23 (number)

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 ${\displaystyle \mathbb {C} _{0}}$. The Leech lattice can be constructed in various ways, which include:

• By means of a matrix of the form ${\displaystyle \scriptstyle {\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix}}}$ where ${\displaystyle I}$ is the identity matrix and ${\displaystyle H}$ 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 ${\displaystyle \mathbb {B} _{24}}$ and Witt design ${\displaystyle \mathbb {W} _{24}}$, 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 ${\displaystyle \mathbb {B} _{23}}$, which has codewords of size 23. ${\displaystyle \mathbb {B} _{23}}$ has Mathieu group ${\displaystyle \mathbb {M} _{23}}$ as its automorphism group, which is the second largest member of the first generation in the happy family of sporadic groups. ${\displaystyle \mathbb {M} _{23}}$ 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, ${\displaystyle \mathbb {M} _{23}}$ is the automorphism group of Mathieu group ${\displaystyle \mathbb {M} _{24}}$, which works through ${\displaystyle \mathbb {W} _{24}}$ 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 ${\displaystyle \mathbb {Q} ({\sqrt {-23}})}$, 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.[28]

Twenty-three four-dimensional crystal families exist within the classification of space groups. These are accompanied by six enantiomorphic forms, which maximizes the total count to twenty-nine crystal families.[29] In three dimensions, five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (counting reflections).[30][31]

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 prism, pentagonal prism, hexagonal prism, octagonal prism, and decagonal prism).

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 ${\displaystyle {\tilde {B}}_{4}}$ cubic group, and 23 five-dimensional uniform polytopes are generated from the ${\displaystyle \mathrm {D} _{5}}$ 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.[32] It is also not constructible with origami, however it is through other traditional methods for all regular polygons.[33]

• 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".[43]
• "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.
• "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,[44] or because he first suffered an asthma attack on 23rd of the month.[45]
• "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.

• 23 is a German film about Karl Koch.[46]
• 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.[47]
• 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 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.[48]

• 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.[49]
• 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.