27 (twenty-seven) is the natural number following 26 and preceding 28.

Mathematics

Including the null-motif, there are 27 distinct hypergraph motifs. Clebsch diagonal cubic surface

There are exactly ''twenty-seven straight lines'' on a smooth

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than ...

, which give a basis of the

fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defi ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

\mathrm

. The unique simple formally real

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan a ...

, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of

quaternion 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. The algebra of quater ...

s, is 27-dimensional; its

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

is the 52-dimensional

exceptional Lie algebra In mathematics, an exceptional Lie algebra is a complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system ...

\mathrm .

There are twenty-seven

sporadic group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

s, if the ''non-strict'' group of Lie type

\mathrm

(with an

irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...

that is twice that of

\mathrm

in 104 dimensions) is included. In

Robin's theorem In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...

for the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

, twenty-seven integers fail to hold

\sigma(n) < e^\gamma n \log \log n

for values

n \leq 5040,

where

\gamma

is the

Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...

; this hypothesis is true

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

this inequality holds for every larger

n.

The

Clebsch surface In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surf ...

has 27 exceptional lines can be defined over the real numbers. It is possible to arrange 27 vertices and connect them with edges to create the

Holt graph In graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex-transitive and edge-transitive graph which is not also symmetric. Such graphs are not common. It is named after Peter ...

. 27 is 3³, and therefore, it is the second

tetration In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...

of 3 (²3).

In other fields

*The

27 club The 27 Club is an informal list consisting mostly of popular musicians, often expanded by artists, actors, and other celebrities who died at age 27. Although the claim of a " statistical spike" for the death of musicians at that age has been r ...

refers to the age when many popular music figures died. It connects the age 27 with connotations of an early end.

Notes

References

External links

Prime Curios! 27
from the

Prime Pages The PrimePages is a website about prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...

Mystery of the number 27
- Large collection of 27 related trivia and facts.
The 27 Project
- collection of sightings of 27 in movies, TV, culture and art {{DEFAULTSORT:27 (Number) Integers

Mathematics

In other fields

See also

Notes

References

Further reading

External links