Because the distance from one point at the circumference through the center to another point at the circumference is always the same, at an infinite set of coordinates along the circle (anywhere, relative to the size of the circle, and always providing an axis which perfectly dissects the circle).
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A cylinder has an infinite number of lines of symmetry (because a circle has an infinite number of lines of symmetry).
Infinite lines because a circle has infinite lines of symmetry.
That number is infinite. Every diameter of a circle is a line of symmetry.
circle
There are infinite planes of symmetry in a cylinder in the same way that a circle has infinite lines of symmetry.