Editing Modular group (section)

===Tessellation of the hyperbolic plane===
[[File:ModularGroup-FundamentalDomain.svg|thumb|400px|right|A typical fundamental domain for the action of {{math|Γ}} on the upper half-plane.]]
The modular group {{math|Γ}} acts on <math display=inline>\mathbb H</math> as a [[discrete group|discrete subgroup]] of <math display=inline> \operatorname{PSL}(2,\mathbb R)</math>, that is, for each {{math|''z''}} in <math display=inline>\mathbb H</math> we can find a neighbourhood of {{math|''z''}} which does not contain any other element of the [[orbit (group theory)|orbit]] of {{math|''z''}}. This also means that we can construct [[fundamental domain]]s, which (roughly) contain exactly one representative from the orbit of every {{math|''z''}} in {{math|'''H'''}}. (Care is needed on the boundary of the domain.)

There are many ways of constructing a fundamental domain, but a common choice is the region

:<math>R = \left\{ z \in \mathbb H \colon \left| z \right| > 1,\, \left| \operatorname{Re}(z) \right| < \tfrac12 \right\}</math>

bounded by the vertical lines {{math|Re(''z'') {{=}} {{sfrac|1|2}}}} and {{math|Re(''z'') {{=}} −{{sfrac|1|2}}}}, and the circle {{math|{{abs|''z''}} {{=}} 1}}. This region is a hyperbolic triangle. It has vertices at {{math|{{sfrac|1|2}} + ''i''{{sfrac|{{radic|3}}|2}}}} and {{math|−{{sfrac|1|2}} + ''i''{{sfrac|{{radic|3}}|2}}}}, where the angle between its sides is {{math|{{sfrac|π|3}}}}, and a third vertex at infinity, where the angle between its sides is&nbsp;0.

There is a strong connection between the modular group and [[elliptic curves]]. Each point <math>z</math> in the upper half-plane gives an elliptic curve, namely the quotient of <math>\mathbb{C}</math> by the lattice generated by 1 and <math>z</math>.  
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group.  Thus, the quotient of the upper half-plane by the action of the modular group is the so-called [[moduli space]] of elliptic curves: a space whose points describe isomorphism classes of elliptic curves.  This is often visualized as the fundamental domain described above, with some points on its boundary identified.

The modular group and its subgroups are also a source of interesting tilings of the hyperbolic plane.  By transforming this fundamental domain in turn by each of the elements of the modular group, a [[tessellation|regular tessellation]] of the hyperbolic plane by congruent hyperbolic triangles known as the V6.6.∞ [[Infinite-order triangular tiling#Related polyhedra and tiling|Infinite-order triangular tiling]] is created.  Note that each such triangle has one vertex either at infinity or on the real axis {{math|Im(''z'') {{=}} 0}}. 

This tiling can be extended to the [[Poincaré_disk_model|Poincaré disk]], where every hyperbolic triangle has one vertex on the boundary of the disk. The tiling of the Poincaré disk is given in a natural way by the [[j-invariant|{{math|''J''}}-invariant]], which is invariant under the modular group, and attains every complex number once in each triangle of these regions.  

This tessellation can be refined slightly, dividing each region into two halves (conventionally colored black and white), by adding an orientation-reversing map; the colors then correspond to orientation of the domain. Adding in {{math|(''x'', ''y'') ↦ (−''x'', ''y'')}} and taking the right half of the region {{math|''R''}} (where {{math|Re(''z'') ≥ 0}}) yields the usual tessellation. This tessellation first appears in print in {{Harv|Klein|1878/79a}},<ref name="lebruyn">{{citation
 | last = Le Bruyn | first = Lieven
 | title = Dedekind or Klein?
 | date = 22 April 2008
 | url = http://www.neverendingbooks.org/dedekind-or-klein
}}</ref> where it is credited to [[Richard Dedekind]], in reference to {{Harv|Dedekind|1877}}.<ref name="lebruyn" /><ref>{{Cite journal
| issn = 0002-9890
| volume = 108
| issue = 1
| pages = 70–76
| last = Stillwell
| first = John
| title = Modular Miracles
| journal = The American Mathematical Monthly| date = January 2001
| jstor = 2695682
| doi = 10.2307/2695682 
}}</ref>

[[File:Morphing of modular tiling to 2 3 7 triangle tiling.gif|thumb|Visualization of the map {{math|(2, 3, ∞) → (2, 3, 7)}} by morphing the associated tilings.<ref name="westendorp">{{cite web |last=Westendorp |first=Gerard |title=Platonic tessellations of Riemann surfaces |url=https://westy31.nl/Geometry/Geometry.html#Modular |website=westy31.nl}}</ref>]]
The map of groups {{math|(2, 3, ∞) → (2, 3, ''n'')}} (from modular group to triangle group) can be visualized in terms of this tiling (yielding a tiling on the modular curve), as depicted in the video at right.

{{Order i-3 tiling table}}