Transcript Ans_Wayne

Problem: Compute the other four permutations
a 
c 
b  Rot
12 0

a 
 
 
a b 
 c 
b
two other
rotations
a 
a 
b  Rot

0  b 
 
 
 c 
 c 
a 
b 
b  Rot
24 0

 c 
 
 
 c 
a 
c
a 
a 
a 
c 
fa
b  Re


 c  two other b  Ref

 b 
 
 
  reflections
 c 
a 
 c 
b 
b
a 
b 
b  Ref

c  a 
 
 
 c 
 c 
Further discussion: there are 24 permutations of
the set {a,b,c,d}, there are 8 symmetries (rotations
and reflections), so there are 16 permutations of its
vertices that can not be obtained by symmetries
Problem: complete the following table

 Rot0
 Rot120
Rot240
 Refa
 Ref
b
 Ref
 c
Rot 0
Rot120 Rot 240
Ref a
Ref b
Rot 0
Rot120 Rot 240
Ref a
Ref b
Rot120 Rot 240
Rot 0
Ref b
Ref c
Rot 240
Rot 0
Rot120
Ref c
Ref a
Ref a
Ref c
Ref b
Rot 0
Rot 240
Ref b
Ref a
Ref c
Rot120
Rot0
Ref c
Ref b
Ref a
Rot 240 Rot120

Ref c 
Ref a 
Ref b 
Rot120 

Rot 240

Rot 0 
Ref c
Further discussion: the patterns in this table
Problem: use the table to show that the set of
symmetries of an equilateral triangle with
composition forms a group
rotation by 0 is the identity since the first row and
first column coincide with the factors that
correspond to the first row and the first column,
inverses exists since every row and every column
contains rotation by 0
Problem: use the associative property to prove
that if A and B and m x m matrices and AB =
identity matrix then B is the inverse of A
I = AB  (inv A) = (inv A) I =
(inv A)(AB) = ((inv A)A)B = IB = B
Problem: for any integer m > 0 show that the set
of m x m matrices with nonzero determinant
forms a group under matrix multiplication and
describe it as a group of transformations
Associativity: follows from the fact that such
matrices are in 1-1 correspondence with the set of
invertible linear transformations of R^n onto itself
and matrix multiplication corresponds to
composition of transformations
Identity: corresponds to the m x m matrix with 1
in the diagonal entries and 0 in all other entries
Inverse: inv M = (1/det(M)) * (cofactor M)
Problem: show that the following set of 3 x 3
matrices forms a subgroup of this group
The matrices permute the standard basis vectors
1
0 
0 
 
0 


a 0
c

b 1
 
 
 
0
1
0
1 0 0
0 0 1 
0 1 0 






Rot 0  0 1 0 , Rot 240  1 0 0 , Rot120  0 0 1






0 0 1
0 1 0
1 0 0
Ref a
1 0 0



0 0 1 , Refb


0 1 0
0 0 1 



0 1 0 , Refc


1 0 0
0 1 0 



1 0 0


0 0 1
Problem: Show that the set of integers Z is a group
under addition and that f is a representation of Z
that Z is a group with identity 0 and that the
inverse of every integer n is –n is clear, for
the doubtful it can be proven using Peano’s
axioms for the natural numbers
0 n  m 0 n 0 m
f ( n  m)  






1  0 1 0 1 
0
f ( n ) f ( m ),
n, m  Z
Problem: Construct 2 &3 dimensional representations
of the group of symmetries of an equilateral triangle
the 3 dimensional representation is described by the
set of 6 permutation matrices, a 2 dimensional
representation is given by computing matrices that
rotate and reflect an equilateral triangle, whose
centroid is the origin, into iteself
 ( Rot
 ( Ref



)  
0






)  
a










1 0
,  ( Rot
0 1




)  
240





1
2
3
2
3 
2 
 ,  ( Rot120 )
1
2 

1 0
,  ( Refb )   ( Ref a )  ( Rot
0 1







 ( Refc )





 



 ( Ref a )  ( Rot




)  
240







)  
120



1
2
3
2

1
2
3
2
1
2

3
2
3 
2 

1
2 





3 
2 

1 
2 
3 
2 

1 
2 
Problem: construct the 2 x 2 matrix that represents this transformation
y  0 1  x 
T

x 


 y 
 x  y  1 1  y 
Further discussion: what is the transpose of this matrix ?
why is it called a symmetric matrix ?
Problem: Compute the other root of this Golden Ration equation
the Golden Mean Equation
1


is equivalent to the quadratic equation
whose roots are

1
   1  0
2
1 5 1 5
,
2
2
so the Golden Mean is the unique positive root
and the other root is

1
2
Problem Show that every vector in
R
can be expressed uniquely as a sum u  w
if
 1 
1 
u  r    V1 and w  s 


1
/



 
with
u V1 , w V2
where r , s  R
1 1  r 
 v1 
then v     u  w  
1  


v
  s
 2

1
r  1 1   v1 
so for any vector v  R we can choose    
1  


  v2 
s 
to obtain v  u  w
2
Problem Show that the k-fold transformation satisfies
T (u  w)   u  (1 /  ) w
k
follows directly from linearity of T
k
k
Problem: Show that the eigenspaces (lines) for T are orthogonal, this
follows necessarily since the matrix that represents T is symmetric
1 

1

ur
 V , w  s 1 

  1
 
 u  w  rs (1   ( 
1

))  0
Problem: How does the Golden Ratio
describe the shape of Nautilus shells?
http://www.geocities.com/CapeCanaveral/Station/8228/spiral.htm
The Nautilus The first diagram is of the outside of a nautilus shell. The
second diagram shows that a spiral can be drawn by putting together
quarter circles, one in each new square. This is the golden spiral. This is
present because the growth of the nautilus is proportional to the size of
the organism. A similar curve to this occurs in the shape of a nautilus
shell. The Fibonacci rectangles spiral increases in size by a factor of Phi
(1.618..) in a quarter of a turn, the nautilus spiral curve takes a whole
turn before points move a factor of 1.618... from the center. The third
diagram is a cross section of a nautilus shell, in which the golden spiral
can be seen. This pattern is also known as the Logarithmic Spiral.
Problem: How does the Golden Ratio describe the
placement of leaves on Calamansi plants ?
http://www.geocities.com/CapeCanaveral/Station/8228/pineandsun.htm
Phyllotaxis is the botanical term for a topic which includes the arrangement of leaves on the stems of plants.
Many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If one looks down on a plant,
the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the
sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
The Fibonacci numbers occur when counting both the number of times one goes around the stem, going from leaf to leaf, as
well as counting the leaves one meets until one encounters a leaf directly above the starting one. If one counts in the other
direction, one gets a different number of turns for the same number of leaves. The number of turns in each direction and the
number of leaves met are three consecutive Fibonacci numbers. When one divides the number of turns by the number of
leaves, one will find the Golden Ratio.
Some of the most common arrangements are in ratios of alternating Fibonacci numbers: 2/5, found in roses and fruit trees,
or 3/8, which is found in plantains, or 5/13, found in leeks, almonds, and pussy willows. In addition, the number of times one
has circled the stalk will be another Fibonacci number.
Problem: How does the Golden Ratio describe the shape of humans?
Leonardo Davinchi is famous for investigating the
proportions of the human body and the link this had with
the circle, pentagon, square and ultimately the "Golden
Ratio". The adaptation of Davinchi's "Vitruvian Man"
(Pictured on the left) illustrates this link.
http://people.bath.ac.uk/ajp24/goldenratio.html#The%20Golden%20Rati
o%20and%20The%20Human%20Body
The Golden Ratio and The Human Body
Careful study of the dimensions of the human body gives numerous examples of the Golden
Ratio and also the number 5, which was mentioned before as to be linked to the Golden Ratio.
(See "The Golden Ratio and Shapes" - "Pentagon")
Scientists have researched into whether a person whose face has many examples of the
"Golden Ratio" is more physically attractive than that of a person whose face does not conform
to the "Golden Ratio" dimensions, many ideas have been put forward but it is still an area of
debate. The following website is an interesting one which discusses work on beauty analysis
and its link to the golden ratio.
http://www.beautyanalysis.com/index2_mba.htm
Problem: Show that the ratios of line segments in a Pentacle (Brown
page 101) all equal PHI (inscribe the Pentacle in a unit circle in the
complex plane so the points are 5-th roots of unity and express the
intersections of lines as convex combinations)
let
  exp( 2i / 5)
so
 1
5
and the corners of the pentacle are the powers of

z can be expressed as the a convex combination as
2
3

z  t1  (1  t )  t  (1  t ) 2

z
therefore a simple calculation yields
|1 z | 1 t
1
4


1






2
|  z |
t
3
and the point
students : work this last = out yourselves !

Pentacle

4
Problem: Show, using a direct computation, that the reciprocal lattice
is spanned by the columns of the transposed inverse of
M
d

b
a c 


T
1
1
M 
M

ad bc 


b d 
 c a 
m


r


T
1
2 and
for w  M


M

L

R
n
s
 
 




w   rm  sn  Z for all m, n  Z
if and only if r , s  Z  w is a linear combination with
1 T
integer coefficients of the column vectors of
M 
Problem Prove this assuming that vib. eig. are eig of symmetric
matrices H and symmetry means that p(g)H = Hp(g) for all g in Group
Define the eigenspace for each vibrational eigenvalue
R
V  { v V : H v   v }
by
Since the vibration matrix is symmetric the famous spectral theorem
implies that V can be decomposed as the sum of eigenspaces
V   V
just like for the case of the matrix that
generates the Fibonacci numbers – wow !
Then observe that for every
v  V and g  Group
H ( g )v   ( g ) Hv   ( g )v   ( g )v
this means that each eigenspace for the vibration matrix is an invariant
subspace for the group representation and therefore can be decomposed
into irreducible representations – each of which is one of the irr. reps that
V decomposes into – this profound fact explains 98.5 % of molecular,
atomic, and nuclear physics and may explain ??? % of string theory