A limit is something towards which you will always get closer and closer. This doesn’t mean you will eventually reach it.
The euclidean length of a segment Δz=(Δx,Δy)Δz=(Δx,Δy) is Δx2+Δy2−−−−−−−−−√Δx2+Δy2 whereas the “taxicab” length of this segment is |Δx|+|Δy||Δx|+|Δy|. “In the limit” this implies that the euclidean circumference of the unit circle is 2π2π, whereas the “taxicab circumference” is 4.
The problem doesn’t have anything to do with pi, or with circles. You can see the same problem with a straight line:
The length of the diagonal is , but by the same logic, the black, red, green and blue lines are all of length , which is bigger.
There are several ways of looking at the paradox. The simplest is to simply note that the red line is not, in fact, a better approximation to the diagonal than the black ones are. It has only one additional point on the line, and an infinite number of ones off it.
You can repeat the operation indefinitely, adding more points, but there will always be more points off the line than on, for the same reason that there are more real numbers than integers. Formulated as a limit, the variable controlling the number of steps is an integer, while the length of the line is given by a real number. Thus, the set of steps is, even in the limit case, longer than the diagonal.
We can ask ourselves, then, why does Archimedes approximation to pi work? The answer is that Archimedes was trying to approximate a curve with a line, and the curve, unlike the steps, really does look flatter and flatter as you get closer and closer. The steps will always be steps, but a smaller and smaller approximation to a curve resembles a line. The limit case of a polygon really is a circle, while steps are always just steps.
- Panels one to four describe a sequence of curves. (Here, “curve” is a generic term referring to any continuous line, be it straight or crooked or curved.)
- Each curve in the sequence has a well-defined length of exactly 4.
These facts are also true:
- The sequence of curves converges uniformly on a limit.
- As panel five correctly states, the limit of the sequence is a circle.
- It is not a sawtoothed curve.
- It is not an “infinitely jagged” sawtoothed curve.
- It is not a “polygon with an infinite number of sides”.
- It is not a fractal.
- The limit is an ordinary, perfectly smooth perfect circle.
- Thus, the length of the limit is exactly π (3.1 or so). (Because it is a perfect circle with diameter 1.)
- It’s not 4!
And so is this final fact:
- None of these facts contradict each other.
The limit of a sequence isn’t necessarily a member of that sequence.
Because of this, the limit of a sequence need not necessarily share any properties with the members of that sequence.
Here, you’ve seen a sequence of curves of length 4, whose limit does not have length 4.
You’ve also seen a sequence of jagged, right-angled curves whose limit is not jagged or right-angled at all, but smooth.
This is not a problem. It’s not a contradiction. It’s just the way it is.
Many of the principles found in fractal geometry  have origins in earlier mathematics. For example scale ability and line lengths have long been associated with geometrical structures. In Elements, Euclid ( 330- 275 B.C. ) proposed lines with infinite lengths to illustrate the concept of parallel lines, there he also used self-similar triangles to show the congruency of triangles see Figure 4.6. Archimedes (287-212 B.C.) used spirals to illustrate repeating transformations. Later the mathematician Jacob Bernoulli (1654-1705) expanded this idea to show that some spirals could be drawn with an infinite length, of which the logarithmic spiral is the most famous see Figure 4.7. Another well known spiral with infinite length is the golden mean spiral derived from the ancient Greek’s golden ratio see Figure 4.8. This spiral closely resembles the sea creature, nautilus seen in Figure 3.89 in Chapter 3.
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexitycomparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.
The essential idea of “fractured” dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline’s measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formalmathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.
One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means arectifiable curve: the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but rather is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which in this case is a number between one and two.
Role of scaling
The concept of a fractal dimension rests in unconventional views of scaling and dimension. As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable stands for the number of sticks, for the scaling factor, and for the fractal dimension:
The symbol above denotes proportionality. This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because =3 when =1/3 as in the example above, =1, and for squares, because =9 when =1/3, =2.
The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, =4 when =1/3, and the value of can be found by rearranging Equation 1:
That is, for a fractal described by =4 when =1/3, =1.2619, a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in. The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, these images themselves are not true fractals because the scaling described by the value of cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.