5

A pentagram, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio, ${\displaystyle \varphi }$. Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. Generally, star polytopes that are regular only exist in dimensions 2 ≤ ${\displaystyle n}$ < 5.

In graph theory, all graphs with 4 or fewer vertices are planar, however, there is a graph with 5 vertices that is not: K₅, the complete graph with 5 vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram.[19] A similar graph is the Petersen graph, which is strongly connected and also nonplanar. It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5.[20][21] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[22] The automorphism group of the Petersen graph is the symmetric group ${\displaystyle \mathrm {S} _{5}}$ of order 5! = 120, equal to the sum of the first 15 non-zero positive integers and 15th triangular number, which in turn is the sum of the first 5 non-zero positive integers, and 5th triangular number.

The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[23]

Illustration by Leonardo da Vinci of a regular dodecahedron, from Luca Pacioli's Divina proportione.

There are ${\displaystyle 5}$ Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[24] The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. There are also ${\displaystyle 5}$:

☆ Regular polyhedron compounds: the stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.[25] Icosahedral symmetry ${\displaystyle \mathrm {I} _{h}}$ is isomorphic to the alternating group on 5 letters ${\displaystyle \mathrm {A} _{5}}$ of order 120, realized by actions on these uniform polyhedron compounds.

☆ Space-filling convex polyhedra: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.[26] While the cube is the only Platonic solid that can tessellate space on its own, the truncated octahedron and the gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can also tessellate space with their own copies.

☆ Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron and elongated dodecahedron, and the hexagonal prism and truncated octahedron.[27] The cube is a special case of a parallelepiped, with the rhombic dodecahedron the only Catalan solid to tessellate space on its own.

☆ Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.[28] They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane.

The 5-cell, or pentatope, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry ${\displaystyle \mathrm {A} _{4}}$ = 120 = 5! and ${\displaystyle \mathrm {S} _{5}}$ group structure, as with the Petersen graph. Its Petrie polygon is a regular pentagon, and its orthogonal projection is equivalent to the complete graph K₅. It is one of six regular 4-polytopes, made of 31 elements: 5 vertices, 10 edges, 10 faces, 5 tetrahedral cells and 1 4-face. 120 5-cells fit inside a 120-cell, the dual polychoron to the 600-cell, both regular 4-polytopes. Also, 5 24-cells fit inside a small stellated 120-cell, the first stellation of the 120-cell. A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only uniform nonconvex duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian product {5}⊗{5/3} and made of 10 pentagrammic crossed antiprisms, 50 tetrahedra, and 10 pentagonal antiprisms. In all, the fourth dimension contains 5 Coxeter groups with a finite number of uniform polychora; four groups are based on regular 4-polytopes, with one general group of 4-prisms of Platonic and Archimedean solids. There are also a total of 5 Coxeter groups that generate non-prismatic Eucledian honeycombs in 4-space, alongside 5 compact hyperbolic Coxeter groups that generate 5 regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb, which has 5 cells around each face.

The 5-simplex is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of ${\displaystyle n}$-simplexes in any ${\displaystyle n}$ dimensions. The 5-simplex has the Coxeter group ${\displaystyle \mathrm {A} _{5}}$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group ${\displaystyle \mathrm {S} _{6}}$, the only finite symmetric group which has an outer automorphism. The 5-cube, made of 10 tesseracts and the 5-cell as its vertex figure, is also regular and one of 31 uniform 5-polytopes under the Coxeter ${\displaystyle \mathrm {B} _{5}}$ hypercubic group. The demipenteract, with 120 cells, is the only fifth-dimensional semiregular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. Analogues of three-dimensional icosahedral symmetry ${\displaystyle \mathrm {I} _{h}}$ (or two-dimensional pentagonal dihedral ${\displaystyle \mathrm {D} _{5}}$ symmetry, and four-dimensional ${\displaystyle \mathrm {H} _{4}}$ symmetry) do not exist in dimensions n ≥ 5; however, there is the uniform polychoral prismatic group ${\displaystyle \mathrm {H} _{4}}$ × ${\displaystyle \mathrm {A} _{1}}$ in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have ${\displaystyle \mathrm {H} _{4}}$ symmetry. Similarly, compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. There are five regular paracompact honeycombs in 5-space, all with infinite facets and vertex figures.[29]

The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as ${\displaystyle \mathrm {M} _{n}}$ multiply transitive permutation groups on ${\displaystyle n}$ objects, with ${\displaystyle n}$ ∈ {11, 12, 22, 23, 24}. In particular, ${\displaystyle \mathrm {M} _{11}}$, the smallest of all sporadic groups, has a rank 3 action on 55 points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with 3 elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with n elements.[30] Of precisely 5 different conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {M} _{11}}$, one is the almost simple symmetric group ${\displaystyle \mathrm {S} _{5}}$ (of order 5!), and another is ${\displaystyle \mathrm {M} _{10}}$, also almost simple, that functions as a point stabilizer which has 5 as its largest prime factor in its group order: 2⁴·3²·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas ${\displaystyle \mathrm {M} _{11}}$ is sharply 4-transitive, ${\displaystyle \mathrm {M} _{12}}$ is sharply 5-transitive and ${\displaystyle \mathrm {M} _{24}}$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. ${\displaystyle \mathrm {M} _{22}}$ has the first five prime numbers as its distinct prime factors in its order of 2⁷·3²·5·7·11, and is the smallest of 5 sporadic groups with 5 distinct prime factors in their order. All Mathieu groups are subgroups of ${\displaystyle \mathrm {M} _{24}}$, which under the Witt design ${\displaystyle \mathrm {W} _{24}}$ of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code ${\displaystyle \mathrm {B} _{24}}$ that has ${\displaystyle \mathrm {M} _{24}}$ as its automorphism group. ${\displaystyle \mathrm {W} _{24}}$ generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of 5 different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional Leech lattice Λ₂₄, which is the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group ${\displaystyle \mathrm {Co} _{0}}$.

There are five solvable groups that are excluded from finite simple groups of Lie type, and more importantly, five exceptional Lie groups: ${\displaystyle {\mathfrak {g}}_{2}}$, ${\displaystyle {\mathfrak {f}}_{4}}$, ${\displaystyle {\mathfrak {e}}_{6}}$, ${\displaystyle {\mathfrak {e}}_{7}}$, and ${\displaystyle {\mathfrak {e}}_{8}}$.

There are five non-supersingular primes: 37, 43, 53, 61, and 67, all smaller than the largest supersingular prime divisor of the friendly giant, 71.[31] All primes greater than 71 are non-supersingular.

5 is the only prime number to end in the digit 5 in decimal because all other numbers written with a 5 in the ones place are multiples of five, which makes it a 1-automorphic number. All multiples of 5 will end in either 5 or 0, and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base. In the powers of 5, every power ends with the number five, and from 5³ onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6. A number ${\displaystyle n}$ raised to the fifth power always ends in the same digit as ${\displaystyle n}$.

• Messier object M5, a magnitude 7.0 globular cluster in the constellation Serpens.[37]
• The New General Catalogue object NGC 5, a magnitude 13 spiral galaxy in the constellation Andromeda.[38]
• The Roman numeral V stands for dwarfs (main sequence stars) in the Yerkes spectral classification scheme.
• The Roman numeral V (usually) stands for the fifth-discovered satellite of a planet or minor planet (e.g. Jupiter V).
• There are five Lagrangian points in a two-body system.

• There are generally considered to be five senses.
• The five basic tastes are sweet, salty, sour, bitter, and umami.[39]
• Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[40]

• 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[41]

• The god Shiva has five faces[42] and his mantra is also called panchakshari (five-worded) mantra.
• The goddess Saraswati, goddess of knowledge and intellectual is associated with panchami or the number 5.
• There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space respectively).
• The most sacred tree in Hinduism has 5 leaves in every leaf stunt.
• Most of the flowers have 5 petals in them.
• The epic Mahabharata revolves around the battle between Duryodhana and his 99 other brothers and the 5 pandava princes—Dharma, Arjuna, Bhima, Nakula and Sahadeva.

• There are traditionally five wounds of Jesus Christ in Christianity: the Scourging at the Pillar, the Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Side Wound of Christ.[43]

• The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
• Five Seals in Sethianism
• Five Trees in the Gospel of Thomas

• The Five Pillars of Islam[44]
• Muslims pray to Allah five times a day[45]
• According to Shia Muslims, the Panjetan or the Five Holy Purified Ones are the members of Muhammad's family: Muhammad, Ali, Fatimah, Hasan, and Husayn and are often symbolically represented by an image of the Khamsa.[46]

• The Torah contains five books—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth").[47]
• The book of Psalms is arranged into five books, paralleling the Five Books of Moses.[48]
• The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.[49]

• The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as panj kakars or the "Five Ks" because they start with letter K representing kakka (ਕ) in the Punjabi language's Gurmukhi script. They are: kesh (unshorn hair), kangha (the comb), kara (the steel bracelet), kachhehra (the soldier's shorts), and kirpan (the sword) (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ).[50] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).

• 5 Elements[51]
• 5 Emperors[52]

• According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca.
• The pentagram, or five-pointed star, bears religious significance in various faiths including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
• In Cantonese, "five" sounds like the word "not" (character: 唔). When five appears in front of a lucky number, e.g. "58", the result is considered unlucky.
• In East Asian tradition, there are five elements: (water, fire, earth, wood, and metal).[53] The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye.[54] Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday.
• In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance.
• Members of The Nation of Gods and Earths, a primarily African American religious organization, call themselves the "Five-Percenters" because they believe that only 5% of mankind is truly enlightened.[55]

• James the Red Engine, a fictional character numbered 5.[56]
• Johnny 5 is the lead character in the film Short Circuit (1986)[57]
• Number Five is a character in Lorien Legacies[58]
• Sankara Stones, five magical rocks in Indiana Jones and the Temple of Doom that are sought by the Thuggees for evil purposes[59]
• The Mach Five Mahha-gō? (マッハ号), the racing car Speed Racer (Go Mifune in the Japanese version) drives in the anime series of the same name (known as "Mach Go! Go! Go!" in Japan)
• In the works of J. R. R. Tolkien, five wizards (Saruman, Gandalf, Radagast, Alatar and Pallando) are sent to Middle-earth to aid against the threat of the Dark Lord Sauron[60]
• In the A Song of Ice and Fire series, the War of the Five Kings is fought between different claimants to the Iron Throne of Westeros, as well as to the thrones of the individual regions of Westeros (Joffrey Baratheon, Stannis Baratheon, Renly Baratheon, Robb Stark and Balon Greyjoy)[61]
• In The Wheel of Time series, the "Emond's Field Five" are a group of five of the series' main characters who all come from the village of Emond's Field (Rand al'Thor, Matrim Cauthon, Perrin Aybara, Egwene al'Vere and Nynaeve al'Meara)
• Myst uses the number 5 as a unique base counting system. In The Myst Reader series, it is further explained that the number 5 is considered a holy number in the fictional D'ni society.
• Number Five is also a character in The Umbrella Academy comic book and TV series adaptation[62]

• Towards the end of the film Monty Python and the Holy Grail (1975), the character of King Arthur repeatedly confuses the number five with the number three.
• Five Go Mad in Dorset (1982) was the first of the long-running series of The Comic Strip Presents... television comedy films[63]
• The Fifth Element (1997), a science fiction film[64]
• Fast Five (2011), the fifth installment of the Fast and Furious film series.[65]
• V for Vendetta (2005), produced by Warner Bros., directed by James McTeigue, and adapted from Alan Moore's graphic novel V for Vendetta prominently features number 5 and Roman Numeral V; the story is based on the historical event in which a group of men attempted to destroy Parliament on November 5, 1605[66]

• Five (group), a UK Boy band[67]
• The Five (composers), 19th-century Russian composers[68]
• 5 Seconds of Summer, pop band that originated in Sydney, Australia
• Five Americans, American rock band active 1965–1969[69]
• Five Finger Death Punch, American heavy metal band from Las Vegas, Nevada. Active 2005–present
• Five Man Electrical Band, Canadian rock group billed (and active) as the Five Man Electrical Band, 1969–1975[70]
• Maroon 5, American pop rock band that originated in Los Angeles, California[71]
• MC5, American punk rock band[72]
• Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas[73]
• The 5th Dimension, American pop vocal group, active 1977–present[74]
• The Dave Clark Five, a.k.a. DC5, an English pop rock group comprising Dave Clark, Lenny Davidson, Rick Huxley, Denis Payton, and Mike Smith; active 1958–1970[75]
• The Jackson 5, American pop rock group featuring various members of the Jackson family; they were billed (and active) as The Jackson 5, 1966–1975[76]
• Hi-5, Australian pop kids group, where it has several international adaptations, and several members throughout the history of the band. It was also a TV show.
• We Five: American folk rock group active 1965–1967 and 1968–1977
• Grandmaster Flash and the Furious Five: American rap group, 1970–80's[77]
• Fifth Harmony, an American girl group.[78]
• Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013[79]
• R5 (band), an American pop and alternative rock group, 2009–2018[80]

• A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[81]
• Modern musical notation uses a musical staff made of five horizontal lines.[82]
• In harmonics – the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
• The number of completed, numbered piano concertos of Ludwig van Beethoven, Sergei Prokofiev, and Camille Saint-Saëns.
• Using the Latin root, five musicians are called a quintet.[83]
• A scale with five notes per octave is called a pentatonic scale.[84]
• Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

Stations

• Channel 5 (UK), a television channel that broadcasts in the United Kingdom[85]
• 5 (TV channel) (formerly known as ABC 5 and TV5) (DWET-TV channel 5 In Metro Manila) a television network in the Philippines.[86]

Series

• Babylon 5, a science fiction television series[87]
• The number 5 features in the television series Battlestar Galactica in regards to the Final Five cylons and the Temple of Five
• Hi-5 (Australian TV series), a television series from Australia[88]
• Hi-5 (UK TV series), a television show from the United Kingdom
• Hi-5 Philippines a television show from the Philippines
• Odyssey 5, a 2002 science fiction television series[89]
• Tillbaka till Vintergatan, a Swedish children's television series featuring a character named "Femman" (meaning five), who can only utter the word 'five'.
• The Five (talk show): Fox News Channel roundtable current events television show, premiered 2011, so-named for its panel of five commentators.
• Yes! PreCure 5 is a 2007 anime series which follows the adventures of Nozomi and her friends. It is also followed by the 2008 sequel Yes! Pretty Cure 5 GoGo!
• The Quintessential Quintuplets is a 2019 slice of life romance anime series which follows the everyday life of five identical quintuplets and their interactions with their tutor. It has two seasons, and a final movie is scheduled in summer 2022.
• Hawaii Five-0, CBS American TV series.[90]

• The Famous Five is a series of children's books by British writer Enid Blyton
• The Power of Five is a series of children's books by British writer and screenwriter Anthony Horowitz
• The Fall of Five is a book written under the collective pseudonym Pittacus Lore in the series Lorien Legacies
• The Book of Five Rings is a text on kenjutsu and the martial arts in general, written by the swordsman Miyamoto Musashi circa 1645
• Slaughterhouse-Five is a book by Kurt Vonnegut about World War II[91]