question: why a fractal has an infinite perimeter
comment: Benoit Mandelbrot, his name is on that famous fractal, asked the seemingly foolish question, "How long is the coastline of England?" It's hard to imagine a more well-mapped coastline, right? Mandelbrot stated that the coastline of England is infinitely long. It turns out that the length of that coastline depends on the length of your yardstick. If you use one half a mile long you get an answer. If you use one a yard long you get a longer answer, because you follow a much "bendier" pathway around the coast, in and out of much smaller details than you could with the longer stick. A much shorter stick and you get a much longer answer because you are down to going around yet smaller and smaller details. As the length of your stick gets closer to infinitely short you get closer to the infinitely long answer according to Mandelbrot. You've never seen a picture of the famous Mandelbrot set, it isn't even possible to create one, the best you can get is an approximation. The set has more details in its perimeter than can be displayed on any monitor. A monitor will show what appear to be tiny little "mini-mandelbrot" satellites all around the main set but it is a mathematically proven fact that all elements of the set are in fact connected. They are all within one perimeter, the connections are just smaller than the limit of your monitor's resolution. No matter how many times you zoom in on any portion of it there will always detail beyond the limit of your resolution. Every time you zoom in you'll still face the same problem *because* the perimeter of the set is infinitely long.
passages: One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.
   The total area covered at the th iteration is:
  
  while the total length of the perimeter is:
  
  which approaches infinity as increases.
  
  Section::::See also.
  
  BULLET::::- List of fractals by Hausdorff dimension
  
  BULLET::::- Gabriel's Horn (infinite surface area but encloses a finite volume)
  
  BULLET::::- Gosper curve (also known as the Peano-Gosper curve or "flowsnake")
  
  BULLET::::- Osgood curve
  
  BULLET::::- Self-similarity
  
  BULLET::::- Teragon
  
  BULLET::::- Weierstrass function
  
  Section::::External links.
  
  BULLET::::- von Koch Curve
  
  BULLET::::- The Koch snowflake in Mathworld
  
  BULLET::::- The Koch Curve poem by Bernt Wahl
  
  BULLET::::- Computing iterations of the Koch curve in WolframAlpha
   Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a fractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the fractal dimension of the fractal.
  
  Section::::Area bisectors.
   Section::::Mathematics.:Geometry and topology.
  
  Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, such as Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
  
  Section::::Mathematics.:Fractals.
  
  The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.
  
  Section::::Mathematics.:Mathematics without infinity.
   BULLET::::- : or : A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottle.
   This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has infinite perimeter.
   The Vicsek fractal has the surprising property that it has zero area yet an infinite perimeter, due to its non-integer dimension. At each iteration, four squares are removed for every five retained, meaning that at iteration "n" the area is formula_2 (assuming an initial square of side length 1). When "n" approached infinity, the area approaches zero. The perimeter however is formula_3, because each side is divided into three parts and the center one is replaced with three sides, yielding an increase of three to five. The perimeter approaches infinity as "n" increases.
   The Koch curve has an infinite length, because the total length of the curve increases by a factor of with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being the length of the segments in the previous stage. Hence, the length of the curve after iterations will be () times the original triangle perimeter and is unbounded, as tends to infinity.
  
  Section::::Properties.:Perimeter of the Koch snowflake.:Limit of perimeter.
  
  As the number of iterations tends to infinity, the limit of the perimeter is:
  
  since ||  1.
   If one removes a piece from a figure, its area decreases but its perimeter may not. In the case of very irregular shapes, confusion between the perimeter and the convex hull may arise. The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. In the animated picture on the left, all the figures have the same convex hull; the big, first hexagon.
  
  Section::::Isoperimetry.
   BULLET::::- "The perimeter of a fractal may be iteratively drawn "ad infinitum"."
  
  BULLET::::- The 17th-century writer Jonathan Swift incorporated the idea of self-similarity in the following lines from his satirical poem "On Poetry: a Rhapsody" (1733):
  
  poem
  
  The vermin only teaze and pinch
  
  Their foes superior by an inch.
  
  So, naturalists observe, a flea
  
  Has smaller fleas that on him prey;
  
  And these have smaller still to bite 'em,
  
  And so proceed "ad infinitum".
  
  Thus every poet, in his kind,
  
  Is bit by him that comes behind/poem
  
  BULLET::::- The mathematician Augustus De Morgan included similar lines in his rhyme "Siphonaptera".
   The fractal dimension of a single contour will be between 1 and 2.
  
  Exponential growth ensures that the contour, in the limit of very
  
  long time integration, becomes fractal.
  
  Fractals composed of a single curve are infinitely long and when
  
  formed iteratively, have an exponential growth rate, just like an
  
  advected contour.
  
  The Koch Snowflake, for instance, grows at a rate of 4/3 per iteration.
  
  The figure below shows the fractal dimension of an advected contour as a function
  
  of time, measured in four different ways. A good method of measuring the
   Squares can be used to generate similar fractal curves. Starting with a unit square and adding to each side at each iteration a square with dimension one third of the squares in the previous iteration, it can be shown that both the length of the perimeter and the total area are determined by geometric progressions. The progression for the area converges to 2 while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve. The resulting area fills a square with the same center as the original, but twice the area, and rotated by radians, the perimeter touching but never overlapping itself.
   A sphere, a surface of constant radius and thus diameter, is a surface of constant width.
  
  Contrary to common belief the Reuleaux tetrahedron is "not" a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width. These shapes were conjectured by to have the minimum volume among all shapes with the same constant width, but this conjecture remains unsolved.
   The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look smoother than the old mesh.
  
  Repeated subdivision results in smoother meshes. It can be shown that the limit surface obtained by this refinement process is at least formula_2 at extraordinary vertices and formula_3 everywhere else (when "n" indicates how many derivatives are continuous, we speak of formula_4 continuity). After one iteration, the number of extraordinary points on the surface remains constant.
   Section::::Statistics.:Measuring a coastline.
  
  More than a decade after Richardson completed his work, Benoit Mandelbrot developed a new branch of mathematics, fractal geometry, to describe just such non-rectifiable complexes in nature as the infinite coastline. His own definition of the new figure serving as the basis for his study is:
   Iteration "n" in the construction adds 2 squares of area formula_1, for a total area of 1. Thus the area of the tree might seem to grow without bound in the limit as "n" → ∞. However, some of the squares overlap starting at the order 5 iteration, and the tree actually has a finite area because it fits inside a 6×4 box.
  
  It can be shown easily that the area "A" of the Pythagoras tree must be in the range 5  "A"  18, which can be narrowed down further with extra effort. Little seems to be known about the actual value of "A".
   However, not all curves can be measured in this way. A fractal is, by definition, a curve whose complexity changes with measurement scale. Whereas approximations of a smooth curve tend to a single value as measurement precision increases, the measured value for a fractal does not converge.
   Section::::Other factors.:Fractal star distribution.
   The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, each scaled down by a linear factor of /2, such that the corners of the squares coincide pairwise. The same procedure is then applied recursively to the two smaller squares, "ad infinitum". The illustration below shows the first few iterations in the construction process.
  
  Section::::Area.
   Section::::"D" is not a unique descriptor.
  
  As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns. The value of "D" for the Koch fractal discussed above, for instance, quantifies the pattern's inherent scaling, but does not uniquely describe nor provide enough information to reconstruct it. Many fractal structures or patterns could be constructed that have the same scaling relationship but are dramatically different from the Koch curve, as is illustrated in Figure 6.
   In "Tilings and Patterns", published in 1987, Branko Grünbaum and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares grew exponentially. For example, the plane can be tiled with different integral squares, but not for every integer, by recursively taking any perfect squared square and enlarging it so that the formerly smallest tile now has the size of the original squared square, then replacing this tile with a copy of the original squared square.
   That is, if the "dimension" of the parameter vector is less than the "number of functions" of the parameter vector in the above representation of the probability density function. Note that most common distributions in the exponential family are "not" curved, and many algorithms designed to work with any exponential family implicitly or explicitly assume that the distribution is not curved.
   The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side).
  
  Section::::Area formulas.:Fractals.
   formula_27
  
  Section::::Upper and lower bounds.
  
  formula_14 has fractal-like behavior with a sharp upper bound for formula_31 given by
  
  The upper bound only contacts formula_14 at 'high-water' points when formula_34.
  
  These are also the generations at which the UWCA based on squares, the Hex–UWCA based on hexagons and the Sierpinski triangle return to their base shape.
  
  Section::::Upper and lower bounds.:Limit superior and limit inferior.
  
  We have
   The modeling of the Earth's rough surfaces via fractional Brownian motion was first proposed by Benoît Mandelbrot.
  
  Because the intended result of the process is to produce a landscape, rather than a mathematical function, processes are frequently applied to such landscapes that may affect the stationarity and even the overall fractal behavior of such a surface, in the interests of producing a more convincing landscape.
  