The sphinx pattern (see this figure) has a twist to its similarity. For the type
X sphinx, there is exactly one smaller type X sphinx inside,
rotated at 
with respect to the larger one. The similarity
that carries the smaller one to the larger one has this rotation as
well as a scale factor of 2. We can express similarities more easily
with complex numbers
corresponding to the point 
. Here i stands for a
choice of 
, i.e. a number satisfying 
. We use
to denote the set of real numbers, and 
to denote the set
of complex numbers. Multiplication by complex numbers captures both
scaling (multiplication by positive real numbers) and rotation. For a
first example, consider
The point (-y,x) is the rotation of (x,y) by 
counterclockwise.
Rather than include pictures of this sort of rotation here, we offer a MAPLE worksheet instead that will allow the readers to try out these sorts of calculations and plots themselves.
Complex Rotations and Spirals worksheetOnce you download this file and initiate MAPLE, you may load the worksheet by selecting File-Open in the MAPLE menus. Then read the text included in the worksheet. You may modify the calculations in the worksheet at will. A good introduction to MAPLE in the framework of the physical sciences is [Bay94]. This book is accompanied by a fine set of MAPLE worksheets (in a library called tmlib) for each chapter. The worksheet for Chapter 8 covers complex number calculations.
More generally, every nonzero complex number can be written in polar coordinates
where r>0 is the length 
of z and 
is
the angle that the line from 0 to z makes with the positive
x-axis. Euler discovered a remarkable formula relating the
trigonometric functions and the exponential. One way to approach
Euler's formula is to study the Taylor series expansions of the three
functions:
The series for 
and 
resemble the odd and even terms,
respectively, of 
.
To account for the signs, we consider the powers 
. These powers
repeat every fourth term in the pattern
Then
The beautiful aspect of this extension of to complex numbers is
that the usual properties of :
can be viewed as consequences of just the algebra of the Taylor series
for . Therefore, these properties also hold for complex
values of z and w. This allows us to see the geometric effect of
multiplication by complex numbers. Choose two complex numbers
and 
in polar coordinates. Then
We are scaling the length 
by the factor r, and we are
adding angle to 
, that is, we rotate w by an angle
counterclockwise.
For our sphinx, we must rotate by 
(or clockwise),
which corresponds to
Therefore, the complex scaling ratio between the small type X
tile and the larger type X tile is 
.
However, simply multiplying by a will not cause the smaller tile to precisely overlap the larger tile. We may have to translate the result. The added bonus of using complex numbers is that translations are also easy to express. Adding fixed amounts to the real and imaginary parts amounts to adding a complex number. That is, a translation of the complex plane takes the form:
where b is a complex number. Combining multiplications and additions, the general similarity of the complex plane is
where 
and b are both complex numbers.
We can find the complex similarity of the sphinx if we introduce
complex coordinates for the corners of the sphinx. We choose the origin
0 to be the lower left corner, and we choose the bottom side to run
along the positive real axis. Finally, suppose that the length
of a short side of the sphinx is exactly 1. Let's walk around
the edge of the sphinx in the clockwise direction starting at 0.
The first vertex we come to is at angle 
or 
radians
exactly 1 unit away from 0. This point is
The remaining vertices are obtained by adding 1, 
, or the
conjugate 
, depending on the direction
chosen. Thus, the vertices in clockwise order are
The smaller X tile has vertices at 
and 
as
well. The sphinx with the vertices so labelled is shown in
this figure. The complex
similarity that maps the smaller sphinx to the larger one must
satisfy:
Subtracting the two equations, we have
Dividing by , we find
as we already knew to be the case. We can also determine
Figure 34: Labelling the vertices of the sphinx by complex coordinates
We may now exploit this similarity and the deflation of the type X and Y sphinxes to obtain a tiling of the whole plane. The first few generations of this method are shown in this figure. We see a spiral pattern emerging. In the last frame, we draw a spiral passing through the images of the vertex at 3 under iteration of the complex similarity T.
Figure 35: Generations of the sphinx
tiling under the complex similarity T.
The spiral converges onto a single point which is the fixed point of the transformation:
Solving, we obtain the solution
This is approximately 1.86 + .988 i. This kind of behavior
is common for complex similarities; if 
, the sequence of iterates
, 
, 
etc., generally lies
on an infinite spiral curve, known as a ``loxodrome.''
If we change coordinates so that the origin becomes the fixed point 
,
the transformation T becomes simply a complex multiplication.
We do this by defining
Then
Iterates of multiplications are easy to write down. We denote the
composition of U with itself n times by 
. Then
. The spirals that result are parametrized
by 
, 
. We can calculate the powers of a complex
number by using polar coordinates 
. Then
the spiral curve has the form 
for some constants c,r>0 and real angles 
.
These are also called exponential spirals.
Our figures of the sphinx tiling in this section were generated with the aid of a MAPLE worksheet. We encourage the reader to examine and experiment with this worksheet which is located at
To review, similarities of the plane are the same as complex linear maps T(z)=az+b. Any structure (tiling or fractal) with self-similarities that involve both scaling and rotations generally exhibits spiralling behavior.