Sherlock Holme's Bike Story and Tangent Lines.

 

Did you know that there was a story starring Sherlock Holmes published back in 1905 in which he deduced the direction of travel from the tracks?  Did you know that some mathematician used this story to determine if he deduced the solution correctly.  

Sherlock was lying in bed, gazing at the ceiling when Dr Watson interrupted him to discuss a murder case. When Dr Watson described the tracks the bicycle left by the suspect, it captured Sherlock's attention and they were off the check the evidence. 

He had to determine which track belonged to the front wheel and which to the back so he could figure out the direction of travel. After reading a couple of new books and exploring differential curves in detail.  Due to the design of the bicycle, the two wheels are independent of each other but are at a fixed distance from each other and always move in conjunction with each other. 

At the end, Sherlock concluded the bicycle in The Adventure of the Priory School was going away from the school because the back wheel impression was deeper than the front wheel. He commented that the back wheel track wiped out parts of the front wheel track as it crossed over it back and forth.

The inaccuracy of his assumption is since the back wheel follows the front wheel, it always crosses the front wheel path.  Always. This is an important point.  The rear wheel always moves towards the front wheel because it cannot turn, it can only go in the direction of the front wheel. Now the important thing to remember is that at any instant, the direction of motion along the curve is tangent to the curve and this comes into play when you don't see the actual rear wheel line being deeper than the front or we can't see the real wheel path crossing over the front wheel one.  Furthermore, the length of the tangent line to the back wheel is always the length between the two wheels. 

If you were to create a graphic representation of the front and back wheels, you have the front wheel which looks like a standard sine wave and the back wheel is also a sine wave with a smaller amplitude but it cuts to the inside of the line representing the front wheel. Now if you were to place a tangent line on the curve of the front wheel, it does not intersect the other line within a bicycle width of the line but if the tangent line is on the back wheel line, it does intersect.  That is how you know which sine wave represents which wheel. If the bicycle is traveling to the left, the front wheel is to the left of the back wheel and if it is going right, the front wheel is to the right of the back wheel. 

So this whole mathematical premise is based on the tractrix formulas formulated by Newton back in 1676.  If a tangent line is drawn from the tractrix line a random point, towards the asymptote, it is the same length as the distance between wheels.  Now in real life, the paths of the back wheel doesn't always follow the tractrix formula because bicycles do not have the same wheelbase.  

This story is wonderful as a way to introduce this mathematical concept since everyone has heard of Sherlock even if they haven't read the book.  It is a great integration of Language Arts and Math. Let me know what you think, I'd love to hear. Have a great day.