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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

 Benoit Mandelbrot


Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory.
Benoît Mandelbrot  (1924-2010)

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Related Links (Outside this Site)

Fractal (Wikipedia)


Doodling in Math Class:  DRAGONS    by  Vi Hart  (2013-08-19).
Fractal Zoom Mandelbrot Corner   by  Gaurav Vohra.
"Fractals: The Colors of Infinity"   by  Arthur Clark   | 1 | 2 | 3 | 4 | 5 | 6 |
Hilbert's Curve:  Usefulness of the InfiniteGrant Sanderson  (2016-01-16).
Chaos Game  by  Ben Sparks  (Numberphile, 2017-04-27).

Fractal Geometry

 Nicolas Oresme 
 (1323-1382) (2011-07-09)   The fractional exponents of Nicolas Oresme
A positive quantity can be raised to the power of any number.

In his unpublished manuscript  Algorismus Proportionum  (probably written between 1356 and 1361)  Nicole d'Oresme (1323-1382)  made several important mathematical innovations, including fractional exponents.

Considering only positive quantities, Oresme realized that the cube root of  x  could usefully be considered as  x  raised to the power of  1/3.  The square of that would then be  x  raised to the power of  2/3.  He thus defined raising to the power of  p/q  as the  q-th  root of a  p-th  power  (or vice-versa).

Oresme went as far as to suggest that the exponent could even be an  irrational  number, because the resulting power could be obtained with arbitrary precision using rational approximation of such an exponent.

At the time, Oresme's innovation was revolutionary.  Powers  (mostly squares and cubes)  had previously been used to obtain the proper measurements of surfaces or solids  (area and volume, respectively)  from linear measurements of their sizes  (length).  What Oresme offered was an analytic perspective on the concept of exponentiation which had no geometrical counterpart...

... or so it seemed for a few centuries.  Read on.

 Helge von Koch 
 (1870-1924) (2011-07-09)   Von Koch's curve has dimension log 4 / log 3
Scaled by a factor of 3, it's congruent to 4 copies of itself.

Informally, the notion of a  measure  is based on the following features:

  • The  measure  of an object is a  nonnegative  number assigned to it.
  • The measure of two disjoint objects is the sum of their measures.
  • Two  congruent  objects have the same measure.

If the first requirement is dropped  (as is the case for the  chi characteristic, which is otherwise similar to a measure)  then no well-defined notion of dimensionality can be based on the resulting  semi-measure.

Refinements of the second of the above statements  (to accomodate infinitely many objects, not necessarily disjoint)  is the crucial concern of classical  measure theory.

The third statement depends entirely on what is meant by  congruent.  At the very least, in linear geometry, two objects that are obtained from each other by  translation  are congruent.  It's not at all trivial that objects obtained from each other by  rotation  are then necessarily congruent.

Arguably, a fourth statement more directly connected to dimensionality should be considered:  The measure of a cartesian product is the product of the measures of the components.  This seems especially important for a consistent generalization of the chi semi-measure.

Loosely speaking, in a vector space, if scaling an object by a factor  k  turns it into an object whose nonzero measure is  k times as large, then that object has  dimension  d.  For example, scaling an ordinary surface makes the measure of its area  k2  times as large and its dimension is thus  d = 2.

Scaling a  Koch curve  by a factor  3  turns it into an object whose measure must be  4  times as large (because it's composed of  4  congruent copies of the original object).  So, the dimension  d  of the  Koch curve, if it is defined at all, must be such that:

3 d   =   4     Therefore:   d   =   log 4 / log 3   =   1.2618595...

Von Koch's  snowflake  is the closed curve obtained by assembling 3 of the above in a triangle:

 Helge von Koch's snowflake 
 by António Miguel de Campos

Incidentally, the  surface area  enclosed by that curve is fairly easy to compute.  Observe that the number of sides of the shape is multiplied by  4  each time; it's equal to  3 4n  after n steps.  At each step, the added surface area consist of as many new triangles as there as sides.  Each such triangle has an area 9 times smaller than the triangle it's built on.  Therefore, the total area is obtained by adding a geometric series to the area of the original triangle  (used as unit ot area):

1  +  3 (1/9)  +  12 (1/9)2  +  ...  +  3 (4/9)n  +  ...   =   1 + (3/9)/(1-4/9)   =   8/5

The area of the snowflake is 60% more than the area of the base triangle.

(2011-07-09)   Hausdorff dimension   (Hausdorff, 1918)

Let  E  be a set of points in a metric space.  For any positive real number  d,  we may consider all the coverings of  E  with open balls.

To every ball of radius R we associate a cost Rd.  The cost of the entire covering is the sum of the costs of all the balls in it  (it may be infinite).

 Come back later, we're
 still working on this one...

Constantin Carathéodory (1873-1950)   |   Felix Hausdorff (1868-1942)
Hausdorff Dimension, Its Properties, and Its Surprises  by  Dierk Schleicher.
The Mathematical Association of America Monthly, 114 (June-July 2007) pp. 509-528.
Wikipedia :   Geometric measure theory  |  Outer measure  |  Hausdorff measure  |  Hausdorff dimension

(2011-07-09)   The Julia set and Fatou set of  f  are complementary.
On the  Complex dynamics  of  Pierre Fatou  and  Gaston Julia.

Pierre Fatou (1878-1929) & Gaston Julia (1893-1978).

 Come back later, we're
 still working on this one...

(2004-06-21)   The Mandelbrot Set
The complex numbers  c  for which  z0 = 0;  zn+1 = zn2 + c   is bounded.

 Come back later, we're
 still working on this one...

The dark side of the Mandelbrot set  by  Burkard Polster  (Mathologer, 2016-03-04).
Wikipedia :  The Mandelbrot set

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