Editing 2821: Path Minimization

{{comic
| number    = 2821
| date      = August 28, 2023
| title     = Path Minimization
| image     = path_minimization_2x.png
| imagesize = 562x559px
| noexpand  = true
| titletext = Of course you get an ice cream cone for the swimmer too! You're not a monster.
}}

==Explanation==

In this comic, it appears that Cueball, standing on shore, is observing a swimmer who is possibly (but not obviously) in distress. The comic illustrates five potential paths that can be taken to reach the swimmer, each with a different reason to make them viable, in the manner of demonstrating different optimal strategies that can be chosen.

The first path is a direct line from Cueball, straight to the swimmer, which allows for the minimum possible distance to be traveled, some on land and the remainder in the water.

The second path travels more obliquely from Cueball to the water and then at a sharper angle to the swimmer. This path would take the shortest amount of time, presuming that Cueball would move faster on land (covering more of the distance) and slower through the water (but less distance). The exact angles would depend on how much faster Cueball is on land than in the water.  The relationship between speeds and angles is the same as that in {{w|Snell's law}} for light passing between two media.

The third path travels at a far more oblique angle to the water, such that the subsequent swimming path is entirely perpendicular to the shoreline, adding to the amount of time spent on land in order to minimize the time spent swimming. Depending on one's swimming ability versus running ability, this could be the safest path to take. It might also be more sensible to keep the target in clear sight for as long as possible, from the land, then aim exactly away from shore when both your head and theirs are barely at wave-height (though currents may complicate this). But this is a completely different reason from the distance or time preferences.

The fourth path travels nearly parallel to the beach. In fact moving slightly ''away'' from the swimmer but towards an intermediate goal: an ice cream stand. After that, the path turns and aims straight towards the swimmer, as all the others eventually do (although it is not made clear at this point if Cueball will also be eating, or just carrying, an ice cream even whilst swimming).

The fifth and final path, barely recognizable as a path, points off the top of the comic and reappears at the bottom. This path presumably travels around the entire world, likely stopping for many, ''many'' rest breaks. It is labeled as the path that ''maximizes'' time. It should be noted that, by the definition given, it is theoretically possible to stretch the maximum time taken out forever by simply walking away and never returning.

You could also fulfill the criteria of reaching the target in finite, but arbitrarily long, time by following a {{w|random walk}}(+swim) or even follow a {{w|space-filling curve}} carefully chosen to be the maximally finite scenario. Or you could simply choose any path, and stop for an arbitrarily long time, or travel at a speed approaching zero. In the comic, however, a requirement for simplicity of path may dictate the use of something close to the opposing {{w|great-circle distance}}, or a variation that has a maximal swim-time even without ''undue'' time-wasting detours, and assume equal speeds of travel on all routes.

Alternatively, the fifth path may be a joke playing on relativity. In special and general relativity, timelike geodesics (locally) maximize the proper time between spacetime events. In a spacetime diagram (in sufficiently nice coordinates), an upwards-directed vertical line would be such a geodesic. Under this interpretation, the fifth path isn't a path around the world or through space at all, but through spacetime.

The comic pokes fun at two famous physical/mathematical problems that are usually stated as happening on a beach. The first is the Lifeguard problem, which Richard Feynman, in his book ''QED'', uses to illustrate {{w|Fermat's principle}}, or principle of least time, which states that the path taken by a light ray between two given points is the path that can be traveled in the least time. This is closely related to {{w|Stationary-action principle}} for mechanical systems. In Feynman's words:

''"Finding the path of least time for light is like finding the path of least time for a lifeguard running and then swimming to rescue a drowning victim: the path of least distance has too much water in it; the path of least water has too much sand in it; the path of least time is a compromise between the two."'' - ''Richard Feynman, QED - The Strange Theory of Light and Matter (1988, Princeton University Press), Chapter 2.''

It is also possible that the comic makes fun of Feynman's idea that a photon (Cueball) would take ''every'' path to reach its destination, including the one that goes around the Earth, so that the paths shown are all being taken instead of being options Cueball is considering (therefore he could bring an ice cream to the swimmer). 

The second problem referenced in this comic is the [https://gametheory101.com/courses/game-theory-101/hotellings-game-and-the-median-voter-theorem Beach Vendor Problem], which is stated as follows. Suppose that on a long beach there are two ice cream vendors. Customers are uniformly distributed on the beach and each person will go get the ice cream at the closest vendor. Each vendor wants to maximize the number of customers that buy at their place. To minimize the customer's walking time, the optimal configuration would be to have one vendor at 1/4 of the beach length and the other at 3/4, but {{w|Hotelling's law}} predicts that the two shops will converge to the middle of the beach, in an attempt to steal as many customers as possible from the competition. This is a case of {{w|Nash equilibrium}} that is also related to the {{w|Median voter theorem}}. If the number of vendors is larger than 2, the problem may become [https://gametheory101.com/tag/hotellings-game/ considerably more complicated].

The title text adds to the ice-cream path the stipulation that you also carry an ice-cream to the target swimmer to 'justify' that choice of route. But how this squares with the reason to rendezvous with the swimmer, or the manner in which this would further complicate the swimming stage, goes unsaid. But it makes it clear that ''not'' doing this isn't considered socially permissible, whether or not he had stopped to eat an ice-cream of his own beforehand.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Cueball stands on the beach, with another person in the water. Ponytail stands on the beach in an ice cream stand in the top right corner.]
:Path that minimizes distance [A straight line from beach cueball to ocean cueball, bearing about 135]
:Path that minimizes time [A line from beach cueball to the waterline closer to horizontal, bearing about 120, then angling towards ocean cueball, bearing about 150]
:Path that minimizes swimming [A line from beach cueball to the waterline closest to ocean cueball, bearing about 115, then angling toward ocean cueball, bearing 180]
:Path that minimizes time until you get ice cream [A line from beach cueball to the ice cream stand manned by Ponytail, bearing about 90, then angling toward ocean cueball, bearing about 190]
:Path that maximizes time [A line from beach cueball away from the shore, bearing 0, fading and disappearing at the top of the panel, and reappearing at the bottom of the panel directly below ocean cueball]

{{comic discussion}}

[[Category:Food]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Ponytail]]
[[Category:Multiple Cueballs]]
[[Category:Physics]]