Pentagonal tiling

, discovered in 2015 In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex (or more) and sphere with three pentagons; the latter produces a tiling that is topologically equivalent to the dodecahedron.

, discovered in 2015]] In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex (or more) and sphere with three pentagons; the latter produces a tiling that is topologically equivalent to the dodecahedron. ==== Type 1==== There are many tiling topologies that contain type 1 pentagons. Five example topologies are given below. ==== Type 2==== These type 2 examples are isohedral. The second is an edge-to-edge variation. They both have pgg (22×) symmetry. If mirror image tiles (yellow and green) are considered distinct, the symmetry is p2 (2222). ==== Types 3, 4, and 5==== === James (1975) Type 10=== 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 ). It is indexed as type 10. The tiling is 3-isohedral and non-edge-to-edge. === Rice (1977) Types 9,11,12,13=== Marjorie Rice, an amateur mathematician, discovered four new types of tessellating pentagons in 1976 and 1977. All four tilings are 2-isohedral. The chiral pairs of tiles are colored in yellow and green for one isohedral set, and two shades of blue for the other set. The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct. The tiling by type 9 tiles is edge-to-edge, but the others are not. Each primitive unit contains eight tiles. === Stein (1985) Type 14=== A 14th convex pentagon type was found by Rolf Stein in 1985. The tiling is 3-isohedral and non-edge-to-edge. It has completely determined tiles, with no degrees of freedom. The exact proportions are specified by \frac{b}{a} = \sqrt{\frac{11 \sqrt{57}-25}{8}} and angle obtuse with \sin B = \frac{\sqrt{57}-3}{8}. Other relations can easily be deduced. The primitive units contain six tiles respectively. It has p2 (2222) symmetry. === Mann/McLoud/Von Derau (2015) Type 15=== 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. It is 3-isohedral and non-edge-to-edge, drawn with 6 colors, 2 shades of 3 colors, representing chiral pairs of the three isohedral positions. The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct. It has completely determined tiles, with no degrees of freedom. The primitive units contain twelve tiles. It has pgg (22×) symmetry, and p2 (2222) if chiral pairs are considered distinct. === No more periodic pentagonal tiling types=== 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. A consequence of this proof is that no convex polygon exists that tiles the plane only aperiodically, since all of the above types allow for a periodic tiling. === Nonperiodic monohedral pentagonal tilings=== Nonperiodic monohedral pentagonal tilings can also be constructed, like the example below with 6-fold rotational symmetry by Michael Hirschhorn. Angles are , , , , . In 2016 Bernhard Klaassen showed that every discrete rotational symmetry type can be represented by a monohedral pentagonal tiling from the same class of pentagons. Examples for 5-fold and 7-fold symmetry are shown below. Such tilings are possible for any type of -fold rotational symmetry with . ===Bibliography=== ; Errata, Forma 25 (1): 49, 2010,

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