24 санының математикалық және геометриялық қасиеттері

24 (twenty four) is the natural number following 23 and preceding 25.

24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2q, 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.

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 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: n+3(Sn) = Z/24Z for all n ≥ 5.

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.

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.

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.