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Inverted Catenaries

|< < Prev #3176 (December 3, 2025)

Title text: 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.

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Explanation

During the winter, in snowy areas, traditionally people need to replace their summer or all-season tires with winter tires made specifically for the cold environment. In this comic, instead of snow, rounded shapes called inverted 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 are capable of rolling smoothly, contrary to how they would act on a normal road. Regular wheels would cause a significantly bumpier ride on this terrain, so Cueball plans to swap them out with square wheels to better suit the season.

Mathematicians have found what types of roads would suit weird wheels the most, and inverted catenary curves are best suited for a square wheel. People have made real tracks demonstrating this.

Note however, 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 also would 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, but 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 is 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 require wheels pasted together from pieces of an equiangular spiral. The all-shape wheel is said to wear out very quickly like low quality all-season used to. (The best modern all-season tires perform better than the average winter tire and have a 62k mile warranty)

Transcript

[Megan and Cueball are walking together as inverted catenary curves fall from the sky. 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.

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