Pentagonal tiling
Bagina (2011) showed that there are only eight edge-to-edge convex types, a result obtained independently by Sugimoto (2012).
Michaël Rao of the École normale supérieure de Lyon claimed in May 2017 to have found the proof that there are in fact no convex pentagons that tile beyond these 15 types.
[3] As of 11 July 2017, the first half of Rao's proof had been independently verified (computer code available[4]) by Thomas Hales, a professor of mathematics at the University of Pittsburgh.
These freedoms include variations of internal angles and edge lengths.
Periodic tilings are characterised by their wallpaper group symmetry, for example p2 (2222) is defined by four 2-fold gyration points.
B. Grünbaum and G. C. Shephard have shown that there are exactly twenty-four distinct "types" of isohedral tilings of the plane by pentagons according to their classification scheme.
The wallpaper group symmetry for each tiling is given, with orbifold notation in parentheses.
A second lower symmetry group is given if tile chirality exists, where mirror images are considered distinct.
If mirror image tiles (yellow and green) are considered distinct, the symmetry is p2 (2222).
Kershner (1968) found three more types of pentagonal tile, bringing the total to eight.
He claimed incorrectly that this was the complete list of pentagons that can tile the plane.
The pgg symmetry is reduced to p2 when chiral pairs are considered distinct.
In 1975 Richard E. James III found a ninth type, after reading about Kershner's results in Martin Gardner's "Mathematical Games" column in Scientific American magazine of July 1975 (reprinted in Gardner (1988)).
Marjorie Rice, an amateur mathematician, discovered four new types of tessellating pentagons in 1976 and 1977.
The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct.
University of Washington Bothell mathematicians Casey Mann, Jennifer McLoud-Mann, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a computer algorithm.
The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct.
In July 2017 Michaël Rao completed a computer-assisted proof showing that there are no other types of convex pentagons that can tile the plane.
The complete list of convex polygons that can tile the plane includes the above 15 pentagons, three types of hexagons, and all quadrilaterals and triangles.
Nonperiodic monohedral pentagonal tilings can also be constructed, like the example below with 6-fold rotational symmetry by Michael Hirschhorn.
They represent special higher symmetry cases of the 15 monohedral tilings above.
By extension of this relation, a plane can be tessellated by a single pentagonal prototile shape in ways that generate hexagonal overlays.
For example: With pentagons that are not required to be convex, additional types of tiling are possible.
In the hyperbolic plane, one can construct regular pentagons that have any interior angle
There are an infinite number of dual uniform tilings in hyperbolic plane with isogonal irregular pentagonal faces.
