Yanxi Liu.

Transcription

1 Yanxi Liu

2 Today s Theme: What is symmetry? From real world to mathematics, and back to the real world

3 Real World Instances of Symmetry

4 Symmetry Patterns from Real World (1)

5 Symmetry Patterns from Real World (2)

6 What is a symmetry? Starting from the somewhat vague notion of symmetry = harmony of proportions, rise to the general idea that of invariance of a configuration of elements under a group of automorphic transformations. --- Hermann Weyl Symmetry, Princeton,1952 Symmetry is an automorphic transformation

7 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.

8 A BIJECTIVE MAPPING A symmetry is a bijective mapping f(a) = B where A == B

9 Definition of Symmetry If g is a distance preserving transformation (isometrty) in Euclidean space R n, and S is a subset of R n, 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 automorphic transformation g.

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

11 Two essential, non-separable ingredients of symmetry 1. a mapping g; 2. a set S g Act upon Invariant of S

12 Definition of Symmetry If g is a distance preserving transformation (isometrty) in Euclidean space R n, and S is a subset of R n, 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 automorphic transformation g.

13 Symmetries and the Set G Act upon Invariant of S {rot1,rot2, ref1, ref2,ref3, id} Reflections about the center line {ref1,id} All rotations about the center of the disk? {id}

14 An Example of Symmetry g(s) = {g(s) s in S} = S g(s) = S a square in R 2 Reflection axis 4-fold rotations S g How many symmetries g the square S has? What are they?

15 Ambiguity on S The symmetries belong to the boundary or the area of the square? a square in R 2 S g S 1 boundary S 2 area S 1 and S 2 have the same set of symmetries!

16 What is the minimum number of symmetries S can have? ONE! What is the maximum number of symmetries S can have? INFINITE! Two types of infinites

17 Symmetries are scale-invariant Why?

18 NO Ambiguity on g For a given set S, its symmetries are uniquely and completely defined.

19 TYPES of SYMMETRIES g What DIFFERENT kinds of symmetries exist in 2-dimensional Euclidean space? Reflection Rotation Translation Glide-reflection ANY MORE? NO MORE!

20 TYPES of SYMMETRIES g What DIFFERENT kinds of symmetries exist in 3-dimensional Euclidean space? Reflection Rotation Translation Glide-reflection ANY MORE? YES!

21 Reflection invariance: the reflection axis With respect to an axis of reflection symmetry

22 Rotation invariance: 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.

23 Translation invariance: none

24 t 1 p2 t 2

25 REGULAR TEXTURE A Tile t 1 t 2

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27 Glide-reflection invariance: the axis of reflection t 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

28 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 distancepreserving mapping definition referring to Euclidean distance? What about other types of distances?

29 Answer: 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 a S b g b g S c d c Rotate S = [a b c d] 90 degrees about the center of the square a d

30 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 distancepreserving mapping definition referring to Euclidean distance? What about other types of distances?

31 Answer: ½t t 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 ½ t 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

32 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 distancepreserving mapping definition referring to Euclidean distance? What about other types of distances?

33 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.

34 What kind of symmetries out there in the real world? Examples obtained by students

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39 Crab Canon By J S Bach The Original Piece L L is reflected w.r.t the midpoint, as if played backwards The two pieces played together

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41 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

42 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/H ubble

43 Yet, another example of unbelievable symmetry /

44 More examples collected by students from previous Computational Symmetry class Check out our PSU Near-regular Texture Database :

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

47 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

48 Why should we care about symmetry? symmetry implies a lower dimensional structure, thus it is a double-sided sward

49 Example I: A dream of my Ph.D. thesis advisor Robin Popplestone: to have a robot who can assemble a wheelbarrow for me automatically

50 Put that cube in the corner! (how many different ways?) 2 1 3

51 Put a ball on a table: how many different ways? Surface 1 translations Surface 2

52 Without a true understanding of symmetry, robots fails to figure out there many equivalent ways

53 Example II: Symmetry in Periodic Patterns

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55 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 )

56 Examples of Seven Frieze Patterns and their symmetry groups I II III IV V VI VII 11 1g m1 12 mg 1m mm

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58 Examples of 17 Wallpaper Patterns and Their Symmetry Groups p1 p2 pm pg cm pmm pmg pgg cmm p4 p4m p4g p3 p31m p3m1 p6 p6m From a web page by: David Joyce, Clark Univ.

59 lattice units of the 17 wallpaper groups

60 p1 p2 pm pg

61 cm pmm pmg pgg

62 cmm p4 p4m

63 p4g p3 p31m p3m1

64 p6 p6m

65 Example III: Reflection Symmetries in Papercut Patterns

66 Symmetry Groups are not just decorative but (computationally and physically) functional reflection symmetry folding line

67 Fold and Cut Example

68 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?