Pentagonal tiling

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In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a specific 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.

Monohedral convex pentagonal tilings[edit]

Fifteen types of convex pentagon are known to tile the plane monohedrally (i.e. with one type of tile).[1] The most recent one was discovered in 2015. It is not known whether this list is complete.[2] Bagina (2011) showed that there are only eight edge-to-edge convex types, a result obtained independently by Sugimoto (2012).

Many of these monohedral tiling types have degrees of freedom. These freedoms include variations of internal angles and edge lengths. In the limit, edges may have lengths that approach zero or angles that approach 180°. Types 1, 2, 4, 5, 6, 7, 8, 9, and 13 allow parametric possibilities with nonconvex prototiles.

The number of degrees of freedom are:

Type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Degrees 5 4 1 2 2 1 1 1 1 1 1 1 1 0 0

Tilings are characterised by their wallpaper group symmetry, for example p2 (2222) is defined by 4 2-fold gyration points. This nomenclature is used in the diagrams below, where the tiles are also colored by their k-isohedral positions within the symmetry. 'a', 'b', 'c', 'd' & 'e' refer to the five edge lengths. 'A', 'B', 'C', 'D' & 'E' refer to the five vertex angles.

Each enumerated tiling family contains pentagons that belong to no other type; however, some individual pentagons may belong to multiple types. In addition, some of the pentagons in the known tiling types also permit alternative tiling patterns beyond the standard tiling exhibited by all members of its type. For example, a type 1 tile (with B+C=180°) has 5 degrees of freedom in an isohedral tiling with a 2-tile lattice. But if, additionally, E=90°, 2A+B=360° and c=b+d, the pentagon has just 2 degrees of freedom in a 2-isohedral tiling with a 4-tile lattice.

Reinhardt (1918)[edit]

Reinhardt (1918) found the first five pentagonal tilings. These all share the property to be isohedral, or "tile transitive", meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the automorphism group acts transitively on the tiles). By contrast, all subsequently found tilings are k-isohedral, with k>1.

All five are isohedral, while type 2 contains glide reflections which if ignored, contain chiral pairs of tiles (2-isohedral), colored here as yellow and green. Types 1, 3 and 4 have higher symmetry special cases with mirror reflections.

The smallest lattices (with translational symmetry, p1) contain two to six tiles.

The five isohedral convex pentagonal tilings (Reinhardt 1918)
1 2 3 4 5
p2 (2222) cmm (2*22) p2 (2222) pgg (22×) p3 (333) p3m1 (*333) p4 (442) p4g (4*2) p6 (632)
P5-type1.png P5-type1 p4g.png P5-type2-chiral coloring.png P5-type2.png P5-type3.png P5-type3 p3m1.png P5-type4.png P5-type4 p4g.png P5-type5.png
Prototile p5-type1.png
B+C=180°
A+D+E=360°
Prototile p5-type1 p4g.png
a=e, b=d
B=C=90°
A+D+E=360°
Prototile p5-type2.png
c=e
B+D=180°
Prototile p5-type3.png
a=b, d=c+e
A=C=D=120°
Prototile p5-type3 p3m1.png
a=b, c=e=d/2
A=C=D=120°
B=E=90°
Prototile p5-type4.png
b=c, d=e
B=D=90°
Prototile p5-type4 p4g.png
b=c=d=e
B=D=90°
Prototile p5-type5.png
a=b, d=e
A=60°, D=120°
Lattice p5-type1.png
2-tile lattice
Lattice p5-type2.png
4-tile lattice
Lattice p5-type3.png
3-tile lattice
Lattice p5-type4.png
4-tile lattice
Lattice p5-type5.png
6-tile lattice
Type 3 has one degree of freedom
p6m (*632) p3 (333) p3m1 (*333)
P5-type3 variations.png
Rhombille tiling limit, rotational, and reflective
Type 4 has two degrees of freedom, including these examples
P5-type4 variations.png
Type 5 has two degrees of freedom, including these examples
P5-type5 variations.png

Kershner (1968) and James (1975)[edit]

Kershner (1968) found three more types of pentagonal tilings for a total of eight known tilings. He claimed incorrectly that this was the complete list of pentagons that can tile the plane. 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)).

Types 6, 7, and 8 are 2-isohedral and type 10 is 3-isohedral. Type 7 and 8 have glide reflections. Ignoring the glides leads to 4-isohedral tilings in both.

The smallest lattices (with translational symmetry, p1) contain four to eight tiles.

2- and 3-isohedral convex pentagonal tilings
Kershner (1968) James (1975)
6 7 8 10
p2 (2222) pgg (22×) p2 (2222) pgg (22×) p2 (2222) p2 (2222) cmm (2*22)
P5-type6.png P5-type6 parallel.png P5-type7.png P5-type7-chiral coloring.png P5-type8.png P5-type8-chiral coloring.png P5-type10.png P5-type10 cmm.png
Prototile p5-type6.png
a=d=e, b=c
B+D=180°, 2B=E
Prototile p5-type6 parallel.png
a=d=e, b=c
B=60°, A=C=D=E=120°
Prototile p5-type7.png
b=c=d=e
B+2E=2C+D=360°
Prototile p5-type8.png
b=c=d=e
2B+C=D+2E=360°
Prototile p5-type10.png
a=b=c+e
A=90, B+E=180°, B+2C=360°
Prototile p5-type10 cmm.png
a=b=2c=2e
A=B=E=90°, C+D=135°
Lattice p5-type6.png
4-tile lattice
Lattice p5-type6 parallel.png
4-tile lattice
Lattice p5-type7.png
8-tile lattice
Lattice p5-type8.png
8-tile lattice
Lattice p5-type10.png
6-tile lattice
Type 6 has one degree of freedom, examples here
P5-type6 variations.png
Type 7 has one degree of freedom, examples here
P5-type7 variations.png
Type 8 has one degree of freedom, examples here
P5-type8 variations.png
Type 10 has one degree of freedom, examples here
P5-type10 variations.png

Rice (1977)[edit]

Marjorie Rice, an amateur mathematician, discovered four new types of tessellating pentagons in 1976 and 1977.[3][4]

All four tilings are 2-isohedral, containing glide reflections. If those are ignored, they become 4-isohedral, with distinct chiral pairs.

The smallest lattices (with translational symmetry, p1) each contain eight tiles.

2-isohedral convex pentagonal tilings (Rice 1976-77)
9 11 12 13
pgg (22×)
P5-type9.png P5-type11.png P5-type12.png P5-type13.png
p2 (2222)
P5-type9-chiral coloring.png P5-type11 chiral coloring.png P5-type12-chiral coloring.png P5-type13-chiral coloring.png
Prototile p5-type9.png
b=c=d=e
2A+C=D+2E=360°
Prototile p5-type11.png
2a+c=d=e
A=90°, 2B+C=360°
C+E=180°
Prototile p5-type12.png
2a=d=c+e
A=90°, 2B+C=360°
C+E=180°
Prototile p5-type13.png
d=2a=2e
B=E=90°, 2A+D=360°
Lattice p5-type9.png
8-tile lattice
Lattice p5-type11.png
8-tile lattice
Lattice p5-type12.png
8-tile lattice
Lattice p5-type13.png
8-tile lattice
Type 9 has one degree of freedom
P5-type9 variations.png
Type 11 has one degree of freedom
P5-type11 variations.png
Type 12 has one degree of freedom
P5-type12 variations.png
Type 13 has one degree of freedom
P5-type13 variations.png

Stein (1985) and Mann/McLoud/Von Derau (2015)[edit]

A 14th convex pentagon type was found by Rolf Stein in 1985.[5] University of Washington Bothell mathematicians Casey Mann, Jennifer McLoud, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a computer algorithm (paper pending as of August 2015).[6]

Types 14 and 15 are 3-isohedral. Type 15 has glide reflections, which if ignored lead to a 6-isohedral coloring. Both of them have completely determined tiles, with no degrees of freedom.

The smallest lattice (with translational symmetry, p1) contains six and twelve tiles respectively.

3-isohedral convex pentagonal tilings
Stein (1985) Mann/McLoud/Von Derau (2015)
14 15
p2 (2222) pgg (22×) p2 (2222)
P5-type14.png P5-type15.png P5-type15-chiral coloring.png
Prototile p5-type14.png
2a=2c=d=e
A=90°, B≈145.34°, C≈69.32°,
D≈124.66°, E≈110.68°
(2B+C=360°, C+E=180°).
Prototile p5-type15.png
a=c=e, b=2a
A=150°, B=60°, C=135°, D=105°, E=90°
Lattice p5-type14.png
6-tile lattice
Lattice p5-type15.png
12-tile lattice

Dual uniform tilings[edit]

There are three isohedral pentagonal tilings generated as duals of the uniform tilings, those with 5-valence vertices. They represent special higher symmetry cases of the 15 monohedral tilings above. Uniform tilings and their duals are all edge-to-edge. These dual tilings are also called Laves tilings. The symmetry of the uniform dual tilings is the same as the uniform tilings. Because the uniform tilings are isogonal, the duals are isohedral.

cmm (2*22) p4g (4*2) p6 (632)
1-uniform 8 dual color1.png 1 uniform 9 dual color1.png 1-uniform 10 dual color1.png
Prismatic pentagonal tiling
Instance of types 1 and 6[7]
Cairo pentagonal tiling
Instance of types 1, 4, and 8[7][8]
Floret pentagonal tiling
Instance of type 5[7]
33344 tiling face purple.png
120°, 120°, 120°, 90°, 90°
V3.3.3.4.4
33434 tiling face green.png
120°, 120°, 90°, 120°, 90°
V3.3.4.3.4
33336 tiling face yellow.png
120°, 120°, 120°, 120°, 60°
V3.3.3.3.6

Dual k-uniform tilings[edit]

The k-uniform tilings with valence-5 vertices also have pentagonal dual tilings, containing the same 3 shaped pentagons as the semiregular duals above, but contain a mixture of pentagonal types. A k-uniform tiling has a k-isohedral dual tiling and are represented by different colors and shades of colors below.

For example these 2, 3, 4, and 5-uniform duals are all pentagonal:[9][10]

2-isohedral 3-isohedral
p4g (4*2) pgg (22×) p2 (2222) p6 (*632)
2-uniform 16 dual color2.png 2-uniform 17 dual color2.png 3-uniform 53 dual color3.png 3-uniform 55 dual color3.png 3-uniform 56 dual color3.png
33344 tiling face purple.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33434 tiling face green.png 33336 tiling face yellow.png
4-isohedral 5-isohedral
pgg (22×) p2 (2222) p6m (*632)
4-uniform 142 dual color4.png 4-uniform 144 dual color4.png 4-uniform 143 dual color4.png 5-uniform 303 dual color5.png 5-uniform 314 dual color5.png
33344 tiling face purple.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33336 tiling face yellow.png
5-isohedral
pgg (22×) p2 (2222)
5-uniform 309 dual color5.png 5-uniform 315 dual color5.png 5-uniform 311 dual color5.png 5-uniform 310 dual color5.png 5-uniform 312 dual color5.png
33344 tiling face purple.png 33344 tiling face purple.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33344 tiling face purple.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33344 tiling face purple.png 33434 tiling face green.png 33434 tiling face green.png 33434 tiling face green.png 33434 tiling face green.png

Pentagonal/hexagonal tessellation[edit]

Pentagonal subdivisions of a hexagon

Pentagons have a peculiar relationship with hexagons. As demonstrated graphically below, some types of hexagons can be subdivided into pentagons. For example, a regular hexagon bisects into two type 1 pentagons. Subdivision of convex hexagons is also possible with three (type 3), four (type 4) and nine (type 3) pentagons.

By extension of this relation, a plane can be tessellated by a single pentagonal prototile shape in ways that generate hexagonal overlays. For example:

Pent-Hex-Type1-2.png
Planar tessellation by a single pentagonal prototile (type 1) with overlays of regular hexagons (each comprising 2 pentagons).
Pent-Hex-Type3-3.png
Planar tessellation by a single pentagonal prototile (type 3) with overlays of regular hexagons (each comprising 3 pentagons).
Pent-Hex-Type4-4.png
Planar tessellation by a single pentagonal prototile (type 4) with overlays of semiregular hexagons (each comprising 4 pentagons).
Pent-Hex-Type3-9.png
Planar tessellation by a single pentagonal prototile (type 3) with overlays of two sizes of regular hexagons (comprising 3 and 9 pentagons respectively).

Non-convex pentagons[edit]

Periodic tiling by the sphinx

With pentagons that are not required to be convex, additional types of tiling are possible. An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep-tile.[11] The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translates of this parallelogram,[11] a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2π.

It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit.[12] A similar method can be used to subdivide squares into four congruent non-convex pentagons, or regular hexagons into six congruent non-convex pentagons, and then tile the plane with the resulting unit.

Regular pentagonal tilings in non-Euclidean geometry[edit]

A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schläfli symbol {5,3}, having three pentagons around each vertex.

In the hyperbolic plane, there are tilings of regular pentagons, for instance order-4 pentagonal tiling, {5,4}, having four pentagons around each vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.

Sphere Hyperbolic plane
Uniform tiling 532-t0.png
{5,3}
Uniform tiling 54-t0.png
{5,4}
Uniform tiling 55-t0.png
{5,5}
Uniform tiling 56-t0.png
{5,6}
Uniform tiling 57-t0.png
{5,7}
Uniform tiling 58-t0.png
{5,8}
...{5,∞}

Irregular hyperbolic plane pentagonal tilings[edit]

There are an infinite number of dual uniform tilings in hyperbolic plane with isogonal irregular pentagonal faces. They have face configurations as V3.3.p.3.q.

Order p-q floret pentagonal tiling
7-3 8-3 9-3 ... 5-4 6-4 7-4 ... 5-5
Ord7 3 floret penta til.png
V3.3.3.3.7
V3.3.3.3.8 V3.3.3.3.9 Order-5-4 floret pentagonal tiling.png
V3.3.4.3.5
V3.3.4.3.6 V3.3.4.3.7 V3.3.5.3.5

References[edit]

  1. ^ Tilings and Patterns, Sec. 9.3 Other Monohedral tilings by convex polygons
  2. ^ Peralta, Eyder (14 August 2015), "With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem", NPR, retrieved 15 August 2015 
  3. ^ Schattschneider 1978.
  4. ^ "Tessellations - Intriguing Tessellations". google.com. Retrieved 22 August 2015. 
  5. ^ Schattschneider 1985.
  6. ^ Bellos, Alex (11 August 2015). "Attack on the pentagon results in discovery of new mathematical tile". The Guardian. 
  7. ^ a b c Reinhardt, Karl (1918), Über die Zerlegung der Ebene in Polygone, Dissertation Frankfurt am Main (in German), Borna-Leipzig, Druck von Robert Noske, pp. 77–81  (caution: there is at least one obvious mistake within this paper, i.e. γ+δ angle sum needs to equal π, not 2π for the first two tiling types defined on page 77)
  8. ^ Cairo pentagonal tiling generated by a pentagon type 4 query and by a pentagon type 2 tiling query on wolframalpha.com (caution: the wolfram definition of pentagon type 2 tiling does not correspond with type 2 defined by Reinhardt in 1918)
  9. ^ Chavey 1989.
  10. ^ n-uniform tilings, Brian Galebach
  11. ^ a b Godrèche 1989.
  12. ^ Gerver 2003.

Bibliography

External links[edit]