Yanxi Liuyanxi@cse.psu.edu

Today’s Theme:

What is symmetry?

From real world to mathematics, and back to the real world

Real World Instances of Symmetry

Symmetry Patterns from Real World (1)Symmetry Patterns from Real World (1)

Symmetry Patterns from Real World (2)Symmetry Patterns from Real World (2)

What is a symmetry?

“Starting from the somewhat vague notion of symmetry = harmony ofproportions, … rise to the general idea …that of invariance of a configuration of elements under a group of automorphictransformations.”

--- Hermann WeylSymmetry, Princeton,1952

Symmetry is an automorphic transformation

Automorphic Transformation == Automorphism

Automorphism:An automorphism is an isomorphism of a system of objects onto itself.

Isomorphism:an isomorphism is bijective morphism

Morphism:A morphism is a map between two objects in an abstract category.

A symmetryis a bijective mapping f(A) = Bwhere A == B

A BIJECTIVE MAPPING

Definition of Symmetry

If g is a distance preserving transformation (isometrty) in Euclidean space Rn , and S is a subset of Rn , then g is a symmetry of S iff

g(S) = {g(s) | for all s in S} = S

I.e.: S is setwise invariant under the automorphictransformation g.

Symmetry Group

All symmetries of a subset S of Euclidean space R have a group structure G, and G is called the symmetry group of S.

n

Two essential, non-separable ingredients of symmetry

g SAct upon

Invariant of

• 1. a mapping g; • 2. a set S

Definition of Symmetry

If g is a distance preserving transformation (isometrty) in Euclidean space Rn , and S is a subset of Rn , then g is a symmetry of S iff

g(S) = {g(s) | for all s in S} = S

I.e.: S is setwise invariant under the automorphictransformation g.

Symmetries and the Set

G SAct upon

Invariant of

All rotations about the center of the disk

{rot1,rot2, ref1, ref2,ref3, id}Reflections about the center line {ref1,id}

? {id}

An Example of Symmetryg(S) = {g(s) | s in S} = S g(S) = S

a square in R2

Reflection axis4-fold rotations

Sg

How many symmetries g the square S has?What are they?

Ambiguity on S• The symmetries belong to the boundary or

the area of the square?a square in R2

Sg

S1 and S2 have the same set of symmetries!

S1 boundary

S2 area

What is the minimum number of symmetries S can have?

What is the maximum number of symmetries S can have?

ONE!

INFINITE! Two types of infinites

Symmetries are scale-invariant

• Why?

NO Ambiguity on g

• For a given set S, its symmetries are uniquely and completely defined.

TYPES of SYMMETRIES g

• What DIFFERENT kinds of symmetries exist in 2-dimensional Euclidean space?– Reflection– Rotation– Translation– Glide-reflection– … ANY MORE? NO MORE!

TYPES of SYMMETRIES g

• What DIFFERENT kinds of symmetries exist in 3-dimensional Euclidean space?– Reflection– Rotation– Translation– Glide-reflection– … ANY MORE? YES!

Reflectioninvariance: the reflection axis

With respect to an axis of reflection symmetry

Rotationinvariance: the center of rotation

N-fold rotational symmetry:Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n does not change the object.

Translationinvariance: none

p2

t1

t2

A Tile

REGULARTEXTURE

t1

t2

• Glide reflection is composed of a translation that is ½ of the smallest translation symmetry t and a reflection r w.r.t. a reflection axis along the direction of the translation

t

Glide-reflectioninvariance: the axis of reflection

Questions• What is “setwise” invariant?• Why is glide-reflection a primitive symmetry

that combines a reflection and a translation? Why can’t other type of symmetries be combined and considered as a primitive symmetry as well?

• In the definition of symmetry, does g have to be an isometry? Is the distance in the “distance-preserving mapping” definition referring to Euclidean distance? What about other types of distances?

Setwise invariant: Under a symmetry operation, the points in the set S are relocated (permuted) but the whole set S remains the same.See the example below, S occupies the same points though the positions of a,b,c,d are changed under the rotation symmetry

Sg

d c

baS

g

d

cb

aRotate S = [a b c d] 90 degrees about the center of the square

Answer:

Questions• What is “setwise” invariant?• Why is glide-reflection a primitive symmetry

that combines a reflection and a translation? Why can’t other type of symmetries be combined and considered as a primitive symmetry as well?

• In the definition of symmetry, does g have to be an isometry? Is the distance in the “distance-preserving mapping” definition referring to Euclidean distance? What about other types of distances?

• If g = tr is a symmetry of S, and t(S) =S and r(S)=S, then g is not a primitive symmetry of S

• In the case of a glide-reflection symmetry, neither ½t or r itself is a symmetry S, when and only when the two are combined they become a symmetry of S. Therefore they form a primitive symmetry of S

t

½ tAnswer:

Questions• What is “setwise” invariant?• Why is glide-reflection a primitive symmetry

that combines a reflection and a translation? Why can’t other type of symmetries be combined and considered as a primitive symmetry as well?

• In the definition of symmetry, does g have to be an isometry? Is the distance in the “distance-preserving mapping” definition referring to Euclidean distance? What about other types of distances?

Answer:

The classic definition of symmetry in all standard mathematics books is defined with respect to Euclidean distance, i.e. Euclidean isometry, and so is our definition in this lecture.

However, extension of the meaning of symmetry beyond Euclidean geometry is possible and will be addressed in later lectures.

What kind of symmetries out there in the real world?

• Examples obtained by students

Crab Canon By J S Bach

The Original Piece L

L is reflected w.r.t the mid-point, as if played ‘backwards’

The two pieces played together

An image of the Red Square nebula surrounding the hot star MWC 922. The picture was taken with infrared adaptive optics imaging at Palomar and Keck Observatories. Credit: Peter Tuthill, Palomar and Keck Observatories

The Red Rectangle is one of the most unusual nebulas known in our Milky Way. Cataloged as HD 44179, this nebula is the result of a dying star. Credit: NASA/ESA/Hubble

Yet, another example of unbelievable symmetry …

• http://www.mustangevolution.com/forum/t26103/

More examples collected by students from previous

“Computational Symmetry” class

• Check out our PSU “Near-regular Texture Database”:

• http://vivid.cse.psu.edu/texturedb/gallery/

Symmetry Group

All symmetries of a subset S of Euclidean space Rn have a group structure, and is called the symmetry group G of S.

What have you used symmetries for in your everyday life?

• Found something wrong with my cat Charlie’s face (asymmetry pathology)

• Successfully found the locations of ‘lady’s room’ around an unknown building multiple times!

• Assemble packaged furniture from IKEA• Solving jigsaw puzzles• …

• symmetry implies a lower dimensional structure, thus it is a double-sided sward

Why should we care about symmetry?

Example I: A dream of my Ph.D. thesis advisor

Robin Popplestone:“…to have a robot who can assemble a wheelbarrow for me automatically”

‘Put that cube in the corner !’ (how many different ways?)

1

23

translations

Surface 2

Surface 1

Put a ball on a table: how many different ways?

Without a true understanding of symmetry, robots fails to figure out

“there many equivalent ways”

Example II: Symmetry in Periodic Patterns

p6

Hilbert’s 18th ProblemQuestion:

In n-dimensional euclidean space is there … only a finite number of essentially different kinds of (symmetry)groups of motions with a fundamental region?

Answer:

Yes! (Bieberbach and Frobenius, published 1910-1912)

11 1g m1 12 mg 1m mm

I II III IV V VI VII

Examples of Seven Frieze Patterns and their symmetry groups

Examples of 17 Wallpaper Patterns and Their Symmetry Groups

From a web page by:David Joyce, Clark Univ.

p1 p2 pm pg cm

pmm pmg pgg cmm p4

p4m p4g p3 p31m p3m1

p6 p6m

lattice units of the 17 wallpaper groups

pm

p1

p2

pg

cm

pmm

pgg

pmg

cmm

p4

p4m

p3

p4g

p3m1

p31m

p6

p6m

Example III: Reflection Symmetries in

Papercut Patterns

Symmetry Groups are not just decorative but (computationally

and physically) functional

reflection symmetry folding line

Fold and Cut Example

Summary • Symmetry is a transformation• Symmetry can only be defined with respect to a set S• Mathematical definition of symmetry g of set S is: g(S) = S• All the symmetries of S form the symmetry group of S • How many different types of primitive symmetries in

Euclidean space?– Only FOUR in 2D-Euclidean space:

• Reflection• Rotation• Translation• Glide-reflection

– What about in 3D-Euclidean space?