Yanxi Liuyanxi@cs.cmu.edu

RI and CALDSchool of Computer Science, Carnegie Mellon University

Today’s Theme:Different Types of Symmetries

and Symmetry Groups

What is a symmetry?

What is a symmetry of S ⊆R3?

DefinitionIf g is a distance preserving transformation in n-dimensional Euclidean space R , and S is a subset of R , then g is a symmetry of S iff g(S) = S, and S is (setwise) invariant under action g.

An example:a square plate in R(2)

Reflection axis

4-fold rotations

nn

DefinitionA group G is a set of elements with a binaryoperation * defined on the set that satisfy:

• * is associative a*(b*c) = (a*b)*c• G is closed, for any a,b in G, a*b also in G• any element in G has an inverse• there is an identity element id in G, id*a = a

a square plate in R(2)

G = symmetries of asquare

* = transformation composition

An example:

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

Examples of group G versus set S

G SAct upon

Invariant of

The Promise:… beyond absolute regularity there are nuances

of decreasing regularities, of reductive invariance: that there is an order to so-called disorder.

Ordering Disorder after K.L. Wolfby W.S. Huff (an architect)

The PerilsTyger! Tyger! burning bright,In the forest of the night,What immortal hand or eyeCould frame thy fearful symmetry?

…

William Blake 1794

What are different types of symmetries?

Symmetry or Near-Symmetry Patterns are ubiquitous

Symmetry or Near-Symmetry Patterns are ubiquitous

a square shape in R2

Symmetry or Near-Symmetry Patterns are ubiquitous

Symmetry or Near-Symmetry Patterns are ubiquitous

Symmetries in 3D

Hierarchy of Symmetries

Which one is more symmetrical?

or

How to organize these different types of

symmetries?

Types of Symmetry Groups G: for all g in G, g(S) =S

•

• point group: All symmetries g in group G leave at least one point in R3 invariant • Space group: symmetry groups in R3

that contains a non-trivial translation symmetry

Euclidean Group

All symmetries of R3 for the symmetry group of the 3D Euclidean space and is called the Euclidean Group

Examples of elements in the Euclidean Group are:Rotations

TranslationsReflections

Glide-reflections…

Proper Euclidean Group

All handedness-preserving isometries form the proper Euclidean group that

excludes reflections

Group relations: Subgroup

Definition:Let (G,*) be a group and let H be subset of G . Then H is a subgroup of G defined under the same operation if H is a group by itself (with respect to * ), that is:• H is closed under the operation. • There exists an identity element e such that for all h, h*e = e*h = h• Let h in H, then there exists an inverse h-1 such that h-1*h = e = h*h-1.

The subgroup is denoted likewise (H, *).We denote H being a subgroup of G by writing H G .

Group Relations: Group Conjugate

G1, G2 are subgroups of G, we call G1 is a conjugate to G2 iff for some g in G, such

that G1 = g * G2 * g-1

Orbits

An orbit of a point x in R3 under group G is G(x) = { g(x)| all g in G}

Types of Symmetry Groups G of S

• G is finite• G is infinite

• G is discrete (Definition 1.1.10)• G is continuous

Definitions of Different Types of Subgroups of the Euclidean Group

Discrete Point Groups asSubgroups of theEuclidean Group

The hierarchy of the subgroups of The Euclidean Group

O orthogonal groupSO special orthogonal groupT translation groupD dihedral groupC cyclic group

S(2), S(3), SO(2), SO(3), T1-3

???continuous

T1C2 …???Non-discrete

Crystallographic groups

Cn, D2n, G of regular solids

discrete

infinitefinite

An Exploration of Hierarchies of Symmetry Groups

• Surface Contact/Motion among solids (robotics assembly planning)

• Periodic Patterns (visual)• Papercut Patterns (fold-then-cut)

Example I: Symmetry in Contact Motions

Solids in Contact-MotionLower-pairs

Insight:

The contacting surface pair from two different solids coincide, thus has the samesymmetry group which determines their relative motions/locations

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

1

23

‘Put that cube in the corner with face 1 on top !’(4 different ways)

Complete and unambiguous task specifications can be tedious for symmetrical objects

Constructive Solid Geometry (CSG) Representation

Different types of surfaces associated with different types of groups

The concept of “conjugated symmetry groups”are used here

Algebraic Surface feature and its coordinates

World coordinates

Solid coordinates

Surface coordinates

l

f

Contacting surfaces

Example II: Symmetry in Visual Form

p2

t1

t2

p6

Hilbert’s 18th Problem

Question:

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

Hierarchy of Frieze 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

p1

p2

p3

p4

p6

pg

pmcm

pgg

pmgpmm

cmm

p4gp4m

p3m1

p31m

p6m

Subgroup Relationship Among the 17 Wallpaper Groups(Coxeter)

P1 camp P2 camp

Where does the symmetry group of an affinely deformed pattern migrate to?

p2

cmmp4m

cmm

cmm

Examples of Symmetry Group Migrationcmcm p1

p3m1

Potential Symmetry= largest Euclidean subgroup in AGA-1An inherent, affine-invariant, non-face-value

property of a patternp6mcmmp2

p4mcmmpmmp2

Example III: Reflection Symmetry for Folding

A Symmetry Group Formalization of Papercut Patterns

SIGGRPHA 2005 Technical Sketch Slides

Symmetry Groups are not just decorative but (computationally and physically) functional

Digital Papercutting

Yanxi Liu and James Hays Carnegie Mellon University

Ying-Qing Xu and Heung-Yeung ShumMicrosoft Research Asia

Motivation

Fold and Cut Example

Contribution• We present a general computational

formalization of this particular art form for the first time.

• We combine the analysis and synthesis of a papercut pattern into one automatic process, as well as the generation of a physically feasible folding plan.

Related work

• Computational geometry– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.

Folding and cutting paper. In JCDCG’98.– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.

Folding and One Straight Cut Suffice SODA'99.

• Papercraft– Mitani, J., and Suziki, H. Making papercraft toys from

meshes using strip-based approximate unfolding. SIGGRAPH 2004.

Formal treatment of papercutting

P S1 S2 S3

Characterization by Symmetry Groups

Frieze Group Dihedral Group

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Fundamental Regions

Dihedral group examples

Fundamental Regions

3 folds 3 folds 4 folds 4 folds 4 folds 4 folds

Dihedral group examples

Fundamental Regions

3 folds 3 folds 4 folds 4 folds 4 folds 4 folds