Wallpaper groups

1. AN EXTENDED UNDERSTANDING OF THE 2-DIMENSIONAL WALLPAPER GROUPS
Lim, Kyuson, Course: Mat392 (Ideas of Mathematics)
Mail: kslchangeoflife.lim@mail.utoronto.ca
Identification of the transitive tilings, with brick walls
The following definitions methodologically classify appropriate motions to identify different types of patterns in the arrangement of brick walls.
As an extension, the surface could be folded from the 2-dimensional equilateral triangle to bond the two sides like an origami to form topological disk.
As an extension, this is one of four Coxeter polygon in the plane where the hexagon never overlaps, by reflections.
First, the images of the regular polygon T0 with the motion of an isometry as the action of the transformation group, 𝐺, fill the plane, g∈G g T0 = R2.
Second, the wallpaper group 𝐺 acts discretely, and does not overlaps on the plane where the tiling have 2 translation vectors.
Third, the action of the wallpaper group 𝐺 is transitive, for g, h ∈ 𝐺, the images g T0 = h T0 if and only if g = h.
The rectangle is observed to have the rotation about a horizontal point and reflect by the vertical and horizontal axes, as to be transitive on the wall.
It is observed only reflections are available for the boding lines being reflected vertically and horizontally, transitive on bricks. Notice, the rotation is
not possible as the boding line also rotate, which leads to an overlap of patterns.
Conclusion
Throughout a concept of kaleidoscope, we have identified the symmetry points of reflections and a right-angled triangle Coxeter polygon.
By the gyration point, we have identified the 3-dimensional topological disk with respect to orbit point to form the surface of pattern, an orbifold.
Within applied understanding of the tilings, it was shown that the technical guidance is applied by the properties to figure out the patterns of groups.
© Copyright 2020. Lim, Kyuson.
Reference
[1] Introduction to Geometry, H.S.M. Coxeter, Wiley Classics Library Edition, 1989.
[2] Geometries, A. B. Sossinsky, AMS(American Mathematical Society), 2012.
[3] The symmetries of the things, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, AK Peters Ltd, 2008.
Kaleidoscope
Before
With the identified
intersection reflection
point, the pattern is
known for the 𝐷6 group
with 6 reflection axes,
transitively generated by
the right-angled triangle.
After
The tessellation is
identified with 6-
fold, 3-fold, 2-fold
kaleidoscopic
points.
Also, 𝐷6 ≅ 𝐷3 × 𝐶2
Extension
A right-angle triangle F in the
plane has all its vertex angles,
π/k for k=2,3, and 6. Then, the
symmetry group of the plane
is generated by the reflection.
The 𝐺𝐹 acts transitively on 𝐹,
as the images 𝑔 𝐹 , 𝑔 ∈ 𝐺𝐹.
A hexagonal shaped lattice with reflection of angle π/6 is recognized as a dihedral group 𝐷6 before the identification of 3 different kaleidoscopic points.
After the identification of 3 kaleidoscopic points that are different, we could find the tilings to be a group p6m among 17 wallpaper groups.
What about the rotation?
A Kaleidoscope is the patterns whose symmetries are defined by reflections.
The classification of reflections becomes apparent with respect to each kaleidoscopic point, conceptually for the pattern of reflections in 6 distinct directions.
The following steps shows how understanding the kaleidoscopic pattern leads to an extension of geometrical knowledge from one of the 2-dimensional
wallpaper group to the identification of a regular convex hexagon of Coxeter geometries in the plane.
The Coxeter polygon (F) is defined as all vertex angles are of the form π/k for k = 2, 3, … and it generates a transitive action of the group GF,
where the GF is the transformation group generated by the reflections in the planes containing the faces of F.
Definition.
(Conway, John Horton) (Conway, John Horton)
(Conway, John Horton)
Gyration
A gyration point corresponds to a rotation point that does not lie on the reflection.
The symmetries of pattern for a classification of surface lead to unique 3-dimensional folding of an orbifolds, with its gyration point as an orbit.
The following process shows how the identified rotation different from reflection leads to the classification of the gyration point and an orbifold of the tile.
Before
The pattern is
identified with
the 𝐶3, 2π/3
rotational
symmetries that
generate the
tessellation.
After
From the 3-fold
kaleidoscopic
point, we also
identify the
independent 3-fold
gyration point, 𝐷3.
Extended
An equilateral triangle shaped lattice is recognized with a rotation angle of 2π/3 before the identification of an independent kaleidoscopic point.
With the identification of a 3-fold gyration point, the combined pattern of tilings leads to a classification to a group p31m among 17 wallpaper groups.
What are orbifolds?
The gyration point is included in the
set of orbits. Hence, the equilateral
triangle is cut off to form the 3 ∗ 3
orbifold that is a topological disk,
where a cone point is a gyration
point and a corner point is the 3-
fold kaleidoscopic point.
Definition.
The orbit of an a point is the set of points to which a point can be moved by the actions of the symmetry group. This orbit-folded version of
the surface is an orbifold. Hence, the orbifold is about how to fold up a pattern into a surface.
Definition.
(Conway, John Horton)
(Conway, John Horton)
(Conway, John Horton)
The orbifold that folds up a pattern into a surface is not restricted to the wallpaper groups, but also applies to the another discrete group, frieze patterns.
Wallpaper group (pm)
Orbifolds
The comparison below emphasizes the difference between the wallpaper groups and the frieze patterns with the definition by an example of a pattern.
Frieze group (𝐹1)
There is the horizontal and
the vertical reflection axes with
2 translations, different from
the frieze patterns which has
the same reflection patterns.
The 𝐹1 consists of only one
translation. The 7 frieze groups
catalog all symmetry groups that
leave design invariant under all
multiples of 1 translation.
The discrete frieze groups are formed by the symmetries of plane patterns that repeat infinitely in one direction only, whose subgroups of
translations are isomorphic to the group 𝑍.
Definition.
(Conway, John Horton) (Conway, John Horton)
Background
As a geometrical classification of the discrete plane symmetry group, including the 17 tilings, the tessellation is an allocation the repeated patterns by
the motions in the isometry to the polygon T0, the fundamental tile, which is an infinite family of tiles T1, T2, … of pairwise non-overlapping copies.
The goal is to identify representative patterns of the tilings by an analysis of an appropriate classified motions of an isometry with the crystallography
restriction theorem for rotations and develop for 3-dimensional orbifold to be compared with others.
As studying for the topics, the knowledge is extended to the understanding of how some tilings are compared to patterns of the frieze groups in the
context of discrete planar groups and developed for understanding the surface of patterns by the orbifolds.
What classify for reflections?
Further understandings of the beauty in classified wallpaper groups includes symmetric patterns of a kaleidoscope and a gyration, and a practical
intention in the brick walls leads to an extension of an artistic intuition and examples of classifications.
Running bond brick walls
Step 2. Find adequate tiles that fill
the plane with translations.
T0 ∶
Step 3. Identify the motions of tiles.
reflection
rotation
Group cmm
Step 1. Try to identify the cut. - Symmetries with the D2.
- Differ from the group pmg with
reflection axes with 2 directions.
(Conway, John Horton)
(Conway, John Horton)
(Conway, John Horton)
(Conway, John Horton)
English bond brick walls Group pmm
- Symmetries with the D2.
Step 1. Try to identify the cut.
- Differ from the group cmm as
all rotation centers on reflection
axes.
reflection
reflection
Step 2. Find adequate tiles that fill
the plane with translations.
T0 ∶
Step 3. Identify the motions of tiles.
(Conway, John Horton)
(Conway, John Horton)
(Conway, John Horton)
(Conway, John Horton)
Spiral bond brick walls
2 points
rotations
Group p2
- Symmetries with the C2.
- Differ from the group pgg with
glide reflections.
2 points
rotations
rotations
Step 1. Try to identify the cut. Step 2. Find adequate tiles that fill
the plane with translations.
T0 ∶
Step 3. Identify the motions of tiles.
(Conway, John Horton)
(Conway, John Horton) (Conway, John Horton)
(Conway, John Horton)
Zigzag bond brick walls
The overall movement of bricks is different from the previous pattern by reflection axes for tessellation of brick walls. We must see that 2 parallel lines
of reflections can not be replaced by the rotation point due to overlapping of the bonding line. Hence, 2 parallel reflection axes form the zigzag pattern.
Group pmg
- Differ from the spiral pattern
with a reflection to begin with.
reflection
rotation
Step 1. Try to identify the cut. Step 2. Find adequate tiles that fill
the plane with translations.
T0 ∶
Step 3. Identify the motions of tiles.
- Symmetries with the D2.
(Conway, John Horton)
(Conway, John Horton) (Conway, John Horton)
(Conway, John Horton)
Crystallography restriction theorem
The proof starts with 2 different points by rotations of an angle
2π
n
to come up with a regular polygon for displacements of points as to form the lattice.
“
As an extension, this is one of four Coxeter polygon in the plane where the right-angled triangle with the vertex angle
𝜋
2
,
𝜋
3
and
𝜋
6
that never overlaps.
As an extension, within the orbit point the 2-dimensional equilateral triangle of tile is folded to bond two sides like an origami to form topological disk.
”
The theorem states a rotational symmetry of the wallpaper groups must be a rotation of order 2,3,4 or 6.
The consequence restrict the possible angles of rotation π,
2π
3
,
π
2
and
π
3
. Also, this leads to two independent translations with period greater than 2.
Definition.
The cut does not show the full movement of the lattice. Must find the horizontal half cut size of the brick is rotated by an angle of π with 4 different
points, horizontally and vertically. It is important to know if there is a reflection axes then no spiral pattern could be formed with the violation of property.
An isometry is a function in the plane that preserves the distance, and the motion is a type of an isometry that preserves the orientation.
Definition.
As for the developed 3 steps of identification, the definition must be satisfied accordingly to check to find how we identify the wallpaper groups.
With the vertical alignment, 2 more patterns exists the group pgg (2 rotations and 2 glide reflections) and p4g (rotations π/2 each with the reflection).