Key words: Polygons
And the ultimate shape wheel is round! What about the polygonal wheels that gradually lead to that that clever invention?
An n-gon Wheel
The approach used for square and triangular wheels can be generalized giving ….
$$ \small y = \csc \frac{\pi}{n} – \cot \frac{\pi}{n}.\cosh\bigl ( \tan \frac{\pi}{n}.x \bigr)$$
Track segments for \( \scriptsize n = 3,4,5,6 10,20\)
The Tracks
In general
- Width of base: \( \small x_{1,2} = \displaystyle \frac{\ln ( \sec(\frac{\pi}{n}) \pm \tan(\frac{\pi}{n}) )}{\tan(\frac{\pi}{n})} \)
- Length: \( \small L=\int_{x_1}^{x_2}\cosh(\tan(\frac{\pi}{n}).x)dx=2 \)
- Height: \( \small y = \csc(\frac{\pi}{n}) -\cot (\frac{\pi}{n}) \)
In agreement with \( \scriptsize n = 3,4\) for the triangular and square wheels as presented earlier.
An Infinigon
As \( \scriptsize n \rightarrow \infty \) the associated polygon tends to a circular wheel and its track is merely the horizontal surface we are accustomed to driving on … smoothly in most cases we hope.
And that’s all folks.
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The ABC series …
1. Arches and Bridges
- Arches, Bridges & Chains
- All Aqueducts lead to Rome
- Arches & Bridges
- Suspension Bridges
2. Hanging Cables or Chains
- Hanging Cables
- Catenary Equations
- Power lines
3. Strange Tracks
- Square Wheels
- Triangular Wheels
- Polygonal Wheels