What is a dog saddle? I got this question as an assignment. The question is why the graph of the function $f(x,y)=x^3y-xy^3$ is called a dog saddle. I am rather confused as I don't know what our professor is really looking for. Isn't it called a dog saddle because it looks like one? Please help me with this. Thanks in advance.
 A: tl; dr: A saddle is a surface resembling a removable seat for horseback riding, which bends downward on the sides and upward to the front and back of the rider. The downward-bending parts may be viewed as "places for legs." A standard mathematical model is the graph $z = x^{2} - y^{2}$. The rider sits at the origin and faces toward the positive $x$-axis.

With characteristic humor, mathematicians speak similarly of a monkey saddle, a surface with three "places" for two legs and a tail if a monkey rides a horse. A standard mathematical model is the graph $z = x^{3} - 3xy^{2}$. Again, the rider sits at the origin and faces toward the positive $x$-axis. (Presumably, the "space for the tail" hangs over the rump of the horse.)

Continuing the pattern, the term dog saddle refers to a surface resembling the seat a dog (with four legs) might use to ride a horse in a universe where such things occur. A standard mathematical model is $z = z^{4} - 6x^{2}y^{2} + y^{4}$, with the same conventions of coordinates as above. (It is not generally noted that images of dogs on horseback are anthropomorphized, with the dog in an "ordinary" saddle. The proper image for a dog saddle with four places for legs instead requires us to imagine the rider splayed flat, like a napping golden retriever, or Bambi on ice.)


Analysis of places for legs: There are multiple ways to analyze where "saddle-like" functions are positive and negative. Among the most straightforward is to factor:

*

*The ordinary saddle has equation
$$
z = x^{2} - y^{2} = (x + y)(x - y).
$$
This is zero on the lines $y = x$ and $y = -x$, and changes sign each time we cross one of the lines. Looking down on the Cartesian plane, $x^{2} - y^{2}$ is positive to the left and right of the crossing lines, i.e., in the region where $|y| < |x|$, and negative above and below, where $|x| < |y|$. The regions where $x^{2} - y^{2} < 0$ are "places for legs."

*The monkey saddle has equation
$$
z = x^{3} - 3xy^{2} = x(x + \sqrt{3}y)(x - \sqrt{3}y).
$$
This is zero on the lines $x = 0$, $y = x/\sqrt{3}$, and $y = -x/\sqrt{3}$. Looking down on the Cartesian plane, these three lines cut the plane into six congruent wedge-shaped regions. In the right-hand wedge, $x^{3} - 3xy^{2}$ is positive, and this expression changes sign each time we cross one of the lines. The monkey saddle therefore has three "places for legs and a tail."

*The dog saddle with preceding equation can be analyzed similarly, but the factorization is more vexing, see below. The dog saddle equation $z = x^{3}y - xy^{3}$ given in the problem factors more readily. It does not seat the rider facing along the positive $x$-axis, however.


Complex powers: These saddles (and their generalizations for arbitrarily many places for legs) all arise naturally in the complex numbers. If we write $w = x + iy$ with $x$, $y$ real, then
\begin{align*}
  w^{2} &= (x + iy)^{2} = (x^{2} - y^{2}) + i(2xy), \\
  w^{3} &= (x + iy)^{3} = (x^{3} - 3xy^{2}) + i(3x^{2}y - y^{3}), \\
  w^{4} &= (x + iy)^{4} = (x^{4} - 6x^{2}y^{2} + y^{4}) + i(4x^{3}y - 4xy^{3}),
\end{align*}
and so ad infinitem. The real parts are precisely our saddle friends; the imaginary parts have congruent graphs to the real parts, but rotated about the $z$-axis through an angle $\pi/n$. To see why, write $w$ in polar form. By Euler's identity $e^{i\theta} = \cos\theta + i\sin\theta$, we have
$$
w^{n} = r^{n}e^{in\theta} = r^{r}(\cos(n\theta) + i\sin(n\theta)).
$$
The rotational symmetry is apparent, as it was not in Cartesian form. (A non-mathematician colleague with access to a 3-D printer kindly made two models of the graph $z = x^{3} - 3xy^{2}$, and was suitably surprised to see them stack perfectly when one was rotated one-third of a turn relative to the other.)
As a final note, we can see why factoring $z = x^{4} - 6x^{2}y^{2} + y^{4}$ is vexing: The four linear factors cut the plane into eight equal-angle wedges at the origin, but these are centered on the coordinate axes, so make angles $\pm\pi/8$ and $\pm3\pi/8$; the cosines and sines of these can be found explicitly using half-angle formulas, but are not as pleasant as the cosines and sines needed for the ordinary or monkey saddle, or for the "imaginary" dog saddle of the original problem.
