4 ms·
Nice and fun, and brings into focus the definition of "distance" and how it interacts with the concept of a metric. Using the L<sub>1</sub> metric gives a diff
by RiderOfGiraffes 16y ago
Nice and fun, and brings into focus the definition of "distance" and how it interacts with the concept of a metric. Using the L<sub>1</sub> metric gives a different value of pi than using the L<sub>2</sub> metric. Obviously.
It's easy enough with arguments like this to show the "length" to be anything you choose. Such demonstrations are instructive.
EDIT: Changed lower-case ell to upper-case for clarity.
- jacobolus 16y agoYour comment should be at the top. Wikipedia has a nice short article about the ℓ₁ norm (also called “taxicab distance” or “Manhattan distance”): http://en.wikipedia.org/wiki/Taxicab_geometry http://en.wikipedia.org/wiki/Taxicab_geometry It explains, > “The use of Manhattan distance leads to a strange concept: when the resolution of the Taxicab geometry is made larger, approaching infinity (the size of division of the axis approaches 0), it seems intuitive that the Manhattan distance would approach the Euclidean metric [...] but it does not. This is essentially a consequence of being forced to adhere to single-axis movement: when following the Manhattan metric, one cannot move diagonally (in more than one axis simultaneously).”
- gfodor 16y agoYes, you've provided the most succinct and accurate explanation here. The root observation of this thought experiment revolves around properties of the l1 norm, and is an important set of concepts to understand property in implementing distance metric-oriented ML algorithms like k-means clustering.