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Supplement to 'Fermat's Spiral and the Line Between Yin and Yang'

Taras Banakh (Lviv National University) and Oleg Verbitsky (Humbodt-Universität zu Berlin) and Yaroslav Vorobets (Texas A&M University)

[Accompanies article to appear in American Mathematical Monthtly.]

[applet written by Tom Leathrum]

The graph here shows the unit circle and the polar curve \(\theta=\pi r^2 T\), a "Fermat spiral," where \(T\) is the value of the "turn" parameter (which can be set with the first slider in the applet). The branch of this curve with \(0\leq r\leq 1\) can be highlighted in red by selecting the "highlight curve" checkbox. The semitransparent blue region is congruent to the gray region, but reflected across the \(x\)-axis and rotated counterclockwise by \(\pi R\) radians, where \(R\) is the value of the "rotate" parameter (which can be set with the second slider in the applet). The blue region can be removed from the graph by unselecting the "show reflected" checkbox (this also disables the "rotate" slider).

Note: In order to view this applet, you must use a browser with the Java plug-in (version 1.6) installed. Math expressions in this page are displayed using jsMath -- you may need to install missing fonts.

Facts about Applet:

  • For any value of the "turn" parameter, the blue region is the reflection of the gray region across the line through the origin which forms an angle half the angle of rotation of the blue region.
  • For any value of the "turn" parameter, the maximal subset of the gray region that is symmetric with respect to reflection through a line is the intersection of the gray region with its reflection through the line, i.e. with the blue region when the angle formed by the line is half the angle of rotation of the blue region.
  • When the "turn" parameter takes the value 1.0, for any angle of rotation of the blue region, the intersection of the gray and blue regions has area exactly 1/4 of the area of the circle.

The first fact is apparent from the graph. The article includes proofs of the other two facts (albeit stated in more general terms), along with other important symmetry properties of the gray and blue regions. The excerpt below provides background for the statement (proven in the article) of an important uniqueness property for the Fermat spiral in the case where the "turn" parameter takes the value 1.0.

Excerpt from Article:

From the mathematical point of view, the yin-yang symbol is a bipartition of the disk \(D\) by a certain curve \(\beta\). We aim at identifying this curve and deriving an explicit mathematical expression for it. Such a project should apparently begin with choosing a set of axioms for basic properties of the yin-yang symbol in terms of \(\beta\):

(A1) \(\beta\) splits \(D\) into two congruent parts.
(A2) \(\beta\) crosses each concentric circle of \(D\) twice.
(A3) \(\beta\) crosses each radius of \(D\) once (besides the center of \(D\), which must be visited by \(\beta\) due to (A2)).

Denote the symmetry group of the disk \(D\) by \(Sym(D)\). As is well known, it consists of reflection and rotation symmetries. Focusing on these intrinsic symmetries of the disk, we will call a set \(X\subseteq D\) symmetric if \(s(X)=X\) for some nonidentity \(s\in Sym(D)\).

Suppose now that \(D\) has unit area. In fact, instead of area we will more often refer to the more general concept of Lebesgue measure. We call a set \(A\subseteq D\) perfect if it has measure 1/2 and any symmetric subset of A has measure at most 1/4.

(A4) \(\beta\) splits \(D\) into perfect sets (from now on it is supposed that \(D\) has unit area).
(A5) \(\beta\) is smooth, i.e., has an infinitely differentiable parameterization \(\beta:[0,1]\to D\) with nonvanishing derivative.
(A6) \(\beta\) is algebraic in polar coordinates.

A classical instance of a curve both smooth and algebraic in polar coordinates is Fermat's spiral. Fermat's spiral is defined by the equation \(a^2r^2=\theta\). The part of it specified by the restriction \(0\leq\theta\leq\pi\) (or, equivalently, \(-\sqrt{\pi}/a\leq r\leq\sqrt{\pi}/a\)) is inscribed in the disk of area \((\pi/a)^2\).

Theorem 1.1 Fermat's spiral \(\pi^2 r^2=\theta\) is, up to congruence, the unique curve satisfying the axiom system (A1)-(A6).

Note that the factor of \(\pi^2\) in the curve equation in Theorem 1.1 comes from the condition that \(D\) has area 1. As all Fermat's spirals are homothetic, we can equally well draw the yin-yang symbol using, say, the spiral \(\theta=r^2\). Varying the range of \(r\), we obtain modificatons as twisted as desired.

[Note: The applet uses a unit circle, which has area \(\pi\). At this scale, the Fermat spiral for Theorem 1.1 is given by the equation \(\pi r^2=\theta\), corresponding to the "turn" parameter taking value 1.0. Extra "twists" as described above can be drawn using higher values for the "turn" parameter.]