In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a planigon is a
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
that can fill the plane with only copies of itself (
isotopic to the
fundamental units of
monohedral tessellations). In the Euclidean plane there are 3 regular planigons;
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
,
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
s, and
regular hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
s; and 8
semiregular planigons; and 4
demiregular planigons which can tile the plane only with other planigons.
All angles of a planigon are whole divisors of 360°. Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along common edges (they coincide).
Tilings made from planigons can be seen as
dual tilings to the
regular, semiregular, and
demiregular tilings of the plane by
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s.
History
In the 1987 book, ''Tilings and Patterns'',
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent[Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...](_blank)
s. Their
dual tilings are called ''Laves tilings'' in honor of
crystallographer Fritz Laves
Fritz Henning Emil Paul Berndt Laves (27 February 1906 – 12 August 1978) was a German crystallographer who served as the president of the German Mineralogical Society from 1956 to 1958. He is the namesake of Laves phases and the Laves tilings; ...
.
They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich.
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
calls the uniform duals ''Catalan tilings'', in parallel to the
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan sol ...
polyhedra.
The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The
tiles
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...
of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. Each vertex has edges evenly spaced around it. Three dimensional analogues of the ''planigons'' are called
stereohedron
In geometry and crystallography, a stereohedron is a convex polyhedron that fills space isohedrally, meaning that the symmetries of the tiling take any copy of the stereohedron to any other copy.
Two-dimensional analogues to the stereohedra are ...
s.
These tilings are listed by their
, the number of faces at each vertex of a face. For example ''V4.8.8'' (or V4.8
2) means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles.
Construction
The
Conway operation of dual interchanges faces and vertices. In
Archimedean solids
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed o ...
and
''k''-uniform tilings alike, the new vertex coincides with the center of each
regular face, or the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
. In the Euclidean (plane) case; in order to make new faces around each original vertex, the centroids must be connected by new edges, each of which must intersect exactly one of the original edges. Since regular polygons have
dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
, we see that these new centroid-centroid edges must be
perpendicular bisectors of the common original edges (e.g. the centroid lies on all edge perpendicular bisectors of a regular polygon). Thus, the edges of ''k''-dual uniform tilings coincide with centroid-to-edge-midpoint line segments of all regular polygons in the ''k''-uniform tilings.
Using the 12-5 Dodecagram (Above)
All 14 uniform usable regular vertex planigons also hail from the
6-5 dodecagram (where each segment subtends
radians, or 150 degrees).
The
incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
of this dodecagram demonstrates that all the 14 VRPs are
cocyclic, as alternatively shown by
circle packings. The ratio of the incircle to the circumcircle is:
and the convex hull is precisely the
regular dodecagons in the
k-uniform tiling
A ''k''-uniform tiling is a tiling of Tessellation, tilings of the plane by convex regular polygons, connected edge-to-edge, with ''k'' types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tilin ...
. The equilateral triangle, square, regular hexagon, and regular dodecagon; are shown above with the VRPs.
In fact, any group of planigons can be constructed from the edges of a
polygram
PolyGram N.V. was a multinational entertainment company and major music record label formerly based in the Netherlands. It was founded in 1962 as the Grammophon-Philips Group by Dutch corporation Philips and German corporation Siemens, to be a ...
, where
and
is the number of sides of sides in the RP adjacent to each involved vertex figure. This is because the
circumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of any regular
-gon (from the vertex to the centroid) is the same as the distance from the center of the polygram to its line segments which intersect at the angle
, since all
polygrams admit incircles of inradii
tangent to all its sides.
Regular Vertices
In ''Tilings and Patterns'',
Grünbaum also constructed the Laves tilings using ''monohedral tiles with'' ''regular vertices''. A vertex is regular if all angles emanating from it are equal. In other words:
# All vertices are regular,
# All Laves planigons are congruent.
In this way, all Laves tilings are unique except for the
square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of the s ...
(1 degree of freedom),
barn pentagonal tiling (1 degree of freedom), and
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemat ...
(2 degrees of freedom):
When applied to higher dual co-uniform tilings, all dual coregular planigons can be distorted except for the triangles (
AAA similarity), with examples below:
Edge-to-Edge Correspondence
Alternatively, ''k''-dual uniform tilings (and all 21 planigons) can be constructed by forming new centroid-edge midpoint line segments of the original regular polygons (dissecting the regular ''n''-gons into ''n'' congruent deltoids or
ortho), and then removing the original edges (leaving the
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
). Complete planigons will form around interior vertices, and line segments of (many possible) planigons will form around boundary vertices, giving a 1-to-1 edge ''k''-dual uniform lattice. On the other hand, centroid-centroid connecting only yields interior planigons, but this construction is nonetheless equivalent to the original in the interior. If the ''k''-uniform tiling fills the entire frame, then so will the ''k''-dual uniform tiling, and the boundary line segments can be ignored (equivalent to original construction).
Affine Linear Expansion
Starting from the regular polygons of a k-uniform tiling, we can scale all regular polygons about their centroids over a linear factor