Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The book falls naturally into two parts. Part I is concerned with the general theory of fractals and their geometry. Firstly, various notions of dimension and methods for their calculation are introduced. Then geometrical properties of fractals are investigated in much the same way as one might study the geometry of classical figures such as circles or ellipses: locally a circle may be approximated by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse, a circle typically intersects a straight line segment in two points (if at all), and so on. There are fractal analogues of such properties, usually with dimension playing a key role. Thus we consider, for example, the local form of fractals, and projections and intersections of fractals.

What's the meaning of the bold text? How does one approximate a circle with a line segment? I've searched a little for it but didn't have success.

share|improve this question
add comment

3 Answers

Locally you can approximate any curve by a line segment. Just stick lots of points on your circle and draw chords connecting consecutive points. You get a polygon (with lots of sides) that approximates your circle. (The degree to which you have a "good" approximation depends on continuity/smoothness of the curve and, of course, how close your points are.)

share|improve this answer
add comment

Locally, any function with a Taylor series can be approximated by a line segment.

In case of the circle, it's easy to visualize. Imagine a regular $n$-gon circumscribed by a circle, and look at what happens as $n \to \infty$. Each local piece of the circle will be approximated by a line segment, and the precision will be better as $n \to \infty$.

The idea of a local approximation is that it only works in a small neighborhood.

share|improve this answer
add comment

If you zoom in on a small arc of a circle, it will appear to be more and more as a straight line interval as your magnification becomes greater and greater. In fact, you can define differentiability this way; see for example Keisler. Fractal curves are not differentiable and they don't "straighten out" to look like line segments when you zoom in. Alternatively, this can be expressed in differential-geometric terms by saying that when you magnify the circle, its geodesic curvature tends to zero.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.