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 ...
of the Euclidean plane, the 3-4-3-12 tiling is one of 20
2-uniform tilings of the
Euclidean plane by regular polygons, containing regular
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s,
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 a ...
s, and
dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...
s, arranged in two
: 3.4.3.12 and 3.12.12.
[Chavey (1989)]
The ''3.12.12'' vertex figure alone generates a
truncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applies to a he ...
, while the ''3.4.3.12'' only exists in this 2-uniform tiling. There are 2
3-uniform tilings that contain both of these vertex figures among one more.
It has
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 a ...
symmetry, p4m,
,4 (*442). It is also called a
demiregular tiling by some authors.
Circle Packing
This 2-uniform tiling can be used as a
circle packing
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated '' packing de ...
. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.12
2 planigon, and pink circles are in contact with 4 other circles (2 cyan, 2 pink), corresponding to the V3.4.3.12 planigon. It is
homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (one dimensional duals to the respective planigons). Both images coincide.
Dual tiling
The dual tiling has
kite
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...
('ties') and
isosceles triangle faces, defined by
s: V3.4.3.12 and V3.12.12. The kites meet in sets of 4 around a center vertex, and the triangles are in pairs making
planigon rhombi. Every four kites and four isosceles triangles make a square of side length
.
This is one of the only dual uniform tilings which only uses planigons (and semiplanigons) containing a 30° angle. Conversely, 3.4.3.12; 3.12
2 is one of the only uniform tilings in which every vertex is contained on a dodecagon.
Related tilings
It has 2 related 3-uniform tilings that include both 3.4.3.12 and 3.12.12 vertex figures:
This tiling can be seen in a series as a lattice of 4''n''-gons starting from 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 th ...
. For 16-gons (''n''=4), the gaps can be filled with isogonal octagons and isosceles triangles.
Notes
References
* Keith Critchlow, ''Order in Space: A design source book'', 1970, pp. 62–67
* Ghyka, M. ''The Geometry of Art and Life'', (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling
* pp. 35–43
* p. 65
* ''Sacred Geometry Design Sourcebook: Universal Dimensional Patterns'', Bruce Rawles, 1997. pp. 36–3
External links
*
*
* {{MathWorld , urlname=DemiregularTessellation , title=Demiregular tessellation
In Search of Demiregular Tilings Helmer Aslaksen
Brian Galebach
Euclidean plane geometry
Tessellation