Big Ideas in Science: Symmetry

Size: px

Start display at page:

Download "Big Ideas in Science: Symmetry"

August Walters
5 years ago
Views:

1 February 7, 05 Big Ideas in Science: Symmetry Perfect Symmetry Tim Burness School of Mathematics

3 Symmetry in science Physics (Next week s lecture!): I I The physical laws of the universe (e.g. conservation of energy) Relativity and quantum physics Chemistry: The symmetry of molecules and crystals Biology: Bilateral symmetry in multicellular organisms Computer science: The design and implementation of algorithms: Symmetry faster, more efficient computation Psychology: Visual symmetry perception etc. etc....

4 Symmetry in mathematics: Perfect symmetry

5 Symmetry in mathematics C A x + y + z =

6 Example: The symmetry of addition Problem: Calculate the sum of the first 50 odd numbers Answer: (00 50)/ = 500 = 50 Generalisation: The sum of the first n odd numbers is (n 3) + (n ) = (n n)/ =n

7 (n 3) + (n ) = n n n 3 n n

8 Problem: Calculate Reversing the summation is not helpful, since = and so on. This broken symmetry is reflected in the complexity of the solution: Answer:

9 Example: Solving equations Problem: Find the solutions to the equation x =0 y 4 y = x x Solutions: x =0

10 The equation x =0 y 3 y = x 3 3 x Solutions: x =and x =

11 The equation x =0 y y = x 3 3 x 3 Solutions: x = p and x = p

12 The equation x +=0 y 5 y = x x By symmetry, we expect to find two solutions, but what are they?

13 p To solve the equation x +=0we need to invent a new number i = p such that i =, so i +=( i) +=0 Solutions: x = i and x = i Note: This is similar to how we solve the equation x +=0, by inventing the number. Negative numbers were not widely accepted until the 6th century!

14 Complex numbers We now have a new number system, the complex numbers C = {a + bi : a and b are real numbers} e.g. 3i and p + i are complex numbers. We add and multiply in a natural way, remembering i =, e.g. ( 3i)+( 4+7i) =( 4) + ( 3i +7i) = +4i ( 3i) ( 4+7i) =( 4) + ( 7i)+( 3i 4) + ( 3i 7i) = 8 + 4i + i i =( 8 + ) + (4i + i) = 3 + 6i

15 The complex plane We can associate the complex number a + bi with the point in the plane with coordinates (a, b). Conversely, any point in the plane corresponds to a complex number. Im 3 3+i 3 Re

16 Applications Complex numbers have fundamental applications throughout mathematics, science, engineering and technology. For example: Quantum physics Relativity Fluid dynamics Electrical engineering Digital signal processing etc. etc.

17 The quadratic formula We can use complex numbers to solve any quadratic equation. Consider ax + bx + c =0 where a, b and c are numbers (with a 6= 0). The solutions are given by the familiar quadratic formula: x = b ± p b 4ac a Example: x 0x + 40 = 0: a =, b = 0, c = 40 x = 0 ± p =5± p 60 = 5 ± p 5 = 5 ± p 5i

18 Symmetry in the solutions Im 4 5+ p 5i Re p 5i

19 Girolamo Cardano

20 Higher degree equations Cardano also studied extensions of the quadratic formula to cubic and quartic equations. The formulae are complicated(!) e.g. x = + 3 v b u 3a + t 3 b 3 7a 3 v u t b 3 7a 3 bc a + d s b 3 + a 4 7a 3 bc a + d s b 3 a 4 7a 3 bc a + d + c a 7 a bc a + d + c a 7 a b 3 3a b 3 3a is a solution of the cubic equation ax 3 + bx + cx + d =0 Problem: Is there a similar formula for solutions of the quintic equation ax 5 + bx 4 + cx 3 + dx + ex + f =0

A mathematical theory of symmetry By studying the symmetries of the solutions, Évariste Galois showed that there is no such formula for the quintic

21 A mathematical theory of symmetry By studying the symmetries of the solutions, Évariste Galois showed that there is no such formula for the quintic equation! By encoding the symmetries in an algebraic object called a group, this incredible breakthrough marked the birth of a mathematical theory of symmetry.

22 Groups and symmetry: An example!!!. Fold paper in half long-ways, then open it out flat. Turn bottom left corner up to touch the fold line, making a sharp point with the bottom right corner, and fold 3. Fold the two red edges together, and then tuck in the top corner

23 4. Label the corners,,3 on both sides, so each corner has the same label front and back 5. Imagine the outline of an equilateral triangle on your desk: 6. Check that there are six different ways (keeping track of the corners) to place your paper triangle onto this outline

24 The six configurations

25 The symmetries of an equilateral triangle I I Identity symmetry 3 3 C A 3 3 C A 3 3 Clockwise rotation Anticlockwise rotation

26 The symmetries of an equilateral triangle T T Top corner flip 3 3 L R 3 3 L R 3 3 Left corner flip Right corner flip

27 Combining symmetries We can multiply two symmetries by performing one after the other, e.g. 3 T 3 A 3 so 3 T?A 3 and the product T?A is itself a symmetry. More precisely, T?A = R

28 The symmetry group? I C A T L R I I C A T L R C C A I R T L A A I C L R T T T L R I C A L L R T A I C R R T L C A I The symmetries of an equilateral triangle are encoded by its symmetry group ({I, C, A, T, L, R},?) Big idea: We can study and compare mathematical objects by investigating the (algebraic) properties of their corresponding symmetry groups.

29 Properties? I C A T L R I I C A T L R C C A I R T L A A I C L R T T T L R I C A L L R T A I C R R T L C A I I?X = X?I = X for any symmetry X Each symmetry occurs exactly once in each row and column In particular, each symmetry has an inverse, e.g. C?A = A?C = I, so A is the inverse of C Order matters, e.g. C?T 6= T?C

30 Group Theory The concept of a symmetry group can be generalised, leading to the notion of an abstract group, which are fundamental objects in Pure Mathematics. Groups arise naturally in many different contexts, e.g. (Z, +) is a group, where Z = {..., 3,,, 0,,, 3,...} (C, +) is a group ({,,i, i}, ) is a group. Here is the group table: i i i i i i i i i i i i Groups of matrices, groups of functions... etc. etc.

31 Simple groups Let G = {I, C, A, T, L, R} be the symmetry group of an equilateral triangle.? I C A T L R I I C A T L R C C A I R T L A A I C L R T T T L R I C A L L R T A I C R R T L C A I Consider the subgroups H = {I, C, A} and K = {I, T}. Every element of G is of the form X?Y, where X is in H and Y is in K, so G = H?K is a factorisation of G.

32 The atoms of symmetry We have factorised G = H?Kas a product of H and K. Here H and K are special because they cannot be factorised any further. Groups like this are called simple groups they play the role of prime numbers in group theory. Key fact: Every group can be factorised as a product of simple groups, so the simple groups are the basic building blocks of all groups. Big idea: Simple groups encode the atoms of symmetry. Big problem: Find all the simple groups!

33 The Classification Theorem The Classification of Finite Simple Groups is one of the most amazing achievements in the history of mathematics! Theorem. Any finite simple group is one of the following:. A group with a prime number of elements. A group of alternating or Lie type 3. One of 6 sporadic groups This problem occupied a global team of mathematicians for several decades the theorem was announced in 980 The proof is incredibly complicated it is over 0000 pages long! The theorem provides us with a periodic table of groups, which gives a complete description of the atoms of symmetry

34 PSLn+(q), Ln+(q) q n(n+)/ n i+ (q ) (n+,q ) i= F but is the (index ) commutator subgroup of It is usually given honorary Lie type status. The groups starting on the second row are the classical groups. The sporadic suzuki group is unrelated to the families of Suzuki groups. Copyright c 0 Ivan Andrus q 36 (q )(q9 )(q8 ) (q6 )(q5 )(q ) (3, q ) F q 63 9 i (, q ) (q ) i= i6=, q 0 30 (q )(q4 ) 0 (q )(q8 )(q4 ) (q )(q8 )(q ) in the upper left are other names by which they may be known. For specific non-sporadic groups these are used to indicate isomorphims. All such isomorphisms appear on the table except the family Bn( m ) = Cn( m ). with the following exceptions: Bn(q) and Cn(q) for q odd, n > ; A8 = A3() and A(4) of order q 4 (q )(q8 ) 6 (q )(q ) q 6 6 (q )(q ) q (q 8 + q 4 + ) 6 (q )(q ) q 36 (q )(q9 + )(q 8 ) (q 6 )(q5 + )(q ) (3, q + ) 3 4 q (q + )(q ) q (q 6 + )(q 4 ) (q 3 + )(q ) q 3 (q 3 + )(q ) On+(q), Wn+(q) q n n i (q ) (, q ) i= q n n i (q ) (, q ) i= q n(n ) (q n ) n i (q ) (4,q n ) q n(n ) (q n n +) i (q ) (4,q n +) i= q n(n+)/ n+ (q i ( (n+,q+) )i ) i= F7, HHM, HTH The Periodic Table Of Finite Simple Groups 0, C, Z Dynkin Diagrams of Simple Lie Algebras An 3 n F4 i C A(4), A(5) A5 A() A(7) Bn h 3 n Dn 3 4 n G i A3(4) B(3) C3(3) D4() D 4 ( ) G() 0 A (9) C3 60 A(9), B() 0 A6 68 G(3) 0 A(8) Cn i 3 n E6,7, B(4) C3(5) D4(3) D 4 (3 ) A (6) 3 C Tits A7 A() E6() E7() E8() F4() G(3) 3D 4 ( 3 ) E 6 ( ) B ( 3 ) F 4 () 0 G (3 3 ) B3() C4(3) D5() D 5 ( ) A (5) C A3() A8 A(3) E6(3) E7(3) E8(3) F4(3) G(4) 3D 4 (3 3 ) E 6 (3 ) B ( 5 ) F 4 ( 3 ) G (3 5 ) B(5) C3(7) D4(5) D 4 (4 ) A 3 (9) C A9 A(7) E6(4) E7(4) E8(4) F4(4) G(5) 3D 4 (4 3 ) E 6 (4 ) B ( 7 ) F 4 ( 5 ) G (3 7 ) B(7) C3(9) D5(3) D 4 (5 ) A (64) C PSpn(q) O + n (q) O n (q) PSUn+(q) Zp An An(q) E6(q) E7(q) E8(q) F4(q) G(q) 3D 4 (q 3 ) E 6 (q ) B( n+ ) F 4( n+ ) G(3 n+ ) Bn(q) Cn(q) Dn(q) D n(q ) A n(q ) Cp n! i= p Alternating Groups Classical Chevalley Groups Chevalley Groups Classical Steinberg Groups Steinberg Groups Suzuki Groups Ree Groups and Tits Group Sporadic Groups Cyclic Groups The Tits group 4 ()0 is not a group of Lie type, 4 (). Alternates Symbol Order For sporadic groups and families, alternate names Finite simple groups are determined by their order M M Sz O NS, O S Suz O N M M3 M Co3 Co Co J(), J() J F5, D HN HJ J LyS Ly HJM J F3, E Th J4 HS M() M(3) Fi Fi McL He F3+, M(4) 0 F Fi4 0 B Ru F, M M

35 Summary Symmetry is a central idea in mathematics, which arises in many different ways Symmetries can be exploited to find simple and elegant solutions Seeking symmetry has led to many fundamental breakthroughs that have revolutionised science and technology Mathematicians have developed the powerful language of group theory to study symmetry in all its forms, with many far reaching applications

36 Further reading Podcast series by Ian Stewart: Video by Marcus du Sautoy: du sautoy symmetry reality s riddle.html Video by Tim Burness and John Conway: Ian Stewart, Why Beauty is Truth: The History of Symmetry, 008 Ian Stewart, Symmetry: A Very Short Introduction, 03 Marcus du Sautoy, Finding Moonshine: A Mathematician s Journey Through Symmetry, 009

Lecture 5.7: Finite simple groups

Lecture 5.7: Finite simple groups Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 410, Modern Algebra M. Macauley (Clemson) Lecture 5.7: