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H Solutions to Problems
3.3 There are only three regular tesselations of the plane: triangles, squares, and
hexagons.
3.4 The rotation generates points that are lying on a circle, perpendicular to the rotation. If the rotational angle is not a rational fraction of a full angle, every time
the rotation is repeated, a new point will be generated. To obtain an integer order, the additional requirement is to be added that the original point is retrieved
after one full turn.
3.5 Consider a subgroup H ⊂ G such that |G|/|H |=2. Then the coset expansion
of G will be limited to only two cosets:
G = H +ˆ gH
Here ˆ
g is a coset generator outside H . The subgroup is normal if the right and
left cosets coincide, Since there is only one coset outside H , it is required that
ˆ
gH = H ˆ
g
Suppose that this equation does not hold. Then this can only mean that there are
elements in H such that
ˆ
h x ˆ
g = ˆ
h y
But then the coset generator must be an element of H , which contradicts the
staring assumption.
3.6 Soccer ball: I h . Tennis ball: D 2d . Basketball: D 2h . Trefoil knot: D 3 .
3.7 The figure (from Wikipedia) shows the helix function for n = 1. One full turn
is realized for t/a = 2π ≈ 6.283. This is a right-handed helix.
H Solutions to Problems
3.3 There are only three regular tesselations of the plane: triangles, squares, and
hexagons.
3.4 The rotation generates points that are lying on a circle, perpendicular to the rotation. If the rotational angle is not a rational fraction of a full angle, every time
the rotation is repeated, a new point will be generated. To obtain an integer order, the additional requirement is to be added that the original point is retrieved
after one full turn.
3.5 Consider a subgroup H ⊂ G such that |G|/|H |=2. Then the coset expansion
of G will be limited to only two cosets:
G = H +ˆ gH
Here ˆ
g is a coset generator outside H . The subgroup is normal if the right and
left cosets coincide, Since there is only one coset outside H , it is required that
ˆ
gH = H ˆ
g
Suppose that this equation does not hold. Then this can only mean that there are
elements in H such that
ˆ
h x ˆ
g = ˆ
h y
But then the coset generator must be an element of H , which contradicts the
staring assumption.
3.6 Soccer ball: I h . Tennis ball: D 2d . Basketball: D 2h . Trefoil knot: D 3 .
3.7 The figure (from Wikipedia) shows the helix function for n = 1. One full turn
is realized for t/a = 2π ≈ 6.283. This is a right-handed helix.