Suppose you have a collection of pairs of shoes and pairs of matching socks. Let’s think about two questions: Can you create a set containing one shoe from each pair? The answer is yes, for example you could take the left shoe from each pair. Can you create a set containing one sock from each…
Tag: mathematics
The pigeonhole principle
Let me tell you about one of my favorite things. The pigeon and dove family, Columbidae, is a group of widespread and common birds. Members of this family were domesticated thousands of years ago and they have had symbolic importance throughout history. Many species continue to live closely with humans. People who keep pigeons are…
The barber shop pole illusion explained with math
This is a well known optical illusion in which rotation appears to be vertical motion. There are several aspects of the barber pole design which make the illusion more effective, including having multiple stripes, making the stripes different colors, and using rotation. We will strip these things away in order to see what is happening…
Metric space
We take for granted that we can measure distances. Most people take for granted that we can represent physical space mathematically and calculate distances between pairs of points. We generally think of ourselves as living in R3, or 3D Euclidean space. Since this is how we represent physical space, it’s generally the most useful (though…
Stories of math notation (part 2): The worst notation in mathematics
Sometimes notation is confusing. One way this can happen is when the same notation is used for multiple different concepts. Usually, if the same notation is used for different things, they occur in different contexts which allows them to be distinguished. In at least one case, the same notation is used for different things in…
Stories of math notation (part 1): y should come before x
Or rather, perhaps both the symbols and positions for x and y should be switched. The convention of writing coordinates as (x, y) is so deeply ingrained that, in my opinion, it causes confusion in situations where x doesn’t naturally come first. Reasons for x first As I see it, the primary reason for writing…
Pathological mathematics
Logic sometimes breeds monsters. Henri Poincaré In math, we often speak of things being “nice,” “well behaved,” “elegant,” or even “beautiful.” The flip side of this is those things in math which often remain in shadow: the unfriendly, the unruly, the awkward, and the ugly. In some cases, inelegant solutions are apparently unavoidable. In other…
The language of balanced parentheses
Grammatically, parentheses must be balanced, meaning every open parenthesis ‘(‘ must eventually be followed by a close parenthesis ‘)’. Parentheses can be nested, meaning there can be parentheses inside parentheses. If we were to take a piece of writing and remove all letters and punctuation except parentheses, we might end up with something like: ()()()(())()(),…
thoughts on mind 