These are obviously point snails rather than snails with a non-zero length, so it's best that this question has been migrated.
Consider n point snails arranged in a regular convex n-gon such that the point snails move in a coordinated manner (i.e., the same angular velocity) and such that at any point in time, each point snail moves in the direction of the neighbor directly in front of it with a constant speed v. Symmetry dictates that the n snails will maintain that regular convex n-gon shape until they simultaneously meet at the center of the n-gon.
The velocity vector of each of the n snails is
$$\begin{aligned}
\vec v &= v\left(-\cos\left(\frac{\pi}2 - \frac{\pi}n\right)\,\hat r + \sin\left(\frac{\pi}2 - \frac{\pi}n\right)\,\hat\theta\right) \\
&= v\left(-\sin\left(\frac{\pi}n\right)\,\hat r + \cos\left(\frac{\pi}n\right)\,\hat\theta\right)
\end{aligned}$$
Equating this with the time derivative of $r\,\hat r = \dot r\,\hat r + r\dot \theta\,\hat\theta$ results in
$$\begin{aligned}
\dot r &= -v\sin\left(\frac{\pi}n\right) && (1)\\
r\dot\theta &= \phantom{-}v\cos\left(\frac{\pi}n\right) && (2)
\end{aligned}$$
Equation (1) tells us that the radial distance decreases linearly with respect to time, which in turn tells us snails will meet at the center in finite time. Equation (2) tells us that the snails will be extremely dizzy when they meet. Their angular velocity diverges to infinity at the time they meet.
Re-expressing $r(t)$ as a function of angle $\theta$, equations (1) and (2) tell us the derivative of $r$ as a function of angle:
$$\frac{\mathrm d r(\theta)}{\mathrm d \theta} = \frac{\dot r}{\dot\theta} = -r \tan\left(\frac{\pi}n\right) $$
The solution to this is the logarithmic spiral
$$r(\theta) = r_0\exp\left(-\theta \tan\left(\frac{\pi}n\right) \right)$$
There's not much to see in the case of three snails. The triangle will have shrunk to about 7% of its original linear size after just $1/4$ of a rotation (90 degrees) and less than $1/2$% of its original linear size after $1/2$ of a rotation. The infinite number of rotations that follow will be pretty much invisible.
