natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
SI prefix
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
for 1024 is
yotta
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
(Y), and for 10−24 (i.e., the reciprocal of 1024)
yocto
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
(y). These numbers are the largest and smallest number to receive an SI prefix to date.
In mathematics
24 is an
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
** Even language, a language spoken by the Evens
* Odd and Even, a solitaire game w ...
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. It is the first number of the form 2''q'', where ''q'' is an
odd prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. It is the smallest number with exactly eight positive
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
highly composite number
__FORCETOC__
A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...
, having more divisors than any smaller number. Furthermore, it is an
abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The ...
, since the sum of its
proper divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
superabundant number In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number ''n'' is called superabundant precisely when, for all ''m'' < ''n''
:\frac 6/5.
Superabundant numbers were defined by . ...
.
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 = × 24
*24 is a
semiperfect number
In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number.
...
, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
*24 is a
practical number
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 ...
, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24.
*24 is a
Harshad number
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base.
Harshad numbers in base are also known as -harshad (or -Niven) numbers.
Harshad numbers ...
, since it is divisible by the sum of its digits in
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
.
*24 is a
highly totient number A highly totient number k is an integer that has more solutions to the equation \phi(x) = k, where \phi is Euler's totient function, than any integer below it. The first few highly totient numbers are
1, 2, 4, 8, 12, 24, 48, 72, 144, 240, ...
, as there are 10 solutions to the equation ''φ''(''x'') = 24, which is more than any integer below 24.
144 144 may refer to:
* 144 (number), the natural number following 143 and preceding 145
* AD 144, a year of the Julian calendar, in the second century AD
* 144 BC, a year of the pre-Julian Roman calendar
* 144 (film), ''144'' (film), a 2015 Indian com ...
(the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
idoneal number In mathematics, Leonhard Euler, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''2 ± ''Dy''2 (where ''x''2 is relativel ...
, and a
tribonacci number In mathematics, the Fibonacci numbers form a sequence defined recursion, recursively by:
:F_n =
\begin
0 & n = 0 \\
1 & n = 1 \\
F_ + F_ & n > 1
\end
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibo ...
.
*24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal ( 5).
*24 is a
compositorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
, as it is the product of composite numbers up to 6.
*24 is a pernicious number, since its
Hamming weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string o ...
in its
binary representation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).
The base-2 numeral system is a positional notation ...
(11000) is prime (2).
*24 is the third
nonagonal number A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular number, triangular and square numbers to the nonagon (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns inv ...
.
*24 is a
congruent number
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of (integer) c ...
, as 24 is the area of a
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
with a
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
number of sides.
*24 is a
semi-meandric number In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.
Meander
Given a fixed oriented li ...
, where an order-6 semi-meander intersects an oriented ray in R2 at 24 points.
*Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
; 24 is the largest number with this property.
*24 is the largest
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
that is divisible by all
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
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
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
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
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be ar ...
; 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
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
Z/''n''Z is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the additive group (Z/2Z)3. This fact plays a role in
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979.
...
.
*:It follows that any number ''n'' relatively prime to 24 (that is, any number of the form 6''K'' ± 1), and in particular any prime ''n'' greater than 3, has the property that ''n''2 – 1 is divisible by 24.
*The
modular discriminant
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
is proportional to the 24th power of the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
: .
In geometry
*24 degrees is the measure of the
central angle
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
and
external angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
of a
pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
Regular pentadecagon
A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol .
A Regular polygon, regular pentadecagon has interior angl ...
.
*An
icositetragon
In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
Regular icositetragon
The '' regular icositetragon'' is represented by Schläfli symbol ...
is a
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
and
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
.
*24 is the Euler characteristic of a
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
: a general elliptic K3 surface has exactly 24 singular fibers.
*24 is the order of the
octahedral group
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
— the group of rotations of the regular octahedron and the group of rotations of the cube. The
binary octahedral group In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group ''O'' or (2, ...
is a subgroup of the 3-sphere ''S''3 consisting of the 24 elements of the binary tetrahedral group along with the 24 elements contained in its coset . These two cosets each form the vertices of a self-dual
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
, 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
Catalan may refer to:
Catalonia
From, or related to Catalonia:
* Catalan language, a Romance language
* Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia
Places
* 13178 Catalan, asteroid #1 ...
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
and
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...
,
truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
,
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, and
snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
, as well as 24
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
in a
deltoidal icositetrahedron
In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icosit ...
triakis octahedron
In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
It can be seen as an octahedron with triangular ...
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
, is the first
stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
uniform star polyhedra
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...
great disnub dirhombidodecahedron
In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron.
It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antipr ...
, also called ''Skilling's figure,'' is a degenerate uniform star polyhedron with a
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of 24, when pairs of coinciding edges are considered to be single edges.
*:Finally, 6
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
elongated square gyrobicupola
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same p ...
(J37), has 24 vertices.
*The
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
has 24 two-dimensional faces (which are all
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
, with 24 octahedral cells and 24 vertices, is a
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
convex regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regu ...
. It possesses 576 (24×24)
rotational symmetries
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
and 1152
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
altogether. It tiles 4-dimensional space in a
24-cell honeycomb
In Four-dimensional space, four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular polytope, regular space-filling tessellation (or honeycomb (geometry), honeycomb) of 4-dimensional Euclidean space by ...
, 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
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
, they are precisely the elements of the subring generated by the
binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...
as represented by the set of 24 quaternions in the D4 lattice. Known as the ring of Hurwitz integral quaternions, this set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere ''S''3 of radius one centered as the origin. ''S''3 is the Lie group ''
Sp(1)
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
'' of unit quaternions (isomorphic to the Lie groups ''
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
'' and ''
Spin(3)
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a Li ...
''), and so the binary tetrahedral group — of order 24 — is a subgroup of ''S''3.
*:The 24 vertices of the 24-cell are contained in the
regular complex polygon
In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real ...
44, 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,
runcination
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.
It is a higher order truncatio ...
s, and
omnitruncation
In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed.
It is a ''shortc ...
s of the 24-cell yield polychora whose
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
s are 24-sided icositetragons; i.e., within the
truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24- ...
,
runcinated 24-cell
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell.
There are 3 unique degrees of runcinations of the 24-cell including with permutations truncati ...
, and
omnitruncated 24-cell
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell.
There are 3 unique degrees of runcinations of the 24-cell including with permutations truncati ...
, amongst others.
*24 is the
kissing number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of ...
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
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
).
*The
Barnes–Wall lattice In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech ...
contains 24
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
s.
*In 24 dimensions there are 24 even positive definite
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...
s, called the
Niemeier lattice
In mathematics, a Niemeier lattice is one of the 24
positive definite even unimodular lattices of rank 24,
which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such latt ...
s. One of these is the exceptional
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
which has many surprising properties; due to its existence, the answers to many problems such as the
kissing number problem
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of ...
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
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
and the
Steiner system
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
''S''(5,8,24) and the
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
''M''24. (One construction of the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
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
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
: ''n''+3(''S''''n'') = Z/24Z for all ''n'' ≥ 5.
In science
* The
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
of
chromium
Chromium is a chemical element with the symbol Cr and atomic number 24. It is the first element in group 6. It is a steely-grey, lustrous, hard, and brittle transition metal.
Chromium metal is valued for its high corrosion resistance and hardne ...
.
* The average number of hours in a day (on
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
), also known as a
mean solar day
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial ti ...
.
* 24! is an approximation (exceeding by just over 3%) of the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
apocalyptic literature
Apocalyptic literature is a genre of prophetical writing that developed in post- Exilic Jewish culture and was popular among millennialist early Christians. '' Apocalypse'' ( grc, , }) is a Greek word meaning "revelation", "an unveiling or unf ...
it represents the complete Church, being the sum of the 12
tribes of Israel
The Twelve Tribes of Israel ( he, שִׁבְטֵי־יִשְׂרָאֵל, translit=Šīḇṭēy Yīsrāʾēl, lit=Tribes of Israel) are, according to Hebrew Bible, Hebrew scriptures, the descendants of the biblical Patriarchs (Bible), patriarch ...
and the 12 Apostles of the Lamb of God. For example, in ''
The Book of Revelation
The Book of Revelation is the final book of the New Testament (and consequently the final book of the Christian Bible). Its title is derived from the first word of the Koine Greek text: , meaning "unveiling" or "revelation". The Book of R ...
'': "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, a ''Tirthankara'' (Sanskrit: '; English language, English: literally a 'Ford (crossing), ford-maker') is a saviour and spiritual teacher of the ''Dharma (Jainism), dharma'' (righteous path). The word ''tirthankara'' signifies the ...
in
Jainism
Jainism ( ), also known as Jain Dharma, is an Indian religions, Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current ...
.
*Number of spokes in the
Ashok Chakra
Ashoka (, ; also ''Asoka''; 304 – 232 BCE), popularly known as Ashoka the Great, was the third emperor of the Maurya Empire of Indian subcontinent during to 232 BCE. His empire covered a large part of the Indian subcontinent, s ...
.
In music
*There are a total of 24 major and minor keys in Western
tonal music
Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is cal ...
, not counting
enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
equivalents. Therefore, for collections of pieces written in each key, the number of pieces in such a collection; e.g., Chopin's 24 Preludes.
association football
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
:
** The
FIFA World Cup
The FIFA World Cup, often simply called the World Cup, is an international association football competition contested by the senior men's national teams of the members of the ' ( FIFA), the sport's global governing body. The tournament ha ...
final tournament featured 24 men's national teams from 1982 to 1994.
** The
FIFA Women's World Cup
The FIFA Women's World Cup is an international association football competition contested by the senior women's national teams of the members of Fédération Internationale de Football Association ( FIFA), the sport's international governing bo ...
final tournament featured 24 national teams in 2015 and 2019.
* In
basketball
Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular Basketball court, court, compete with the primary objective of #Shooting, shooting a basketball (ball), basketball (appr ...
:
** In the
NBA
The National Basketball Association (NBA) is a professional basketball league in North America. The league is composed of 30 teams (29 in the United States and 1 in Canada) and is one of the major professional sports leagues in the United St ...
, the time on a
shot clock
A shot clock is a countdown timer used in a variety of games and sports, proving a set amount of time that a team may possess the object of play before attempting to score a goal. Shot clocks are used in several sports including basketball, wat ...
is 24 seconds.
In other fields
24 is also:
* The number of
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s a computer needs to represent
24-bit color
In computer architecture, 4-bit integers, or other data units are those that are 4 bits wide. Also, 4-bit central processing unit (CPU) and arithmetic logic unit (ALU) architectures are those that are based on registers, or data buses of that siz ...
images (for a maximum of 16,777,216 colours—but greater numbers of bits provide more accurate colors).
* The number of
karat
The fineness of a precious metal object (coin, bar, jewelry, etc.) represents the weight of ''fine metal'' therein, in proportion to the total weight which includes alloying base metals and any impurities. Alloy metals are added to increase hardne ...
s representing 100% pure
gold
Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile met ...
.
* The number of cycles in the Chinese solar year.
* The number of years from the start of the
Cold War
The Cold War is a term commonly used to refer to a period of geopolitical tension between the United States and the Soviet Union and their respective allies, the Western Bloc and the Eastern Bloc. The term '' cold war'' is used because the ...
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
Persistence of vision traditionally refers to the optical illusion that occurs when visual perception of an object does not cease for some time after the rays of light proceeding from it have ceased to enter the eye.
The illusion has also been d ...
.
* The number of letters in both the modern and classical
Greek alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as we ...
. For the latter reason, also the number of chapters or "books" into which
Homer
Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the ''Iliad'' and the ''Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of the ...
's ''
Odyssey
The ''Odyssey'' (; grc, Ὀδύσσεια, Odýsseia, ) is one of two major Ancient Greek literature, ancient Greek Epic poetry, epic poems attributed to Homer. It is one of the oldest extant works of literature still widely read by moder ...
'' and ''
Iliad
The ''Iliad'' (; grc, Ἰλιάς, Iliás, ; "a poem about Ilium") is one of two major ancient Greek epic poems attributed to Homer. It is one of the oldest extant works of literature still widely read by modern audiences. As with the ''Odysse ...
'' came to be divided.
* The number of runes in the
Elder Futhark
The Elder Futhark (or Fuþark), also known as the Older Futhark, Old Futhark, or Germanic Futhark, is the oldest form of the runic alphabets. It was a writing system used by Germanic peoples for Northwest Germanic dialects in the Migration Peri ...
.
* The number of points on a
backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...
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 Most Noble Order of the Garter is an order of chivalry founded by Edward III of England in 1348. It is the most senior order of knighthood in the British honours system, outranked in precedence only by the Victoria Cross and the George C ...
.
* The number of the French department
Dordogne
Dordogne ( , or ; ; oc, Dordonha ) is a large rural department in Southwestern France, with its prefecture in Périgueux. Located in the Nouvelle-Aquitaine region roughly half-way between the Loire Valley and the Pyrenees, it is named af ...
.
* Four and twenty is the number of blackbirds baked in a pie in the traditional English nursery rhyme "
Sing a Song of Sixpence
"Sing a Song of Sixpence" is an English nursery rhyme, perhaps originating in the 18th century. It is listed in the Roud Folk Song Index as number 13191. The sixpence in the rhyme is a British coin that was first minted in 1551.
Origins
The r ...