Editing Heptagon (section)

==Tiling and packing==
{{multiple image|align=right
|image1=3.7.42 vertex.png|caption1=Triangle, heptagon, and 42-gon vertex
|image2=Heptagonal tiling.svg|caption2=Hyperbolic heptagon tiling
|total_width=480}}
A regular triangle, heptagon, and 42-gon can completely [[Vertex (geometry)#Of a plane tiling|fill a plane vertex]]. However, there is no tiling of the plane with only these polygons, because there is no way to fit one of them onto the third side of the triangle without leaving a gap or creating an overlap. In the [[Hyperbolic geometry|hyperbolic plane]], tilings by regular heptagons are possible. There are also concave heptagon tilings possible in the Euclidean plane.<ref> Sycamore916, ed. "Heptagon." Polytope Wiki. Last modified November 2023.
     Accessed January 20, 2024. https://polytope.miraheze.org/wiki/Heptagon. </ref>

[[File:2-d heptagon packing.svg|thumb|The densest [[double lattice]] packing of the Euclidean plane by regular heptagons, conjectured to have the lowest maximum packing density of any convex set]]
The regular heptagon has a [[double lattice]] packing of the Euclidean plane of packing density approximately 0.89269. This has been conjectured to be the lowest density possible for the optimal double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set.<ref>{{cite journal
 | last = Kallus | first = Yoav
 | arxiv = 1305.0289
 | doi = 10.2140/gt.2015.19.343
 | issue = 1
 | journal = Geometry & Topology
 | mr = 3318753
 | pages = 343–363
 | title = Pessimal packing shapes
 | volume = 19
 | year = 2015}}</ref>