Below are some examples of relative equilibria in the planar four-vortex problem with vortex circulations

**1) Asymmetric: ** This family of asymmetric relative equilibria is convex for **m < 0** and
concave for **m > 0**, with three of the vortices becoming collinear at **m = 0**.
The configuration for **m = 1** is symmetric and also degenerate. It consists of three vortices
located at the vertices of an equilateral triangle and a fourth vortex at the center.
Other than some asymmetric collinear configurations, these are the only asymmetric solutions for these
(symmetric) choices of circulations.

**2) Kites: ** Above are two families of kite configurations where vortices 3 and 4 are on the axis of symmetry.
Both families begin at **m = 1** with the equilateral triangle plus a vortex at the center solution.
The family on the left exists only for **0 < m ≤ 1**. As **m** decreases toward 0, the distance between
the third and fourth vortex goes to 0, while the shape of the exterior triangle becomes equilateral. The
family on the right exists for **-1/2 < m ≤ 1**. It is concave for **m > 0** and convex
for **m < 0**, with three vortices becoming collinear at **m = 0**.
The convex configurations all have three vortices very close to
being collinear, and as **m** decreases toward **-1/2**, the distance between the third and fourth vortex
goes to infinity.

**3) Isosceles Trapezoid: ** The isosceles trapezoid family above exists for **0 < m ≤ 1**. Beginning with a square
at **m = 1**, the length of the shorter base monotonically decreases to **0** as **m** decreases,
ending in an equilateral triangle with vortices 3 and 4 colliding at **m = 0**. The ratio of the common
side-length to the larger base starts at **1** at the square configuration and decreases to approximately
**0.904781** when **m ≈ 0.234658**, before increasing back to **1** when **m = 0**.
This family is always stable.

**4) Rhombii: ** Above are two families of rhombii relative equilibria. The family on the
left exists for **-1 ≤ m ≤ 1**, with the larger strength vortices lying on
the shorter diagonal. Starting with a square at **m = 1**, the ratio of the longer diagonal to the shorter increases in length as **m** decreases, ending with a maximum length of
**$\sqrt{2}$ + 1**
at **m = -1**. This family is linearly stable for
** -2 +$\sqrt{3}$ < m ≤ 1**.
When **m = -2 +$\sqrt{3}$**,
the total angular vortex momentum is zero and the relative equilibrium is degenerate.
For **-1 < m < -2 +$\sqrt{3}$**,
it is unstable.

The rhombus family on the right exists for ** -1 ≤ m < 0 **, with the larger
(positive) strength vortices situated on the longer diagonal. This family emerges from
a collinear collision (Euler) configuration where vortices 3 and 4 have merged together
at a point equidistant from vortices 1 and 2. As **m** decreases, the ratio of the
shorter diagonal to the longer grows in size
monotonically until reaching a maximum of **$\sqrt{2}$ - 1** at **m = -1**. Interestingly,
the rhombus becomes an actual equilibrium at **m = -2 +$\sqrt{3}$**,
and for **m < -2 +$\sqrt{3}$**,
the direction of rotation flips. This family is always unstable. However, it undergoes a pitchfork
bifurcation at **m = m***, where **m*** is the only real root of the cubic
**9m ^{3} + 3m^{2} + 7m + 5**. As