# 24 (number)

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{{Redirect|Number 24|the Norwegian film|Number 24 (film)}}
{{Infobox number
| number =  24
|cardinal=Twenty-four|ordinal=24th|ordinal text=Twenty-fourth| numeral = tetravigesimal
| divisor =  1, 2, 3, 4, 6, 8, 12, 24
}}

'''24''' ('''twenty-four''') is the [natural number](/source/natural_number) following [23](/source/23_(number)) and preceding [25](/source/25_(number)). It is equal to two [dozen](/source/dozen) and one sixth of a [gross](/source/Gross_(unit)).

==In mathematics==
24 is the number of [permutation](/source/permutation)s of four items. Thus it is the order of the [symmetric group](/source/symmetric_group) on four symbols, and is the [factorial](/source/factorial) of 4. This is the full symmetry group of the [regular tetrahedron](/source/regular_tetrahedron), and also the rotational symmetry group of the [cube](/source/cube) (or [octahedron](/source/octahedron)).<ref>{{citation |last=Armstrong |first=M. A. |title=Groups and Symmetry |series=Undergraduate Texts in Mathematics |publisher=Springer-Verlag |location=New York |year=1988 |isbn=978-0-387-96675-5 |doi=10.1007/978-1-4757-4034-9}}, page 108.</ref><ref>{{cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Regular Polytopes |edition=3rd |publisher=Dover Publications |location=New York |year=1973 |isbn=978-0-486-61480-9}}</ref>

24 is an [even](/source/Parity_(mathematics)) [composite number](/source/composite_number), a [highly composite number](/source/highly_composite_number), an [abundant number](/source/abundant_number), a [practical number](/source/practical_number), and a [congruent number](/source/congruent_number). The many ways 24 can be constructed inspired a [children's mathematical game](/source/24_(puzzle)) involving the use of any of the four standard operations on four numbers on a card to get 24.

24 is also part of the only nontrivial solution pair to the [cannonball problem](/source/cannonball_problem), along with [70](/source/70_(number)).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Cannonball Problem |url=https://mathworld.wolfram.com/CannonballProblem.html |access-date=2020-08-19 |website=mathworld.wolfram.com |language=en}}</ref> It is also the [kissing number](/source/kissing_number) in [4-dimensional space](/source/Four-dimensional_space). This fact can be used in the construction of the 24-dimensional [Leech lattice](/source/Leech_lattice). An [icositetragon](/source/icositetragon) is a [regular polygon](/source/regular_polygon) with 24 sides. A [tesseract](/source/tesseract) has 24 two-dimensional [square](/source/square) [faces](/source/Face_(geometry)). 

The [24-cell](/source/24-cell), consisting of 24 octahedra and having 24 vertices, is a special polytope that only exists in four dimensions. The vertices of the 24-cell are the [root vector](/source/root_vector)s of the [<math>D_4</math> root system](/source/D4_(root_system)):
<math display="block">\{\pm e_i \pm e_j : 1\leq i<j\leq 4\}</math>
in four-dimensional Euclidean space. In quaternionic form, the same configuration may be identified with the 24 unit [Hurwitz quaternion](/source/Hurwitz_quaternion)s, which form the [binary tetrahedral group](/source/binary_tetrahedral_group).<ref name="CoxeterRegularPolytopes">{{cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Regular Polytopes |edition=3rd |publisher=Dover Publications |location=New York |year=1973 |isbn=978-0-486-61480-9}}</ref><ref name="HumphreysLie">{{cite book |last=Humphreys |first=James E. |author-link=James E. Humphreys |title=Introduction to Lie Algebras and Representation Theory |series=Graduate Texts in Mathematics |volume=9 |publisher=Springer-Verlag |location=New York |year=1972 |isbn=978-0-387-90053-7}}</ref><ref name="StillwellNaiveLie">{{cite book |last=Stillwell |first=John |author-link=John Stillwell |title=Naive Lie Theory |series=Undergraduate Texts in Mathematics |publisher=Springer |location=New York |year=2008 |isbn=978-0-387-78214-0}}</ref>

The optimal [sphere packing problem](/source/sphere_packing_problem) has been solved in dimension 24, one of the only dimensions where this has been solved (the others being dimensions 1–3, and 8).<ref>{{cite journal |last1=Cohn|first1=Henry|last2=Kumar|first2=Abhinav|last3=Miller|first3=Stephen|last4=Radchenko|first4=Danylo|last5=Viazovska|first5=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 24|url=http://annals.math.princeton.edu/2017/185-3/p08|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=1017–1033|arxiv=1603.06518|doi=10.4007/annals.2017.185.3.8}}</ref>

The number 24 appears prominently in the theory of [modular form](/source/modular_form)s through the [Dedekind eta function](/source/Dedekind_eta_function)<ref>{{cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |title=Modular Functions and Dirichlet Series in Number Theory |series=Graduate Texts in Mathematics |volume=41 |edition=2nd |publisher=Springer-Verlag |location=New York |year=1990 |chapter=The Dedekind eta function |pages=47–73 |isbn=978-0-387-97127-8}}</ref>
<math display="block">\eta(\tau)=q^{1/24}\prod_{n>0}(1-q^n),\qquad q=e^{2\pi i\tau}.</math>
The eta function has weight <math>1/2</math> and transforms with a [multiplier system](/source/multiplier_system) of order 24. Its 24th power is the [modular discriminant](/source/modular_discriminant)
<math display="block">\Delta(\tau)=\eta(\tau)^{24}=q\prod_{n>0}(1-q^n)^{24},</math>
a [cusp form](/source/cusp_form) of weight 12 for <math>\mathrm{SL}_2(\mathbb Z)</math>.

The number 24 also occurs in [coding theory](/source/coding_theory) and finite group theory through the [extended binary Golay code](/source/binary_Golay_code), a self-dual binary linear code with parameters <math>[24,12,8]</math>. The supports of its codewords of weight 8, called ''octads'', form the [Steiner system](/source/Steiner_system) <math>S(5,8,24)</math>, also known as the [Witt design](/source/Witt_design). The automorphism group of the extended binary Golay code, equivalently of this Steiner system, is the [Mathieu group <math>M_{24}</math>](/source/Mathieu_group_M24), one of the [sporadic simple groups](/source/sporadic_simple_groups). The extended Golay code and the Witt design are also used in standard constructions of the [Leech lattice](/source/Leech_lattice).<ref name="HuffmanPless">{{cite book |last1=Huffman |first1=W. Cary |last2=Pless |first2=Vera |title=Fundamentals of Error-Correcting Codes |publisher=Cambridge University Press |year=2003 |chapter=The Golay codes |isbn=978-0-521-78280-7 |doi=10.1017/CBO9780511807077}}</ref><ref name="ConwaySloane">{{cite book |last1=Conway |first1=John H. |author-link1=John Horton Conway |last2=Sloane |first2=N. J. A. |author-link2=Neil Sloane |title=Sphere Packings, Lattices and Groups |series=Grundlehren der mathematischen Wissenschaften |volume=290 |edition=3rd |publisher=Springer |location=New York |year=1999 |chapter=The Golay Codes and the Mathieu Groups |isbn=978-0-387-98585-5 |doi=10.1007/978-1-4757-6568-7}}</ref>

==In religion==
*In Christian [apocalyptic literature](/source/apocalyptic_literature) it represents the complete Church, being the sum of the 12 [tribes of Israel](/source/tribes_of_Israel) and the 12 [Apostles](/source/Apostles_in_the_New_Testament) of the Lamb of God. For example, in ''[The Book of Revelation](/source/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."<ref>{{cite web|url=http://bible.cc/revelation/4-4.htm |title=Revelation 4:4, New International Version (1984) |publisher=Bible.cc |access-date=2013-05-03}}</ref>
*Number of [Tirthankaras](/source/Tirthankaras) in [Jainism](/source/Jainism).
*Number of spokes in the [Ashoka Chakra](/source/Ashoka_Chakra).

==In culture==
*In [Brazil](/source/Brazil), twenty-four is associated with [homosexuality](/source/homosexuality) as it is the number that stands for the [deer](/source/deer) in a game known as “[jogo do bicho](/source/jogo_do_bicho)”.
*There are 24 hours in a [day](/source/day).

==References==
{{reflist}}

==External links==
{{Commons category}}
* [https://math.ucr.edu/home/baez/numbers/index.html#24 My Favorite Numbers: 24], [John C. Baez](/source/John_C._Baez)

{{Integers|zero}}

{{DEFAULTSORT:24 (Number)}}
Category:Integers

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Adapted from the Wikipedia article [24 (number)](https://en.wikipedia.org/wiki/24_(number)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/24_(number)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.