171 (number)

In mathematics

171 is the 18th triangular number and a Jacobsthal number.

There are 171 transitive relations on three labeled elements, and 171 combinatorially distinct ways of subdividing a cuboid by flat cuts into a mesh of tetrahedra, without adding extra vertices.{{cite journal

The diagonals of a regular decagon meet at 171 points, including both crossings and the vertices of the decagon.

There are 171 faces and edges in the 57-cell, an abstract 4-polytope with hemi-dodecahedral cells that is its own dual polytope.

Within moonshine theory of sporadic groups, the friendly giant \mathbb {M} is defined as having cyclic groups ⟨ m ⟩ that are linked with the function, :f_{m}(\tau) = q^{-1} + a_{1}q + a_{2}q^{2} + ... , \text{ } a_{k} ∈ \mathbb{Z}, \text{ } q = e^{2\pi i \tau}, \text{ } \tau0; where q is the character of \mathbb {M} at m. This generates 171 moonshine groups within \mathbb {M} associated with f_{m} that are principal moduli for different genus zero congruence groups commensurable with the projective linear group \operatorname{PSL_2}(\mathbb{Z}).

References

References

1. {{cite OEIS. A000217. Triangular numbers
2. {{cite OEIS. A001045. Jacobsthal sequence
3. {{cite OEIS. A006905. Number of transitive relations on n labeled nodes
4. {{cite OEIS. A007569. Number of nodes in regular n-gon with all diagonals drawn
5. (2002). "Abstract Regular Polytopes". Cambridge University Press.
6. (2004). "On the Discrete Groups of Moonshine". Proceedings of the American Mathematical Society.