Editing 3176: Inverted Catenaries

{{comic
| number    = 3176
| date      = December 3, 2025
| title     = Inverted Catenaries
| image     = inverted_catenaries_2x.png
| imagesize = 317x317px
| noexpand  = true
| titletext = Some tires are marketed as 'all-shape tires,' but if driven in a climate with both inverted catenary falls and triangle falls, they wear out really fast.
}}

==Explanation==
During the winter, in snowy areas, people may replace their car's summer or all-season tires with winter tires made specifically for the cold environment. In this comic, instead of snow, rounded shapes called inverted {{w|Catenary|catenary curves}} fall from the skies. On a plane covered in inverted catenaries all the same size, square wheels whose side length matches the arc length of the catenary [https://mathtourist.blogspot.com/2011/05/riding-on-square-wheels.html are capable of rolling smoothly], contrary to how they would act on a normal road. People have made real tracks demonstrating this. Regular wheels would cause a significantly bumpier ride on this terrain, so Cueball plans to swap them out with {{w|square wheel}}s to better suit the season.

Note, however, that this assumes the catenaries are arranged periodically with no spacing between them, fully cover the surface, and are consistent in shape and orientation. The orientation would also restrict the direction of travel, effectively meaning your vehicle would be traveling on rails. Changes in direction could be managed using catenaries whose arc length was consistent but whose segment length varied, with the variations in vertical size being accommodated by vehicles' suspension systems. Letting the direction changes be controlled by drivers (e.g. branching roads) would require complex 3D road surface shapes.

The title text mentions all-shape tires (as a play on all-season tires), which are advertised to supposedly fit any shape road. However, different shapes would require very different wheels; for example, falling triangles would form a sawtooth road, for which one would optimally use wheels pasted together from pieces of an equiangular spiral. The all-shape tire is said to wear out very quickly like low quality all-season tires used to. (The best modern all-season tires perform better than the average winter tire and have a 62k mile warranty.)

The comic might be a comment on the aforementioned experiments. While mathematically interesting and a good example of experimental mathematics (see [[3180]]) the "real live" applications of this are rather... limited. At least in this literal form.

==Transcript==
{{incomplete transcript|Don't remove this notice too soon.}}
:[Megan and Cueball are walking together as inverted catenary curves fall from the sky (presumably from an "inverted catenary tree"). A few have landed in a regular formation, all flat-side down and evenly spaced, with some touching each other.]
:Cueball: Oh wow, the first inverted catenary fall of the year!
:Cueball: Time to swap out my all-season tires for square ones.

{{comic discussion}}<noinclude>
[[Category:Weather]]
[[Category:Geometry]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Megan]]