. AKKA was a weapon of mass destruction and the secret of it was entrusted to a series of women. AKKA was used in the past to overthrow the Purple tyranny. It was also used to wipe out most of the Medusae, though they had tried to steal the secret. When they were wiped out, the Moon where they had established their base was erased out of existence. At the end of the story, John Ulnar falls in love with the keeper of AKKA, Aladoree Anthar, and marries her. Aladoree Anthar is described as a young woman with lustrous brown hair and gray eyes, beautiful as a goddess.

Williamson next wrote The Cometeers which takes place twenty years after The Legion of Space in which the same characters battle another alien race, this one of different origin.

In this second tale, they fight The Cometeers who are an alien race of energy beings controlling a "comet" which is really a giant force field containing a swarm of planets populated by their slaves. The slave races are of flesh and blood, but none are remotely similar to humans. The Cometeers cannot be destroyed by AKKA, as they are incorporeal from the Universe's point of view and exist for the most part in an alternate reality. The ruling Cometeers feed on their slaves and literally absorb their souls, leaving disgusting, dying hulks in their wake. It is said that they do so, as they were once fleshly entities themselves of various species. Hence, the ruling Cometeers keep other intelligent beings as slaves and "cattle". They fear AKKA, though, as it can erase all their possessions.

They are defeated by the skills of Giles Habibula. Giles broke into a secret chamber guarded by complex locks and force fields that the incorporeal Cometeers could not penetrate. In it the ruler of the Cometeers had kept its own weapon of mass destruction, one that would cause the Cometeers to disintegrate. The ruling Cometeer kept this weapon to enforce its rule over the others of its kind. Once the Cometeers were destroyed, their slaves were ordered by the Legion to take the comet and leave the Solar System, and never return.

Another novel, One Against the Legion, tells of a Purple pretender who sets up a robotic base on a world over seventy light years from Earth, and tries to conquer the Solar System via matter transporter technology he has stolen. In this story robots are outlawed, as they are in Dune. The story also features Jay Kalam, lobbying to allow the New Cometeers to leave the Solar System in peace, as many people were demanding that AKKA be used to obliterate the departing swarm of planets once and for all.

In 1983, Williamson published a final Legion novel, The Queen of the Legion. Giles Habibula reappears in this final novel, which is set after the disbanding of the Legion.

Series

 * Legion of Space Series
 * 1) The Legion of Space (1947; six-part serial in Astounding, 1934)
 * 2) The Cometeers (1950; four-part serial in Astounding, 1936, plus One Against the Legion, three-part serial in Astounding, 1939)
 * 3) One Against the Legion (1967; three-part serial in Astounding, 1939, plus "Nowhere Near")
 * 4) * Three from the Legion (1980; omnibus of three novels plus "Nowhere Near")
 * 5) The Queen of the Legion (1983)

 * Humanoids Series
 * 1) "With Folded Hands" (1947; in Astounding)
 * 2) The Humanoids (1949; three-part serial as "...And Searching Mind" in Astounding, 1949)
 * 3) The Humanoid Touch (1980)
 * 4) * The Humanoids / With Folded Hands (1996; omnibus)

An editor suggested that Williamson combine the ideas of contraterrene matter (antimatter) and asteroid mining, which inspired the Seetee (C-T) series of short stories written as Will Stewart.
 * Seetee series
 * "Collision Orbit" (short story, as by Will Stewart; from Astounding, 1942)
 * Seetee Shock (1949; as by Will Stewart; from Astounding, 1949)
 * Seetee Ship (1951; as by Will Stewart; from previously published stories 1942–3)
 * Seetee Ship/Seetee Shock (1971, omnibus volume of both).

 * Undersea Trilogy (with Frederik Pohl)
 * 1) Undersea Quest (1954)
 * 2) Undersea Fleet (1956)
 * 3) Undersea City (1958)
 * 4) The Undersea Trilogy (1992; omnibus)

 * Saga of Cuckoo (with Frederik Pohl)
 * 1) Farthest Star (1975)
 * 2) Wall Around A Star (1983)

 * Starchild Trilogy (with Frederik Pohl)
 * 1) The Reefs of Space (1964)
 * 2) Starchild (1965)
 * 3) Rogue Star (1969)
 * 4) The Starchild Trilogy (1977; omnibus)

Novels

 * The Girl from Mars (1930, with Miles J. Breuer)
 * The Green Girl (1930)
 * Golden Blood (1933)
 * Xandulu (1934)
 * The Blue Spot (1935)
 * Islands of the Sun (1935)
 * Reign of Wizardry (1940) (loosely based on the story of Theseus from Greek mythology)
 * Darker Than You Think (1948)
 * Dragon's Island  (1951; also known as The Not-Men)
 * Star Bridge (1955, with James E. Gunn)
 * The Dome Around America (1955; also known as Gateway to Paradise)
 * The Trial of Terra (1962; from four previously published stories, 1951–1962)
 * Bright New Universe (1967)
 * Trapped in Space (1968)
 * The Moon Children (1972)
 * The Power of Blackness (1975)
 * Brother to Demons, Brother to Gods (1979; from five previously published stories, 1977–78)
 * Manseed (1982)
 * Lifeburst (1984)
 * Firechild (1986)
 * Land's End (1988, with Frederik Pohl)
 * Mazeway (1990)
 * The Singers of Time (1991, with Frederik Pohl)
 * Beachhead (1992)
 * Demon Moon (1994)
 * The Black Sun (1997)
 * The Fortress of Utopia (1998; originally in Startling Stories, 1939)
 * The Silicon Dagger (1999)
 * The Stone from a Green Star (1999; originally in Amazing Stories, 1931)
 * Terraforming Earth (2001; Co-winner of 2002 John W. Campbell Memorial Award)
 * The Stonehenge Gate (2005)

Collections

 * The Legion of Time, and After World's End (1952)
 * The Pandora Effect (1969)
 * People Machines (1971)
 * The Early Williamson, 1975
 * The Best of Jack Williamson (1978)
 * The Alien Intelligence (1980)
 * Millions de Soleils (1988)
 * Into the Eighth Decade (1990)
 * The Prince of Space/The Girl from Mars (1998; TGFM written with Miles J. Breuer)
 * The Collected Stories of Jack Williamson, Volume One, The Metal Man and Others (1999)
 * The Collected Stories of Jack Williamson, Volume Two, Wolves of Darkness (1999)
 * The Blue Spot, and Entropy Reversed (Released Entropy) (2000; both from Astounding, 1937)
 * The Collected Stories of Jack Williamson, Volume Three, Wizard's Isle (2000)
 * Dragon's Island and other stories (2002; novel and two shorts)
 * The Collected Stories of Jack Williamson, Volume Four, Spider Island (2002)
 * Seventy-Five: The Diamond Anniversary of a Science Fiction Pioneer, Stephen Haffner & Richard A. Hauptmann, eds. (2004)
 * The Collected Stories of Jack Williamson, Volume Five, The Crucible of Power (2006)
 * In Memory of Wonder's Child Stephen Haffner, ed. (2007)
 * The Worlds of Jack Williamson: A Centennial Tribute (1908–2008), Stephen Haffner, ed. (2008)
 * The Collected Stories of Jack Williamson, Volume Six, Gateway to Paradise (2008)
 * With Folded Hands… And Searching Mind, The Collected Stories of Jack Williamson, Volume Seven (2010)
 * At the Human Limit, The Collected Stories of Jack Williamson, Volume Eight (2011)

Short stories

 * The Metal Man (1928)
 * The Cosmic Express (1930)
 * The Meteor Girl (1931)
 * The Lake of Light (1931)
 * The Doom from Planet 4 (1931)
 * The Moon Era (1931, 1932); also published as separate novelette
 * The Pygmy Planet (1932; Cover feature in Astounding Stories February 1932)
 * Salvage in Space (1933) Cover Story
 * Born of the Sun (1934)
 * Star Bright (1939)
 * The Angel From Hell (1939; in Marvel Tales, writing as Nils O. Sonderlund)
 * Hindsight (1940)
 * Collision Orbit (1942; writing as Will Stewart; into Seetee Ship)
 * Minus Sign (1942; writing as Will Stewart; into Seetee Ship)
 * Opposites—React! (1943; writing as Will Stewart; into Seetee Ship)
 * With Folded Hands... (1947, Awarded Prometheus Hall of Fame in 2018)
 * The Man from Outside (1951)
 * Beans (1958)
 * A Planet for Plundering (1962)
 * The Masked World (1963)
 * Jamboree (1969)
 * The Highest Dive (1976)
 * The Humanoid Universe (1980)
 * The Firefly Tree (1997)
 * The Pet Rocks Mystery (1998)
 * Eden Star (2000)
 * The Ultimate Earth (2000; awarded the Hugo for Best Novella in 2001)

Other

 * Will Acedeme Kill Science Fiction, Asimov's Choice Comets & Computers (Dale Books, 1978, ISBN 0-89559-022-0)

Autobiography

 * Wonder's Child: My Life in Science Fiction. Bluejay Books, New York, 1984. (Hardcover)
 * Wonder's Child: My Life in Science Fiction. Benbella Books, Dallas, 2005. (Paperback, updated with new photographs and epilogue)

Audio

 * NPR audio broadcast on Jack Williamson
 * Dimension X: "With Folded Hands"
 * Jack Williamson was interviewed by The Science Fiction Radio Show in the early 1980s
 * Jack Williamson was interviewed by The Science Fiction Radio Show in the early 1980s

title: Jack Williamson


text:
22 (number)

22 (twenty-two) is the natural number following 21 and preceding 23.

Properties
22 is a palindromic number. 22 is the sixth distinct semiprime, and the fourth of the form $$2 \times q$$ where $$q$$ is a higher prime. It is the second member of the second cluster of discrete biprimes (21, 22), where the next such cluster is (38, 39). It contains an aliquot sum of 14 (itself semiprime), within an aliquot sequence of four composite numbers (22, 14, 10, 8, 7, 1, 0) that are rooted in the prime 7-aliquot tree.

Twenty-two is also:

22 is also a Perrin number, from a sum of 10 and 12, and the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number.

22 can also read as "two twos", which is the only fixed point of John Conway's look-and-say function. In other words, "22" generates the infinite repeating sequence "22, 22, 22, ..."

Permutable and unique primes
There are 22 permutable primes in decimal: $$ \begin{align} & (2,3,5,7), \\ & (R_{2},13,17,31,37,71,73,79,97), \\ & (113,131,199,311,337,373,733,919,991) \\ \end{align} $$ that precede the infinite (conjectured) sequence of prime repunits $$R_{19}, R_{23}, R_{317}, R_{1031}, R_{49081}, \ldots$$, where $$R_{2}$$ represents $$11.$$

The twenty-second unique prime in base ten is notable for having starkly different digits compared to its preceding (and latter) unique primes, as well as for the similarity of its digits to those of the reciprocal of $$7$$, equal to $$0.\overline{142857}.$$

Being 84 = 7 × 12 digits long with a period length of 294 = 14 × 21 digits, it is the number:
 * $$142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143.$$

The sum of all two-digit permutable primes in decimal — that are pairs, without including $$R_{2}$$ — is 418, which is the sum of the digits of the twenty-second unique prime in base ten (all repunit primes are unique, where 3 and 37 are permutable as well as unique).

Free polyominoes
22 is the number of distinct tilings of a four by four square with free 4-polyominoes.

This sequence has as its first terms {1, 1, 2, 22, 515, …}, where the sum of the first five terms yields the 100th prime number 541.

Polytopes
All regular polygons with $$n$$ < $$22$$ edges can be constructed with an angle trisector, with the exception of the 11-sided hendecagon.

There is an elementary set of twenty-two single-orbit convex tilings that tessellate two-dimensional space with face-transitive, edge-transitive, and/or vertex-transitive properties: eleven of these are regular and semiregular Archimedean tilings, while the other eleven are their dual Laves tilings. Twenty-two edge-to-edge star polygon tilings exist in the second dimension that incorporate regular convex polygons: eighteen involve specific angles, while four involve angles that are adjustable. Finally, there are also twenty-two regular complex apeirohedra of the form p{a}q{b}r: eight are self-dual, while fourteen exist as dual polytope pairs; twenty-one belong in $$\mathbb{C}^2$$ while one belongs in $$\mathbb{C}^3$$.

There are twenty-two different subgroups that describe full icosahedral symmetry, that is based on the regular icosahedron. Three groups are generated by particular inversions, five groups by reflections, and nine groups by rotations, alongside three mixed groups, the pyritohedral group, and the full icosahedral group.

There are 22 finite semiregular polytopes through the eighth dimension, aside from the infinite families of prisms and antiprisms in the third dimension and inclusive of 2 enantiomorphic forms. Defined as vertex-transitive polytopes with regular facets, there are:
 * 15 Archimedean semiregular solids in 3-space, which include two chiral forms, one from the snub cube and one from the snub dodecahedron. In other words, from symmetries of 13 distinct semiregular polyhedra, two of which have mirror images.
 * 3 semiregular polychora in 4-space: the rectified 5-cell, the rectified 600-cell and the snub 24-cell.
 * 4 semiregular polytopes from 5-space through 8-space that are part of the family of k21 polytopes: the demipenteract, 221, 321, and 421; with the final figure representing the root vectors of simple Lie group E8.
 * The family of k21 polytopes can be extended backward to include the rectified 5-cell and the three-dimensional triangular prism, which is the simplest semiregular polytope.
 * On the other hand, k22 polytopes are a family of five different polytopes up through the eighth dimension, that include three finite polytopes and two honeycombs. Its root figure is the first proper duoprism, the 3-3 duoprism (-122), which is made of six triangular prisms. The second figure is the birectified 5-simplex (022), and the last finite figure is the 6th-dimensional 122 polytope. 122 is highly symmetric, whose 72 vertices represent the root vectors of the simple Lie group E6. 322 is a paracompact infinite honeycomb that contains 222 Euclidean honeycomb facets under Coxeter group symmetry $${\bar{T}}_7$$, with 222 made of 122 facets, and so forth. The Coxeter symbol for these figures is of the form kij, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k-length sequence of branches.

There are twenty-two Coxeter groups in the sixth dimension that generate uniform polytopes: four of these generate uniform non-prismatic figures, while the remaining eighteen generate uniform prisms, duoprisms and triaprisms.

Sporadic groups
The number 22 appears prominently within sporadic groups. Mathieu group M22 is one of 26 such sporadic finite simple groups, defined as the 3-transitive permutation representation on 22 points. It is the monomial of the McLaughlin sporadic group, McL, and the unique index 2 subgroup of the automorphism group of Steiner system S(3,6,22). Mathieu group M23 contains M22 as a point stabilizer, and has a minimal irreducible complex representation in 22 dimensions, like McL. M23 has two rank 3 actions on 253 points, with 253 equal to the sum of the first 22 non-zero positive integers, or the 22nd triangular number. Both M22 and M23 are maximal subgroups within Mathieu group M24, which works inside the lexicographic generation of Steiner system S(5,8,24) W24, where single elements within 759 octads of 24-element sets occur 253 times throughout its entire set. On the other hand, the Higman–Sims sporadic group HS also has a minimal faithful complex representation in 22 dimensions, and is equal to 100 times the group order of M22, $|HS|$ = 100$|M_{22}|$. Conway group Co1 and Fischer group Fi24 both have 22 different conjugacy classes.

Binary and ternary Golay codes
The extended binary Golay code $$\mathbb B_{24}$$, which is related to Steiner system W24, is constructed as a vector space of F2 from the words:

 * $$c_j = e(\overline{\rm x})\cdot\overline{\rm x}^j (j=0,...,22),\text{ }$$ and $$\text{ }$$$$c_{23} = \sum_{i=0}^{22}\overline{\rm x}^{i}+\overline{\rm x}^\infty$$

 * with $$c\in F_2$$, and $$e(\overline{\rm x})$$ the quadratic residue code of the binary Golay code $$\mathbb B_{23}$$ (with $$\overline{\rm x}^\infty$$ its parity check). M23 is the automorphism group of $$\mathbb B_{23}$$.

The extended ternary Golay code [12, 6, 6], whose root is the ternary Golay code [11, 6, 5] over F3, has a complete weight enumerator value equal to:

 * $$x^{12}+y^{12}+z^{12}+22\left(x^6y^6+y^6z^6+z^6x^6\right)+220\left(x^6y^3z^3+x^3y^6z^3+x^3y^3z^6\right).$$

Calculations for $φ(n)$
$$\frac{22}{7} = 3.14{\color{red}28}\ldots$$ is a commonly used approximation of the irrational number $\pi$, the ratio of the circumference of a circle to its diameter, where in particular 22 and 7 are consecutive hexagonal pyramidal numbers. Also,

 * $$\sqrt[4]{\frac{2143}{22}} = 3.141\;592\;65{\color{red}2\;582\;\ldots}$$ from an approximate construction of the squaring of the circle by Srinivasa Ramanujan, correct to eight decimal places.

On the other hand, where twenty-two is the second-smallest numerator in convergents of pi, the difference between the next two larger numerators (355, 333) is also 22. In the true approximation for $$\pi$$, the first six digits of its fractional part in decimal representation sum to 22, strings of digits which also end in 2. Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for $$\pi$$ for all integers $$n = \{1, 2, 3, 4 ..., 22\}$$.

In science

 * 22 is the atomic number of titanium.
 * 22 is the number of proteinogenic amino acids.
 * 22 is the number of bones in the human skull: 14 belong to the facial skeleton and 8 to the neurocranium.

In aircraft

 * 22 is the designation of the USAF stealth fighter, the F-22 Raptor.

In music

 * "Twenty Two" is a song by:
 * Karma to Burn (2007)
 * The Vicar (2013)
 * Jordan Sweeney (2008)
 * The Good Life (2000)
 * Sweet Nectar (1996)
 * American Generals (2004)
 * Dan Anderson (2007)
 * Bad Cash Quartet (2006)
 * Millencolin (1999)
 * Enter the Worship Circle (2005)
 * Blank Dogs (2008)
 * Al Candello (2002)
 * Amen Dunes (2018)
 * In Jay-Z's song "22 Two's", he rhymes the words: too, to, and two, 22 times in the first verse.
 * "22 Acacia Avenue" is a song by Iron Maiden on the album The Number of the Beast.
 * Catch 22 is an album by death metal band Hypocrisy.
 * "22" is a song by Lily Allen on the album It's Not Me, It's You.
 * 22 Dreams is a song and album by Paul Weller. The album has 22 songs on it.
 * The Norwegian electronica project Ugress uses 22 as a recurring theme. All four albums feature a track with 22 in the title.
 * "22" is a song by Taylor Swift on her fourth album Red.
 * "The Number 22" is a song by Ashbury Heights on the album The Looking Glass Society.
 * 22, A Million is an album by Bon Iver. The first track of the album is called "22 (OVER SOON)".
 * Cubic 22 was a Belgian techno duo.
 * "22" is a song by the English alternative rock band Deaf Havana on their album Old Souls.
 * "22" is a song by the Irish singer Sarah McTernan. She represented Ireland with this song at Eurovision 2019.

In other fields

 * Catch-22 (1961), Joseph Heller's novel, and its 1970 film adaptation gave rise to the expression of logic "catch-22".
 * Revista 22 is a magazine published in Romania.
 * There are 22 stars in the Paramount Pictures logo.
 * "Twenty Two" (February 10, 1961) is Season 2–episode 17 (February 10, 1961) of the 1959–1964 TV series The Twilight Zone, in which a hospitalized dancer has nightmares about a sinister nurse inviting her to Room 22, the hospital morgue.
 * Traditional Tarot decks have 22 cards with allegorical subjects. These serve as trump cards in the game. The Fool is usually a kind of wild-card among the trumps and unnumbered, so the highest trump is numbered 21. Occult Tarot decks usually have 22 similar cards which are called Major Arcana by fortune-tellers. Occultists have related this number to the 22 letters of the Hebrew alphabet and the 22 paths in the Kabbalistic Tree of Life.
 * "22" is the number assigned to the unborn soul who serves as a prominent character in the Pixar film Soul.

In computing and technology

 * 22 is the standard port number for the Secure Shell protocol.

In culture and religion

 * There are 22 letters in the Hebrew alphabet.
 * In the Kabbalah, there are 22 paths between the Sephirot.
 * There are 22 chapters in the book: The Revelation of Jesus Christ, Book of Revelation.
 * 22 is a master number in numerology.

In sports

 * In both American football and association football, a total of 22 players (counting both teams) start the game, and this is also the maximum number of players that can be legally involved in play at any given time.
 * In men's Australian rules football, each team is allowed a squad of 22 players (18 on the field and 4 interchanges).
 * The length of a cricket pitch is 22 yards.
 * In rugby union, the "22" is a line in each half of the field which is 22 meters from the respective try line. It has significance in a number of laws particularly relating to kicking the ball away.
 * A snooker game (called a "frame") starts with 22 coloured balls at specified locations on the table (15 red balls and 7 others).

In weights and measures

 * The number of yards in a chain.

In other uses
Twenty-two may also refer to:
 * 22 is the number of the French department Côtes-d'Armor
 * "22" is a common name for the .22 calibre .22 Long Rifle cartridge.
 * In French jargon, "22" is used as a phrase to warn of the coming of the police (typically "22, v'là les flics !" (In English: "5-0! Cops!")
 * In photography, f/22 is the largest f-stop (and thus smallest aperture) available on most lenses made for single-lens reflex cameras
 * In Spanish lottery and bingo, 22 is nicknamed los dos patitos after its shape.

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

title: 22 (number)


text:
Edward Whymper

Edward Whymper FRSE (27 April 1840 – 16 September 1911) was an English mountaineer, explorer, illustrator, and author best known for the first ascent of the Matterhorn in 1865. Four members of his climbing party were killed during the descent. Whymper also made important first ascents on the Mont Blanc massif and in the Pennine Alps, Chimborazo in South America, and the Canadian Rockies. His exploration of Greenland contributed an important advance to Arctic exploration. Whymper wrote several books on mountaineering, including Scrambles Amongst the Alps.

Early life
Edward Whymper was born at Lambeth Terrace on Kennington Road in London on 27 April 1840 to the artist and wood engraver Josiah Wood Whymper and Elizabeth Whitworth Claridge. He was the second of eleven children, his older brother being the artist and explorer Frederick Whymper. He was trained to be a wood-engraver at an early age. In 1860, he made extensive forays into the central and western Alps to produce a series of commissioned alpine scenery drawings. Among the objects of this tour was the illustration of an unsuccessful attempt made by Professor Bonney's party to ascend Mont Pelvoux, at that time believed to be the highest peak of the Dauphiné Alps.

In 1861, Whymper successfully completed the ascent of Mont Pelvoux, the first of a series of expeditions that threw much needed light on the topography of an area which at the time was very poorly mapped. From the summit of Mont Pelvoux, Whymper discovered that it was overtopped by a neighbouring peak, subsequently named the Barre des Écrins, which, before the annexation of Savoy added Mont Blanc to the possessions of France, was the highest point in the French Alps. Whymper climbed the Barre des Écrins in 1864 with Horace Walker, A. W. Moore and guides Christian Almer senior and junior.

The years 1861 to 1865 were filled with new expeditions in the Mont Blanc massif and the Pennine Alps, among them the first recorded ascents of the Aiguille d'Argentière and Mont Dolent in 1864, and the Aiguille Verte, the Grand Cornier and Pointe Whymper on the Grandes Jorasses in 1865. That same year he also made the first crossing of the Moming Pass. According to his own words, his only failure was on the west ridge of the Dent d'Hérens in 1863. As a result of his Alpine experience, he designed a tent which came to be known as the "Whymper tent" and tents based on his design were still being manufactured 100 years later.

The Matterhorn
Professor John Tyndall and Whymper emulated each other in determined attempts to reach the summit of the Matterhorn by the south-western, or Italian, ridge. In 1865, Whymper, who had failed eight times already, attempted unsuccessfully to climb a couloir on the south-east face with Michel Croz. After Croz left for a prior engagement with Charles Hudson, Whymper was unable to secure the services of Val Tournanche guide Jean Antoine Carrel, and instead planned to try the eastern face with Lord Francis Douglas and the two Zermatt guides, Peter Taugwalder father and son.

Whymper was convinced that the Matterhorn's precipitous appearance when viewed from Zermatt was an optical illusion, and that the dip of the strata, which on the Italian side formed a continuous series of overhangs, should make the opposite side a natural staircase. This party of four was joined by Hudson and Croz, and the inexperienced Douglas Hadow. Their attempt by what is now the normal route, the Hörnli ridge, met with success on 14 July 1865, only days before an Italian party. On the descent, Hadow slipped and fell onto Croz, dislodging him and dragging Douglas and Hudson to their deaths; the rope parted, saving the other three.

A controversy ensued as to whether the rope had actually been cut, but a formal investigation could not find any proof, and Peter Taugwalder was acquitted. The rope had snapped between Taugwalder and Lord Francis Douglas. Whymper asked Taugwalder to show him the rope. To his surprise, he saw that it was the oldest and weakest of the ropes they brought, and one which had been intended only as a reserve. All those who had fallen had been tied with a Manila rope, or with a second and equally strong one, and consequently it had been only between the survivors and those who had fallen where the weaker rope had been used. Whymper also had suggested to Hudson that they should have attached a rope to the rocks on the most difficult place, and held it as they descended, as an additional protection. Hudson approved the idea, but it was never done. It can be deduced that Taugwalder had no other choice but to use a weaker rope, as the stronger rope was not long enough to connect Taugwalder to Douglas. The account of Whymper's attempts on the Matterhorn occupies the greater part of his book, Scrambles amongst the Alps (1871), in which the illustrations are engraved by Whymper himself. The accident haunted Whymper: "Every night, do you understand, I see my comrades of the Matterhorn slipping on their backs, their arms outstretched, one after the other, in perfect order at equal distances—Croz the guide, first, then Hadow, then Hudson, and lastly Douglas. Yes, I shall always see them ..."

Exploration in Greenland
Whymper's 1865 campaign had been planned to test his route-finding skills in preparation for an expedition to Greenland in 1867. The exploration in Greenland resulted in an important collection of fossil plants, which were described by Professor Heer and deposited in the British Museum. Whymper's report was published in the report of the British Association of 1869. Though hampered by a lack of supplies and an epidemic among the local people, he proved that the interior could be explored by the use of suitably constructed sledges, and thus contributed an important advance to Arctic exploration.

Another expedition in 1872 was devoted to a survey of the coastline.

South American exploration
Whymper next organised an expedition to Ecuador, designed primarily to collect data for the study of altitude sickness and the effect of reduced pressure on the human body. His chief guide was Jean-Antoine Carrel, who later died from exhaustion on the Matterhorn after bringing his employers to safety through a snowstorm.

During 1880, Whymper made two ascents of Chimborazo (6,267m), including its first ascent; though Alexander von Humboldt had climbed on the volcano in 1802, he did not reach the summit. Whymper spent a night on the summit of Cotopaxi and made first ascents of Sincholagua, Antisana, Cayambe, Sara Urco and Cotacachi. In 1892, he published the results of his journey in a volume entitled Travels amongst the Great Andes of the Equator.

His observations on altitude sickness led him to conclude that it was caused by a reduction in atmospheric pressure, which lessens the value of inhaled air, and by expansion of the air or gas within the body, causing pressure upon the internal organs. The effects produced by gas expansion may be temporary and dissipate when equilibrium has been restored between the internal and external pressure. The publication of his work was recognised on the part of the Royal Geographical Society by the award of the Patron's medal.

His experiences in South America having convinced him of certain serious errors in the readings of aneroid barometers at high altitudes, he published a work entitled How to Use the Aneroid Barometer, and succeeded in introducing important improvements in their construction. He afterwards published two guide-books to Zermatt and Chamonix.

While in Ecuador, Whymper made a collection of amphibians and reptiles that he handed over to George Albert Boulenger at the British Museum. The collection received some praise from Boulenger, who said that "though containing no striking novelties", the collection was "interesting on account of the care bestowed by its collector in recording the exact locality from which every specimen was obtained". Boulenger described four new species from the materials, three of them named after Whymper: the snake Coronella Whymperi (now a junior synonym of Saphenophis boursieri) and the frogs Prostherapis Whymperi, Phryniscus elegans, and Hylodes Whymperi (now a junior synonym of Pristimantis curtipes).

Canadian Rockies
In the early 1900s, Whymper visited the Canadian Rockies several times and made arrangements with the Canadian Pacific Railway (CPR) to promote the Canadian Rockies and the railway in his talks in Europe and Asia. In exchange, the CPR agreed to pay transportation costs for him and his four guides. According to the surveyor and mountaineer A. O. Wheeler, Whymper was hired to “conduct explorations and surveys in the interests of the Canadian Pacific railway company” (Wheeler, 1905). In 1901, Whymper and his four guides (Joseph Bossoney, Christian Kaufmann, Christian Klucker and Joseph Pollinger) made the first ascents of Mount Whymper and Stanley Peak in the Vermilion Pass area of the Canadian Rockies.

His brother Frederick also has a mountain in British Columbia named after him, from his days as artist illustrator with the Robert Brown's Vancouver Island Exploring Expedition in 1864.

Illustrator
When not climbing, Whymper pursued his profession as an engraver of illustrations for books and periodicals. Among the books he illustrated was his fellow-mountaineer Florence Crauford Grove's The Frosty Caucasus (1875) Whymper also illustrated and engraved John Tyndall's "Hours of Exercise in The Alps" (1871). He illustrated books for Isabella L. Bird but his brother Charles Whymper was the designer of the Henrietta Amelia Bird memorial clock tower in Tobermory, Isle of Mull, Scotland. It was built in 1905, funded by Isabella Bird (Mrs. Bishop) in memory of her sister.

Final years
On 25 April 1906, aged 65, Whymper married Edith Mary Lewin aged 23 (born 1883) at Emmanuel Church in Forest Gate, Essex (now London). The service was presided over by Canon J. M'Cormick, who had assisted the mountaineer after the Matterhorn accident. The marriage produced one daughter, Ethel. The couple were separated in 1910. Edith remarried in 1913 and died the following year from complications of pregnancy.

Shortly after returning to Chamonix from another climb in the Alps, Whymper became ill, locked himself in his room at the Grand Hotel Couttet, and refused all medical treatment. Whymper died alone on 16 September 1911, at the age of 71. A funeral was held four days later. He is buried in the English cemetery in Chamonix.

Works

 * Scrambles Amongst the Alps: In the Years 1860–69. London: John Murray, 1871. ISBN 978-0898150438.
 * Winter Pictures: By Poet and Artist. London: Religious Tract Society, 1875.
 * The Ascent of the Matterhorn. London: John Murray, 1880. ISBN 978-0862993474.
 * How to Use the Aneroid Barometer. London: John Murray, 1881.
 * Travels Amongst the Great Andes of the Equator. London: John Murray, 1891. ISBN 978-0879052812.
 * Chamonix and the Range of Mont Blanc: A Guide. London: John Murray, 1896.
 * The Valley of Zermatt and the Matterhorn: A Guide. London: John Murray, 1897.
 * The Apprenticeship of a Mountaineer: Edward Whymper's London Diary, 1855–1859. Ed. Ian Smith. London: London Record Society, 2008. ISBN 978-0900952432.
 * Among the Tibetans by Isabella L. Bird (1894, as illustrator)

title: Edward Whymper


text:
24 (number)

24 (twenty-four) is the natural number following 23 and preceding 25. It is one sixth of a gross, or two dozens.

In mathematics
24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2$q$q, where q is an odd prime. It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12 and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors (36) is greater than itself, as well as a superabundant number.

In number theory and algebra

 * 24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index:
 * 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = $5⁄2$ × 24
 * 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
 * 24 is a practical number, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24.
 * 24 is a Harshad number, since it is divisible by the sum of its digits in decimal.
 * 24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
 * 24 is a twin-prime sum, specifically the sum of the third pair of twin primes $$(11 + 13)$$.
 * 24 is a highly totient number, as there are 10 solutions to the equation φ(x) = 24, which is more than any integer below 24. 144 (the square of 12) and 576 (the square of 24) are also highly totient.
 * 24 is a polite number, an amenable number, an idoneal number, and a tribonacci number.
 * 24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal (5).
 * 24 is a compositorial, as it is the product of composite numbers up to 6.
 * 24 is a pernicious number, since its Hamming weight in its binary representation (11000) is prime (2).
 * 24 is the third nonagonal number.
 * 24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24.
 * 24 is a congruent number, as 24 is the area of a right triangle with a rational number of sides.
 * 24 is a semi-meandric number, where an order-6 semi-meander intersects an oriented ray in R2 at 24 points.
 * 24 is the number of digits of the fifth and largest known unitary perfect number, when written in decimal: 146361946186458562560000.
 * Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a prime number; 24 is the largest number with this property.
 * 24 is the largest integer that is divisible by all natural numbers no larger than its square root.
 * The product of any four consecutive numbers is divisible by 24. This is because, among any four consecutive numbers, there must be two even numbers, one of which is a multiple of four, and there must be at least one multiple of three.
 * 24 = 4!, the factorial of 4. It is the largest factorial that does not contain a trailing zero at the end of its digits (since factorial of any integer greater than 4 is divisible by both 2 and 5), and represents the number of ways to order 4 distinct items:
 * (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,1,2,4), (3,1,4,2), (3,2,1,4), (3,2,4,1), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,1,3,2), (4,2,1,3), (4,2,3,1), (4,3,1,2), (4,3,2,1).
 * 24 is the only nontrivial solution to the cannonball problem; that is, 12 + 22 + 32 + … + 242 is a perfect square (702).
 * 24 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 12, 24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = {±1, ±5, ±7, ±11} is isomorphic to the additive group (Z/2Z)3.  This fact plays a role in monstrous moonshine.
 * It follows that any number n relatively prime to 24 (that is, any number of the form 6K ± 1), and in particular any prime n greater than 3, has the property that n2 – 1 is divisible by 24.
 * The modular discriminant $Δ(τ)$ is proportional to the 24th power of the Dedekind eta function $η(τ)$: $Δ(τ) = (2π)^{12}η(τ)^{24}$.

In geometry

 * 24 degrees is the measure of the central angle and external angle of a pentadecagon.
 * An icositetragon is a regular polygon with 24 sides and Dih24 symmetry of order 48. It can fill a plane-vertex alongside a triangle and octagon.
 * 24 is the Euler characteristic of a K3 surface: a general elliptic K3 surface has exactly 24 singular fibers.
 * 24 is the order of the octahedral group — the group of rotations of the regular octahedron and the group of rotations of the cube. The binary octahedral group is a subgroup of the 3-sphere S3 consisting of the 24 elements {±1, ±i, ±j, ±k, (±1±i±j±k)/2} of the binary tetrahedral group along with the 24 elements contained in its coset {(±1±i)/√2, (±1±j)/√2, (±1±k)/√2, (±i±j)/√2, (±i±k)/√2, (±j±k)/√2}. These two cosets each form the vertices of a self-dual 24-cell, and the two 24-cells are dual to each other. (See point below on 24-cell).
 * 24 is the count of different elements in various uniform polyhedron solids. Within the family of Archimedean and Catalan solids, there are 24 edges in a cuboctahedron and rhombic dodecahedron, 24 vertices in a  rhombicuboctahedron, truncated cube, truncated octahedron, and snub cube, as well as 24 faces in a deltoidal icositetrahedron, tetrakis hexahedron, triakis octahedron, and pentagonal icositetrahedron. The cube-octahedron compound, with a rhombic dodecahedral convex hull, is the first stellation of the cuboctahedron, with a total of 24 edges.
 * There are 12 non-prismatic uniform polyhedron compounds (UC01, UC03, UC08, UC10, UC12, UC30, UC42, UC46, UC48, UC50, UC52, and UC54) and 12 uniform star polyhedra (U03, U13, U14, U15, U17, U18, U19, U21, U36, U37, U41, and U58) with a vertex, edge, or face count of 24. The great disnub dirhombidodecahedron, also called Skilling's figure, is a degenerate uniform star polyhedron with a Euler characteristic of 24, when pairs of coinciding edges are considered to be single edges.
 * Finally, 6 Johnson solids (J17, J27, J37, J45, J61, and J90) also have vertex, edge, or face counts of 24. The pseudo great rhombicuboctahedron, one of two known pseudo-uniform polyhedra alongside the elongated square gyrobicupola (J37), has 24 vertices.
 * The tesseract has 24 two-dimensional faces (which are all squares). Its dual four-dimensional polytope is the 16-cell, which has 24 edges.
 * The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope. It possesses 576 (24×24) rotational symmetries and 1152 isometries altogether. It tiles 4-dimensional space in a 24-cell honeycomb, in which each 24-cell is surrounded by 24 24-cells.
 * The vertices of the 24-cell honeycomb can be chosen so that in 4-dimensional space, identified with the ring of quaternions, they are precisely the elements of the subring (the ring of "Hurwitz integral quaternions") generated by the binary tetrahedral group as represented by the set of 24 quaternions $$\{\pm 1,\pm i,\pm j,\pm k,\tfrac{1}{2}(\pm 1 \pm i \pm j \pm k)\}$$ in the D4 lattice. This set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere S3 of radius one centered at the origin. S3 is the Lie group Sp(1) of unit quaternions (isomorphic to the Lie groups SU(2) and Spin(3)), and so the binary tetrahedral group — of order 24 — is a subgroup of S3.
 * The 24 vertices of the 24-cell are contained in the regular complex polygon 4{3}4, or  of symmetry order 1152, as well as 24 4-edges of 24 octahedral cells (of 48). Its representation in the F4 Coxeter plane contains two rings of 12 vertices each.
 * Truncations, runcinations, and omnitruncations of the 24-cell yield 4-dimensional polytopes whose Petrie polygons are 24-sided icositetragons; i.e., within the truncated 24-cell, runcinated 24-cell, and omnitruncated 24-cell, amongst others.
 * 24 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a 24-cell).
 * The Barnes–Wall lattice contains 24 lattices.
 * In 24 dimensions there are 24 even positive definite unimodular lattices, called the Niemeier lattices. One of these is the exceptional Leech lattice which has many surprising properties; due to its existence, the answers to many problems such as the kissing number problem and densest lattice sphere-packing problem are known in 24 dimensions but not in many lower dimensions. The Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S(5,8,24) and the Mathieu group M24. (One construction of the Leech lattice is possible because 12 + 22 + 32 + ... + 242 = 702.)
 * 24 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: $\pi$n+3(Sn) =  Z/24Z for all n ≥ 5.

In science

 * The atomic number of chromium.
 * The average number of hours in a day (on Earth), also known as a mean solar day.
 * 24! is an approximation (exceeding by just over 3%) of the Avogadro constant.
 * 24 dm3 is the volume of one mole of any gas at room temperature and pressure.

In religion

 * The number of books in the Tanakh.
 * In Christian apocalyptic literature it represents the complete Church, being the sum of the 12 tribes of Israel and the 12 Apostles of the Lamb of God. For example, in The Book of Revelation: "Surrounding the throne were twenty-four other thrones, and seated on them were twenty-four elders. They were dressed in white and had crowns of gold on their heads."
 * Number of Tirthankaras in Jainism.
 * Number of spokes in the Ashok Chakra.

In music

 * There are a total of 24 major and minor keys in Western tonal music, not counting enharmonic equivalents. Therefore, for collections of pieces written in each key, the number of pieces in such a collection; e.g., Chopin's 24 Preludes.

In sports

 * Four-and-Twenty was an American racehorse.
 * In association football:
 * The FIFA World Cup final tournament featured 24 men's national teams from 1982 to 1994.
 * The FIFA Women's World Cup final tournament featured 24 national teams in 2015 and 2019.
 * In basketball:
 * In the NBA, the time on a shot clock is 24 seconds.
 * In ice hockey
 * In the NHL, 24 is the record number of most Stanley Cups won in NHL history, which the Montreal Canadiens have achieved.

In other fields
24 is also:
 * The number of bits a computer needs to represent 24-bit color images (for a maximum of 16,777,216 colours—but greater numbers of bits provide more accurate colors).
 * The number of karats representing 100% pure gold.
 * The number of cycles in the Chinese solar year.
 * The number of years from the start of the Cold War until the signing of the Seabed Arms Control Treaty, which banned the placing of nuclear weapons on the ocean floor within certain coastal distances.
 * The number of frames per second at which motion picture film is usually projected, as this is sufficient to allow for persistence of vision.
 * The number of letters in both the modern and classical Greek alphabet. For the latter reason, also the number of chapters or "books" into which Homer's Odyssey and Iliad came to be divided.
 * The number of runes in the Elder Futhark.
 * The number of points on a backgammon board.
 * A children's mathematical game involving the use of any of the four standard operations on four numbers on a card to get 24 (see 24 Game).
 * The maximum number of Knight Companions in the Order of the Garter.
 * The number of the French department Dordogne.
 * Four and twenty is the number of blackbirds baked in a pie in the traditional English nursery rhyme "Sing a Song of Sixpence".
 * In Brazil, twenty-four is associated with homosexuality as it is the number that stands for the deer in a game known as “jogo do bicho”.

title: 24 (number)


text:
23 (number)

23 (twenty-three) is the natural number following 22 and preceding 24.

In mathematics
Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the next to last member of the first Cunningham chain of the first kind (2, 5, 11, 23, 47), and the sum of the prime factors of the second set of consecutive discrete semiprimes, (21, 22). 23 is the smallest odd prime to be a highly cototient number, as the solution to $$x-\phi(x)$$ for the integers 95, 119, 143, and 529.

 * 23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime, and a happy number.
 * The sum of the first nine primes up to 23 is a square: $$2 + 3 + \dots + 23 = 100 = 10^{2}$$ and the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
 * It is the fifth factorial prime, and since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime.
 * In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
 * 23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem.
 * The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers.
 * Otherwise, $$46 = 23 \times 2$$ is the largest even number that is not the sum of two abundant numbers.

 * 23 is the second Woodall prime, and an Eisenstein prime with no imaginary part and real part of the form $$3n-1.$$ It is the fifth Sophie Germain prime and the fourth safe prime.
 * 23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees.
 * The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae.
 * 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.
 * 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.
 * 23 is the smallest prime $$p$$ such that the largest consecutive pair of $$p-$$smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of $$(p-1)-$$smooth numbers.
 * According to the birthday paradox, in a group of 23 or more randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday.
 * A related coincidence is that 365 times the natural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22nd triangular number, 253.

 * The first twenty-three odd prime numbers (between 3 and 89 inclusive), are all cluster primes $$p$$ such that every even positive integer $$k \le p - 3$$ can be written as the sum of two prime numbers that do not exceed $$p$$.
 * 23 is the smallest discriminant of imaginary quadratic fields with class number 3 (negated), and it is the smallest discriminant of complex cubic fields (also negated).
 * The twenty-third permutable prime in decimal $$R_{19}$$ is also the second to be a prime repunit (after $$R_{2}$$), followed by $$R_{23}$$ and $$R_{1031}$$.

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

Mersenne numbers
The first Mersenne number of the form $$2^{n} - 1$$ that does not yield a prime number when inputting a prime exponent is $$ 2047 = 23 \times 89,$$ with $$n=11.$$

On the other hand, the second composite Mersenne number contains an exponent $$n$$ of twenty-three:

$$M_{23} = 2^{23} - 1 = 8\;388\;607 = 47 \times 178\;481$$

The twenty-third prime number (83) is an exponent to the fourteenth composite Mersenne number, which factorizes into two prime numbers, the largest of which is twenty-three digits long when written in base ten:

$$M_{83} = 967...407 = 167 \times 57\;912\;614\;113\;275\;649\;087\;721$$

Further down in this sequence, the seventeenth and eighteenth composite Mersenne numbers have two prime factors each as well, where the largest of these are respectively twenty-two and twenty-four digits long,

$$ \begin{align} M_{103} & = 101 \ldots 007 = 2\;550\;183\;799 \times 3\;976\;656\;429\;941\;438\;590\;393 \\ M_{109} & = 649 \ldots 511 = 745\;988\;807 \times 870\;035\;986\;098\;720\;987\;332\;873 \\ \end{align} $$

Where prime exponents for $$M_{23}$$ and $$M_{83}$$ add to 106, which lies in between prime exponents of $$M_{103}$$ and $$M_{109}$$, the index of the latter two (17 and 18) in the sequence of Mersenne numbers sum to 35, which is the twenty-third composite number.

$$23!$$ is twenty-three digits long in decimal, and there are only three other numbers $$n$$ whose factorials generate numbers that are $$n$$ digits long in base ten: 1, 22, and 24.

In geometry
The Leech lattice Λ24 is a 24-dimensional lattice through which 23 other positive definite even unimodular Niemeier lattices of rank 24 are built, and vice-versa. Λ24 represents the solution to the kissing number in 24 dimensions as the precise lattice structure for the maximum number of spheres that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located at deep holes of radii $√2$ in lattice points around its automorphism group, Conway group $$\mathbb C_{0}$$. The Leech lattice can be constructed in various ways, which include:

 * By means of a matrix of the form $$\scriptstyle\begin{pmatrix} Ia&H/2\\H/2&Ib\end{pmatrix}$$ where $$I$$ is the identity matrix and $$H$$ is a 24 by 24 Hadamard matrix (Z/23Z ∪ ∞) with a = 2 and b = 3, and entries X(∞) = 1 and X(0) = -1 with X(n) the quadratic residue symbol mod 23 for nonzero n.
 * Through the extended binary Golay code $$\mathbb B_{24}$$ and Witt design $\mathbb W_{24}$, which produce a construction of the 196,560 minimal vectors in the Leech lattice. The extended binary Golay code is an extension of the perfect binary Golay code $$\mathbb B_{23}$$, which has codewords of size 23. $$\mathbb B_{23}$$ has Mathieu group $\mathbb M_{23}$ as its automorphism group, which is the second largest member of the first generation in the happy family of sporadic groups. $$\mathbb M_{23}$$ has a minimum faithful complex representation in 22 dimensions and group-3 actions on 253 objects, with 253 equal to the number of pairs of objects in a set of 23 objects. In turn, $$\mathbb M_{23}$$ is the automorphism group of Mathieu group $\mathbb M_{24}$, which works through $$\mathbb W_{24}$$ to generate 8-element octads whose individual elements occur 253 times through its entire block design.
 * Using Niemer lattice D24 of group order 223·24! and Coxeter number 46 = 2·23, it can be made into a module over the ring of integers of quadratic field $$\mathbb{Q}(\sqrt{-23})$$, whereby multiplying D24 by a non-principal ideal of the ring of integers yields the Leech lattice.

Conway and Sloane provided constructions of the Leech lattice from all other 23 Niemeier lattices.

Twenty-three four-dimensional crystal families exist within the classification of space groups. These are accompanied by six enantiomorphic forms, maximizing the total count to twenty-nine crystal families. Five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (with reflections).

There are 23 three-dimensional uniform polyhedra that are cell facets inside uniform 4-polytopes that are not part of infinite families of antiprismatic prisms and duoprisms: the five Platonic solids, the thirteen Archimedean solids, and five semiregular prisms (the triangular, pentagonal, hexagonal, octagonal, and decagonal prisms).

23 Coxeter groups of paracompact hyperbolic honeycombs in the third dimension generate 151 unique Wythoffian constructions of paracompact honeycombs. 23 four-dimensional Euclidean honeycombs are generated from the ${\tilde{B}}_4$ cubic group, and 23 five-dimensional uniform polytopes are generated from the $\mathrm D_{5}$ demihypercubic group.

In two-dimensional geometry, the regular 23-sided icositrigon is the first regular polygon that is not constructible with a compass and straight edge or with the aide of an angle trisector (since it is neither a Fermat prime nor a Pierpont prime), nor by neusis or a double-notched straight edge. It is also not constructible with origami, however it is through other traditional methods for all regular polygons.

In science and technology

 * The atomic number of vanadium.
 * The atomic mass number of the stable isotope of sodium.
 * Normal human sex cells have 23 chromosomes. Other human cells have 46 chromosomes, arranged in 23 pairs.
 * Scientific notation for the Avogadro constant is written as.
 * 23 is the width of the Arecibo message, sent to space in search for extraterrestrial intelligence.
 * 23 is the TCP/IP port used for telnet and is the default for the telnet command.
 * The Earth's axis is tilted at approximately 23°.

In religion

 * In Biblical numerology, it is associated with Psalm 23, also known as the Shepherd Psalm. It is possibly the most quoted and best known Psalm. Psalms is also the 23rd book in the Douay–Rheims Bible.
 * In Islam, the Qur'an was revealed in a total of 23 years to Muhammed.
 * Muslims believe the first verses of the Qur'an were revealed to the Islamic prophet Muhammad on the 23rd night of the 9th Islamic month, though, its disputed.
 * In I Ching divination, Hexagram 23 is 剝 (bō), "peeling" or "splitting apart".
 * Principia Discordia, the sacred text of Discordianism, holds that 23 (along with the discordian prime 5) is one of the sacred numbers of Eris, goddess of discord.

Music

 * Alfred Harth uses the number 23 in his artist name Alfred 23 Harth, or A23H, since the year 1+9+8+5 = 23.
 * Twentythree is the name of Tristan Prettyman's debut album
 * Twentythree an album by Carbon Based Lifeforms
 * "Viginti Tres" (Latin for twenty-three) is a song by Tool on their album 10,000 Days
 * Blink-182's song "What's My Age Again?" includes the lyrics "nobody likes you when you're 23."
 * 23 is an album and title track by Blonde Redhead
 * The Incubus song "Pardon Me" includes the lyrics "A decade ago, I never thought I would be, at 23, on the verge of spontaneous combustion, woe is me!" Frontman Brandon Boyd was 23 years old when he wrote the song and described himself as being "kind of obsessive about that number".
 * "23" is a song by Jimmy Eat World, on their album Futures. The number also appears in the songs "Christmas Card" and "12."23".95" as well as on some items of clothing produced by the band.
 * Four tet and Yellowcard both have songs titled "Twenty-Three".
 * Dear 23, an album by The Posies
 * Untitled 23, an album by The Church
 * Noah23 has several albums which reference the number 23, such as Neophyte Phenotype, Rock Paper Scissors, and Upside Down Bluejay, all of which have 23 tracks. His stage name also references the number.
 * "23 Minutes in Brussels", a song by Luna on their album Penthouse.
 * The composer Alban Berg had a particular interest in the number 23, using it to structure several works. Various suggestions have been made as to the reason for this interest: that he took it from the Biorhythms theory of Wilhelm Fliess, in which a 23-day cycle is considered significant, or because he first suffered an asthma attack on 23rd of the month.
 * "23" is a single by Mike Will Made It
 * On the cover of The Beatles' 1969 album Yellow Submarine the number 23 is displayed on the chest of one of the Blue Meanies.
 * Network 23 refers to members of the Spiral Tribe. Sometimes 23 used to discretely mark the spots of a freetekno rave.
 * The number 23 is used a lot throughout the visuals and music by the band Gorillaz, who have even devoted a whole page of their autobiography Rise Of The Ogre to the 23 enigma theory.

Film and television

 * 23 is a German film about Karl Koch.
 * In Jeepers Creepers, the Creeper appears every 23 years for 23 days to feast on human body parts
 * In L: Change the World, the protagonist L signs his own name in the Death Note notebook and somehow knows that he has given himself 23 days to live, revealing a 23-day rule for the maximum number of days a person may live after they are added to the Japanese god of death's Death Note.
 * The 1980s TV series Max Headroom was set at Network 23.
 * In The Big Lebowski, the main characters deliberately use only lane 23 at the bowling alley.
 * In Final Destination 2, 23 is the number of a Route.
 * In The Matrix Reloaded, the Architect tells Neo it is of utmost importance to choose 23 people to repopulate Zion.
 * In the TV series Lost, 23 is one of the 6 reoccurring numbers (4, 8, 15, 16, 23, 42) that appear frequently throughout the show.
 * The Number 23 is a 2007 film starring Jim Carrey about a man who becomes obsessed with the 23 enigma.

Other fields

 * 23 skidoo (phrase) (sometimes 23 skiddoo) is an American slang phrase popularized during the early 20th century. 23 skidoo has been described as "perhaps the first truly national fad expression and one of the most popular fad expressions to appear in the U.S".
 * The 23 enigma, proposed by William S. Burroughs plays a prominent role in the plot of the Illuminatus! Trilogy by Robert Shea and Robert Anton Wilson.
 * The 23, in South Africa, refers to the 23 conscientious objectors who publicly refused to do military service in the Apartheid army in 1987. The following years the number increased to 143 (in 1988) and 771 (in 1989), with Apartheid being dismantled from 1990 onwards.
 * X-23 is a character in the Marvel Universe. She is named for being the 23rd attempt to create a female genetic twin of Wolverine after attempts to create a male clone failed.
 * 23 is the number of times Julius Caesar was stabbed in the Theatre of Pompey.

In sports

 * Each national team competing in the FIFA Women's World Cup is allowed a 23-player squad. This squad size has been in place since 2015.

title: 23 (number)


text:
25 (number)

25 (twenty-five) is the natural number following 24 and preceding 26.

In mathematics

It is a square number, being 52 = 5 × 5, and hence the third non-unitary square prime of the form p2.

It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76.

25 has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in (1 and 0).

It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem.

25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9.

25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number.

25 percent (%) is equal to $1⁄4$.

It is the smallest decimal Friedman number as it can be expressed by its own digits: 52.

It is also a Cullen number and a vertically symmetrical number. 25 is the smallest pseudoprime satisfying the congruence 7n = 7 mod n.

25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate.

According to the Shapiro inequality, 25 is the smallest odd integer n such that there exist x$1$, x$2$, ..., x$n$ such that

 * $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} < \frac{n}{2}$$

where x$n + 1$ = x$1$, x$n + 2$ = x$2$.

Within decimal, one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00, 25, 50, or 75.

There are 25 primes under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

F4, H4 symmetry and lattices Λ24, II25,1
Twenty-five 24-cells with $$\mathrm {F_{4}}$$ symmetry in the fourth dimension can be arranged in two distinct manners, such that

The 24-cell can be further generated using three copies of the 8-cell, where the 24-cell honeycomb is dual to the 16-cell honeycomb (with the tesseract the dual polytope to the 16-cell).

On the other hand, the positive unimodular lattice $$\mathrm {II_{25,1}}$$ in twenty-six dimensions is constructed from the Leech lattice in twenty-four dimensions using Weyl vector

 * $$(0,1,2,3,4,\ldots,24|70)$$

that features the only non-trivial solution, i.e. aside from $$\{0, 1\}$$, to the cannonball problem where sum of the squares of the first twenty-five natural numbers $$\{0, 1, 2,\ldots,24\}$$ in $$\mathbb {N_{0}}$$ is in equivalence with the square of $$70$$ (that is the fiftieth composite). The Leech lattice, meanwhile, is constructed in multiple ways, one of which is through copies of the $$\mathbb {E_{8}}$$ lattice in eight dimensions isomorphic to the 600-cell, where twenty-five 24-cells fit; a set of these twenty-five integers can also generate the twenty-fourth triangular number, whose value twice over is $$600 = 24 \times 25.$$

In science

 * The Standard Model of physics features a total of 25 elementary particles: 12 fermions (made of 6 quarks and 6 leptons) and 13 bosons (made of 12 gauge bosons and 1 scalar boson).
 * The atomic number of manganese.
 * The average percentage DNA overlap of an individual with their half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin (offspring of identical twins), or double cousin.

In religion

 * In Ezekiel's vision of a new temple: The number twenty-five is of cardinal importance in Ezekiel's Temple Vision (in the Bible, Ezekiel chapters 40–48).
 * In Islam, there are 25 prophets mentioned in the Quran.

In sports

 * Before 2020, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expanded their roster to 40 players.
 * The size of the playing roster on a Nippon Professional Baseball team for a particular game. Active NPB rosters consist of 28 players, but prior to each game, managers must designate three players who will be ineligible for that game.
 * In baseball, the number 25 is typically reserved for the best slugger on the team. Examples include Mark McGwire, Barry Bonds, Jim Thome, and Mark Teixeira.
 * The number of points needed to win a set in volleyball under rally scoring rules (except for the fifth set), so long as the losing team's score is two less than the winning team's score (i.e., if the winning team scores 25 points, the losing team can have no more than 23 points)
 * In U.S. college football, schools that are members of NCAA Division I FBS are allowed to provide athletic scholarships to a maximum of 25 new football players (i.e., players who were not previously receiving scholarships) each season.

In other fields
Twenty-five is:
 * The number of words in the American game show 25 Words or Less
 * Part of the name of the LSD-25 molecule.
 * The number of years of marriage marked in a silver wedding anniversary.
 * The minimum age of candidates for election to the United States House of Representatives. "Under 25" provides a common cut-off point for designating youth.
 * The number of cents in a quarter.
 * The usual TCP port for SMTP.
 * The name of the national board game of India (Pachisi — Hindi for 25).
 * The name of an Irish card game, see Spoil Five.
 * The per-second frame rate of the PAL video standard.
 * The name of Adele's 2015 album 25, named for her age in that year.
 * The number of the French department Doubs.
 * The temperature, in Celsius, at which some characteristics of semiconductors are specifically given in a datasheet.
 * The number of panels on the weekly TV Asahi quiz show Panel Quiz Attack 25.

Slang names

 * Pony (British slang for £25)

title: 25 (number)


text:
26 (number)

26 (twenty-six) is the natural number following 25 and preceding 27.

In mathematics

 * 26 is the seventh discrete semiprime ($$2 \times 13$$) and the fifth with 2 as the lowest non-unitary factor thus of the form (2.q), where q is a higher prime.
 * 26 is the smallest even number n such that both n + 1 and n − 1 are composite.
 * With an aliquot sum of 16, within an aliquot sequence of five composite numbers (26,16,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
 * 26 is the only integer that is one greater than a square (5$2$ + 1) and one less than a cube (3$3$ − 1).
 * 26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections.
 * There are 26 sporadic groups.
 * The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a simple way as a subquotient.
 * 26 is the smallest number that is both a nontotient and a noncototient number.
 * There are 26 faces of a rhombicuboctahedron.
 * When a 3 × 3 × 3 cube is made of 27 unit cubes (e.g. Rubik's Cube), 26 of them are viewable as the exterior layer. Related to this, 26 is the total of directions from the center of a cube when adding the following: 6 flat faces + 12 edges + 8 vertices.
 * A 26-sided polygon is called an icosihexagon.
 * φ(26) = φ(σ(26)).

Properties of its positional representation in certain radixes
Twenty-six is a repdigit in base three (2223) and in base 12 (2212).

In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome.

In science

 * The atomic number of iron.
 * The number of spacetime dimensions in bosonic string theory.

Astronomy

 * Messier object M26, a magnitude 9.5 open cluster in the constellation Scutum.
 * The New General Catalogue object NGC 26, a spiral galaxy in the constellation Pegasus.

In religion

 * 26 is the gematric number, being the sum of the Hebrew characters (יהוה) being the name of the god of Israel – YHWH (Yahweh).
 * GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26).
 * The Greek Strongs number G26 is "Agape", which means "Love".
 * The expression "For His mercy endures forever" is found verbatim in English and the original Hebrew 26 times in Psalm 136. The expression is found once in each of the 26 verses.
 * According to Jewish chronology, God gave the Torah in the 26th generation since Creation.

In sports

 * The number of complete miles in a marathon (26 miles and 385 yards).
 * Each national team competing in the FIFA World Cup is allowed a 26-player squad. This squad size has been in place since 2022.
 * Effective in 2020, the size of the full roster on a Major League Baseball team, except for regular-season games on or after September 1, when teams expand their rosters to 28 players.
 * The number of holes in a floorball ball.
 * The maximum capacity of cars allowed to compete in an official Formula One Grand Prix race.

In other fields
Twenty-six is:
 * A 2003 novel by Leo McKay, Jr. The title refers to the number of miners killed in the Westray Mine explosion.

26 is:
 * The sixth track of After Laughter, the fifth studio album by American alternative band Paramore.
 * The eighth track of The Balcony, the first studio album by British modern rock band Catfish and the Bottlemen.
 * In Microsoft Windows list of Alt Codes, the combination +26 yields the "right arrow" symbol →.
 * The number of the last letter of the English alphabet, Z.
 * The name of the Australian alternative rock band, 26.
 * The number of letters in the basic Latin alphabet, if capital letters are not distinguished from lowercase letters.
 * The total number of cases on most versions of Deal or No Deal.
 * Often (in the United States) the number of episodes in a television program each year; this allows one new show per week for half the year, and one rerun per week for the rest of the year.
 * The alias of punk rock singer Doc Corbin Dart.
 * The age at which males could no longer be drafted in the United States.
 * A dice game popular in the Midwest United States from the 1930s to 1950s; players had to roll a chosen number 26 times or more, exactly 13 or fewer than 10.
 * In a normal deck of cards, there are 26 red cards and 26 black cards.
 * The number of Cantons of Switzerland.
 * The number of bones in the normal human foot and ankle.
 * XXVI Holdings, Inc, named after the Roman numerals for the number 26, is a holding company that is part of Alphabet, Inc., the structure that ultimately owns Google.
 * The number of counties in the Republic of Ireland.

title: 26 (number)