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A vertex is defined as a 'meeting point of two lines that form an angle'.

When I increment the circumference of a circle into infinitesimal small increments I get something like shown below in the picture.

This infinitesimal small increment of the circumference can then be divided into left and right which is shown by the black arrows in the picture.

My question is, if the part of the circumference is infinitesimal small, the left and right part of it would not be curved and therefor the above mentioned description of a vertex would be true.

sketch

  • At an infinitesimal scale, what you are zooming onto is a line that is perpendicular to the radius. This is why this isn't true. – sreysus Sep 17 '23 at 13:43

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You are assuming that if you zoom in far enough on the circle, then you will be able to observe a "corner" where two lines meet to form a vertex. In other words, you are treating a line segment with infinitesimal length as if it were a line segment with a finite but very, very, small length. Infinitesimal is not a number in the conventional sense; it is more of a concept. In the same way inifinity refers to a quantity that increases without limit, infinitesimal refers to a quantity that decreases without limit. So, as you zoom in on the circle, you will NEVER be able to observe a corner where two lines meet to form a vertex.

RyRy the Fly Guy
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