I have read about this topic a while ago and when I saw a workshop on this at the One-Day Conference for the Teacher of Secondary Mathematics I wanted to participate. Taxi-Cab Geometry looks at things in a completely different way than what we are used to. If you are used with different metric spaces, than this is going to be just a quick game for you.
In our day to day life, we measure distance by virtue of the theorem of Pythagoras. The theorem provides a metric, but if we change the way we measure (in other words, if we change the metric), do we change the appearance of things? Is the square going to look like a square, or the circle like a circle? It sounds a little strange first, but you may not realize that the way we measure, or calculate, the space in which we live fixes the shapes that we see around us. Thus if we change the way we measure or calculate we may well change many other things.
Defining taxi-cab geometry is not very hard. The definition Wikipedia offers is quite good: Taxicab geometry, considered by Hermann Minkowski in 19th-century Germany, is a form of geometry in which the usual distance function of metric or Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. But the best way to see this type of geometry is to literally consider its name. Think about it as if you are a Taxi driver and the way you move into a town where the roads are the lines from a square paper, or more easily, think about it as if you live in a city with no round-abouts. To better understand the difference between Euclidean geometry (our day to day geometry) and Taxi-Cab geometry, take a look at the image bellow ( in taxicab geometry, the red, yellow, and blue paths all have the shortest length of 12. In Euclidean geometry, the green line has length approximate 8.49 and is the unique shortest path).
During the workshop we played a little with finding distances between points and what different shapes would look like, which was quite interesting and relaxing. To get an idea, here is an image which shows you different shapes in taxi-cab geometry. Also, I would totally encourage you to try and drawn a circle, you will be surprised.
After the workshop I was thinking about the fact that this type of geometry could be extremely useful in our society and it could be used more. It could really help us position some buildings, such as hospitals and police, so that they are at equal distances from different points of the city. Also, building schools could be another interesting application of this. I was quite interested by problems like the following:
I hope I have made you more curios about this topic and its applications. If you want to find out more, I recommend Taxicab Geometry: An Adventure in Non-Euclidean Geometry (Dover Books on Mathematics) by Eugene F. Krause:
If you want to read more about my day at the conference, read One Day in Stirling. Also, if you are interested in some new way to teach mathematics in a more graphical way, I recommend reading about another workshop from the conference, Graphical Method for teaching Maths.
Hope you enjoyed this post and that you are excited for the future ones. Have a great week. You can find me on Facebook, Tumblr, Google+, Twitter and Instagram. I will try to post there as often as possible.
Don’t forget that maths is everywhere! Enjoy!