two other rotations

1. Problem: Compute the other four permutations two otherrotations a b c two otherreflections 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

2. Problem: complete the following table Further discussion: the patterns in this table

3. 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

4. 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)

5. Problem: show that the following set of 3 x 3 matrices forms a subgroup of this group The matrices permute the standard basis vectors

6. 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

7. 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

8. Problem: construct the 2 x 2 matrix that represents this transformation Further discussion: what is the transpose of this matrix ? why is it called a symmetric matrix ?

9. Problem: Compute the other root of this Golden Ration equation the Golden Mean Equation is equivalent to the quadratic equation whose roots are so the Golden Mean is the unique positive root and the other root is

10. Problem Show that every vector in can be expressed uniquely as a sum with if and where then so for any vector we can choose to obtain Problem Show that the k-fold transformation satisfies follows directly from linearity of T

11. Problem: Show that the eigenspaces (lines) for T are orthogonal, this follows necessarily since the matrix that represents T is symmetric

12. 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.

13. 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.

14. 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%20Ratio%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

15. 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 so and the corners of the pentacle are the powers of and the point can be expressed as the a convex combination as therefore a simple calculation yields students : work this last = out yourselves ! Pentacle

16. Problem: Show, using a direct computation, that the reciprocal lattice is spanned by the columns of the transposed inverse of and for for all if and only if is a linear combination with integer coefficients of the column vectors of

17. 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 by Since the vibration matrix is symmetric the famous spectral theorem implies that V can be decomposed as the sum of eigenspaces just like for the case of the matrix that generates the Fibonacci numbers – wow ! Then observe that for every and 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