From Wikipedia, the free encyclopedia
Natural number
171 (one hundred [and] seventy-one ) is the
natural number following
170 and preceding
172 .
In mathematics
171 is a
triangular number
[1] and a
Jacobsthal number .
[2]
There are 171
transitive relations on three labeled elements,
[3] and 171 combinatorially distinct ways of subdividing a
cuboid by flat cuts into a mesh of
tetrahedra , without adding extra vertices.
[4]
The diagonals of a regular
decagon meet at 171 points, including both crossings and the vertices of the decagon.
[5]
There are 171
faces and
edges in the
57-cell , an
abstract
4-polytope with hemi-
dodecahedral
cells that is its own
dual polytope .
[6]
Within
moonshine theory of
sporadic groups , the
friendly giant
M
{\displaystyle \mathbb {M} }
is defined as having
cyclic groups ⟨
m
{\displaystyle m}
⟩ that are linked with the function,
f
m
(
τ
)
=
q
−
1
+
a
1
q
+
a
2
q
2
+
.
.
.
,
a
k
{\displaystyle f_{m}(\tau )=q^{-1}+a_{1}q+a_{2}q^{2}+...,{\text{ }}a_{k}}
∈
Z
,
q
=
e
2
π
i
τ
,
τ
>
0
;
{\displaystyle \mathbb {Z} ,{\text{ }}q=e^{2\pi i\tau },{\text{ }}\tau >0;}
where
q
{\displaystyle q}
is the
character of
M
{\displaystyle \mathbb {M} }
at
m
{\displaystyle m}
.
This generates 171 moonshine groups within
M
{\displaystyle \mathbb {M} }
associated with
f
m
{\displaystyle f_{m}}
that are
principal moduli for different
genus zero congruence groups
commensurable with the
projective linear group
P
S
L
2
(
Z
)
{\displaystyle \operatorname {PSL_{2}} (\mathbb {Z} )}
.
[7]
See also
References
^
Sloane, N. J. A. (ed.).
"Sequence A000217 (Triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A001045 (Jacobsthal sequence)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A006905 (Number of transitive relations on n labeled nodes)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Pellerin, Jeanne; Verhetsel, Kilian; Remacle, Jean-François (December 2018). "There are 174 subdivisions of the hexahedron into tetrahedra". ACM Transactions on Graphics . 37 (6): 1–9.
arXiv :
1801.01288 .
doi :
10.1145/3272127.3275037 .
S2CID
54136193 .
^
Sloane, N. J. A. (ed.).
"Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
McMullen, Peter ;
Schulte, Egon (2002).
Abstract Regular Polytopes . Encyclopedia of Mathematics and its Applications. Vol. 92. Cambridge: Cambridge University Press. pp. 185–186, 502.
doi :
10.1017/CBO9780511546686 .
ISBN
0-521-81496-0 .
MR
1965665 .
S2CID
115688843 .
^
Conway, John ;
Mckay, John ; Sebbar, Abdellah (2004).
"On the Discrete Groups of Moonshine" (PDF) . Proceedings of the American Mathematical Society . 132 (8): 2233.
doi :
10.1090/S0002-9939-04-07421-0 .
eISSN
1088-6826 .
JSTOR
4097448 .
S2CID
54828343 .
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