Geometria fractal

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The Fractal Geometry of Numbers
David Gibson

Figure 1a

Figure 1b

Figure 1c

Figure 1d

Micromath Spring 2002

21

In August 1994, Wired Magazine published an interview entitled the ÔThe Geometric Dreams of Benoit MandelbrotÕ. Mandelbrot coined the term ÔfractalÕ in the 1970Õs and the stunning beauty of his dreams has created much common ground for mathematics, art and nature.Fractal images resonate deeply with the subconscious mind. Indeed the Mandelbrot Set (Fig 1a) has been called ÔThe Thumbprint of GodÕ. It can be magnified without end revealing innumerable natural archetypes (Fig 1b-c). The interview begins as follows: ÒYour book is called ÔThe Fractal Geometry of NatureÕ. What is the fractal geometry of natureÓ? Mandelbrot: ÒThe geometry of Nature is fractal tothe extent that if you look at many shapes in nature Ð clouds, trees etc. small parts are the same as big parts; thatÕs the definition of ÔfractalÕ.Ó If we had no computers but an abacus, what would we know about fractals? Mandelbrot: ÒNothingÓ In fact, there is more to the abacus than meets the eye Ðthere are myriads of fractal patterns which have lain buried in its beads for the last five thousandyears. Graphics are only a trigger for an infinite image Ð but the imagination can endlessly embroider the boundaries. The ÔFractal AbacusÕ uses computer-generated images to make these patterns visible. Adults are often surprised at the way children readily grasp the full significance of these complex, yet beautiful images of number. Children lack adult preconceptions. There is simplicity withinthis complexity, and a resonance between form and function which fires the imagination. It is a bridge between geometry and number, and between the aesthetic and the functional. The Fractal Abacus is more than a classroom resource Ð it is the gateway to another world. It helps to develop the conceptual framework which enables us to check our change at the shop, and at the same time presentsopportunities to enlarge the imagination beyond the linear boundaries of blocks and rods. It is an opportunity (as William Blake might have put it) ÒTo hold infinity in the palm of your hand.Ó We can represent Base Ten ÔentitiesÕ as shown in Fig 2 a fractal pattern is already beginning to form. This representation of ÔoneÕ as a white ten pointed star is, however, a very great simplification
22Micromath Spring 2002

Numbers as fractals
Ð one should be represented as ten tenths, or a
ONE

TEN

HUNDRED

THOUSAND

Figure 2

hundred hundredths, or a thousand thousandths etc. (Fig 3) On this basis it is important to remember that these

(and all of the graphic images which follow) are simplifications, since they have to stop somewhere, but the imagination can carry on embroideringthem with

FORTY FOUR

A HALF

ONE AS A HUNDRE D HUNDREDTHS

Figure 3

arbitrary colours. When a number is not an exact power of ten it should, strictly speaking, be represented as a broken pattern (Fig 4). In practice magnetic ÔbeadsÕ are used. A box ensures that no more than nine counters can be placed in a column (Fig 5). Circular ÔTens FramesÕ are used to change ten ÔonesÕ for aÔtenÕ, or ten ÔtensÕ for a hundred, or vice-versa (Fig 6). Many of us assume that God created our Base Ten

A QUARTER
Figure 4
Micromath Spring 2002 23

number system (ie. hundreds, tens and units) in the Beginning. He didnÕt, He did, however, create humans with five ÔdigitsÕ on each hand. Without fractals,
Figure 5

Variations of ÔCantorÕs CombÕ (Fig 8b) and The ÔNumber LineÕ fractal (Fig 8c),serve any number base. Base three, six and ten (so called Ôtriangular

ONE

SIX

ABACUS BOX No more than nine counters per column are allowed

Figure 6

THIRTY SIX

CHANGING CIRCLES The ten counters on the outside can be changed for the inside counter (or vice-versa)

TWO HUNDRED AND SIXTEEN

Number bases
Earthlings struggle with other systems. The software version of the...
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