Sherlock Holmes’ Most Difficult Math Problem: Bicycle Tracks

A bicycle leaves two different types of tracks behind, belonging to two different types of tires. One must belong to the front tire, and the other to the rear.
The tires of a bicycle are independent of each other. Due to a bicycle’s design, the front and rear tires are always in tandem with each other at a fixed distance and move in conjunction with each other. | Source: Mathematical Impressions: Bicycle Tracks by Simon Foundations
When you laterally move the chain to which the watch was attached in either direction, the watch follow the path of the chain and get ever closer but never quite reach where the chain is.
Asymptotic | Louie Zong’s love song, with math puns!
*** The Latin root word “trac” means to pull, as in the word “tractor.” [Source]
Newton’s tractrix formula.
The distance from the top point of the tractrix to the asymptote and the length of any tangent line from any point on the tractrix to the asymptote are always the same.
The wheelbase is the horizontal distance between the centers of the front and rear wheels. | Wikipedia
The tire with the smaller amplitude would have to belong to the rear tire. Source: Mathematical Impressions: Bicycle Tracks by Simon Foundations
Upon selecting a few random points from the rear tire’s tracks and drawing tangent lines in a particular direction that reached those of the front tire, all tangent lines had to be of equal length. That would show that the bicycle had taken the direction of the tangent lines. However, if the tangent lines were of unequal length, it would mean that the bicycle had taken a direction opposite of the tangent lines drawn from the rear tire to the front.
If someone were skilled enough to turn a bicycle in a perfect circle, then the bike would leave tracks of two perfect circles on the ground. But of course, the smaller of the two circles would undoubtedly belong to the rear tire. However, because the circle is perfectly symmetrical, it would be impossible to distinguish whether the bicycle was turning clockwise or counterclockwise.
Drawing tangent lines to decide the direction of a bicycle.
*** When you apply this method in the real world, you may notice that the rear tire does not always precisely follow the tractrix. This is because bicycles do not always have the same wheelbase. There are many types of bicycles; some are for racing, and some are for leisure. That results in differences in the design of their frames.Designers usually emplace the front fork and axle on a bicycle with great care and precision regarding its angle. The slightest change in the design can result in apparent differences in the bicycle’s continuity and wheelbase when turning or leaning.Though simple mathematical models such as this one often omit such real-world factors and considerations, they still offer us greater insight into how such principles work

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.

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