MATH 0052 GROUPS AND GEOMETRY FINAL ONLINE
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MATH 0052 GROUPS AND GEOMETRY FINAL ONLINE
Question 1. The diagram shows a cube with each face bisected by a line parallel to two of its sides:
Use the orbit-stabiliser theorem to calculate the order of the of this figure (reflections allowed).
symmetry group
5 marks
Question 2.
(a) Let P be a regular heptagon (seven sides). What is the symmetry group of P (including reflections)?
5 marks
(b) A two-colouring of P is a colouring of each edge of P either yellow or blue. Two two-colourings are called equivalent if one is obtained from the other by a symmetry of P . How many inequivalent two- colourings of P are there?
5 marks
Question 3.
(a) In two dimensions a glide reflection in the direction of the x-axis is the composite of a translation in the direction of the x-axis and a reflection in the x-axis. Express such a glide reflection as a product of reflections.
5 marks
(b) Is it true that every rotation of R4 has an axis? Justify your answer with a proof or a counterexample.
5 marks
Question 4.
(a) Let p and q be integers > 2. Let T be a triangle with angles π/2, π/p and π/q. For which values of p and q is T spherical, euclidean, or hyperbolic?
5 marks
(b) Suppose that a finite set of spherical line-segments divide the surface of S2 into N congruent equilateral triangles and that the angle α at any vertex of any triangle lies in (0, π). Classify all such tesselations of S2 by equilateral triangles, giving the values of N and α in each case.
10 marks
Question 5. Show that if ρ < 1, the set
|z - i|
is a hyperbolic circle
{z e H : d扛 (z, i) = R}
(where d扛 is hyperbolic distance in the upper half-space model of the hy- perbolic plane). Calculate R in terms of ρ .
5 marks
Question 6. Consider the action of the group PSL(2, C) on the Riemann sphere S2 by M¨obius transformations. Suppose that C c S2 is a circle and that g e PSL(2, C) maps C to itself, g(C) = C. Show that the trace of g must be real or imaginary.
10 marks
Question 7.
(a) Let R1 and R2 be reflections in Rn in orthogonal mirrors Π 1 and Π2 , which you may take to contain the origin. Show that R1R2 = R2R1 . Explain why this is a half-turn about the axis Π 1 n Π2 .
5 marks
(b) In R3, describe, as precisely as you can, the composite of two half- turns about distinct axes, distinguishing carefully the different pos- sible cases. [If the two axes do not meet and are not parallel, you may find it helpful to make use of the unique line which meets both axes at 90О .]
10 marks
In answering this question, bear in mind that the composite of two half- turns must be an isometry. If, for example, this isometry comes out to be a rotation, when you ‘describe’ it you will need to give its axis and angle. Similar information needs to be given for other types of isometries.
One approach is to write the half-turns as products of reflections in ap- propriately chosen pairs of orthgonal planes—but you may use any method.
TOTAL: 70 marks
2022-05-18