This exercise shows that the group of symmetries of a certain type of geometric figure may depend on the dimension of the space in which we consider the figure to lie.
a. Describe all symmetries of a point in the real line ℝ; that is, describe all isometries of ℝ that leave one point fixed.
b. Describe all symmetries (translations, reflections, etc.) of a point in the plane ℝ2.
c. Describe all symmetries of a line segment in ℝ.
d. Describe all symmetries of a line segment in ℝ2.
e. Describe some symmetries of a line segment in ℝ3.
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