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Yanxi [email protected]
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