Each row and column, as well as the two main diago-
nals, sum to 34, and so this array represents a fourth-
order magic square. It is the first example of a magic
square ever recorded in Western Europe. (It has the
added feature of including the year the engraving was
completed in the two middle cells of the bottom row.)
Dürer died in Nürnberg, Germany, on April 6,
1528. His theory of proportions allowed artists who
succeeded him to easily perform the transformations on
figures needed to translate them across a canvas and
maintain the correct sense of perspective. (Prior to
Dürer, artists accomplished this feat purely by intuition
or by trial and error.) In doing so, Dürer had provided
a well-thought-out theory of geometric perspective that
was also valued by mathematicians.
dyadic Any quantity related to the concept of base 2 is
sometimes referred to as dyadic. For example, the dyadic
rationals are those fractions whose denominators are
powers of 2. Thus = and = , for
instance, are dyadic rationals, but is not.
Folding a strip of paper 1 ft long in half produces
a crease at the position of the dyadic rational 1/2. If
one continues to fold the left or right end of the strip
to previously constructed crease marks, then crease
marks appear at all the dyadic rationals (and only the
dyadic rationals).
This paper-folding activity is intimately connected
to the construction of
BINARY NUMBERS
. For instance,
to create a crease mark along a strip of paper at the
position of the dyadic rational 13/16, write the numer-
ator 13 in binary:
13 = 11012
(This is equivalent to writing the fraction 13/16 as a
binary “decimal:” 13/16 = .1101 in base 2.) Now read
the binary expansion backwards, interpreting the digit
1 as the instruction “lift right and fold” and the digit 0
as “lift left and fold.” In this example we have:
1 : Lift the right end of the strip and fold to produce
a crease at position 1/2.
0 : Lift the left end and fold to the previous crease.
This produces a new crease at position 1/4.
1 : Lift the right end and fold to the previous crease.
This produces a new crease at position 5/8.
1 : Lift the right end and fold to the previous crease.
This produces a new crease at position 13/16,
as desired.
In general:
The binary representation of the numerator of
any dyadic rational represents instructions for
the construction of that dyadic rational along a
strip of paper.
In some limiting sense, this procedure also works for
fractions that are not dyadic. For example, the number
1/3 written as decimal in base 2 is .010101… If one
were to read this as a set of instructions to “fold right
and fold left indefinitely,” then the sequence of creases
produced do indeed converge to the position 1/3.
dynamical system Any process in which each suc-
cessive state is a function of the preceding state is
called a dynamical system. For instance, the feedback
from a microphone as part of a public announcement
system is a dynamical system: the amplifier transmits
minute erroneous sounds, which the microphone
hears and amplifies, which it then hears and amplifies,
and so on.
In mathematics, if fis a mapping from a space Xto
itself, then the
ITERATION
of fdefines a dynamical sys-
tem—the successive states of the system are the iterates
of farising from a given starting point x:
x,f(x),f(f(x)),f(f(f(x))),…
For example, the mapping f(x) = xof the real number
line to itself, starting with x= 1, gives the iterates 1, ,
, ,…. If Xis a circle in the plane and fis the function
that rotates that circle 10°clockwise, then the iterates of
any point on the circle constitute 36 evenly spaced
points on that circle.
The “orbit” of a point xis the sequence of iterates
it produces. If the system reaches an equilibrium, that
is, tends toward a stable state, or if it cycles between a
number of states, then the equilibrium point (or sets of
equilibrium points) are called “attractors” of the sys-
1
–
8
1
–
4
1
–
2
1
–
2
1
–
3
173
–
—
210
173
–—
1024
3
–
22
3
–
4
150 dyadic