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Time for some more geometry! In my previous post I challenged you to solve Cookie Cutters, which asks us to scale the vertices of a polygon so that it has a certain prescribed area. It’s possible to solve this just by looking up an algorithm for computing the area of a polygon (see the “shoelace formula”). But the way to get good at solving geometry problems is not by memorizing a bunch of formula
Kenny Foner and I have a paper accepted to ICFP on subtracting bijections. Here’s the basic problem: suppose you have a bijection between two sum types, , and another bijection . Of course and must have the same size, but can you construct an actual bijection between and ? This problem and its solution has been well-studied in the context of combinatorics; we were wondering what additional insight
The title pretty much says it all: I have finally finished (I hope) updating the Typeclassopedia to reflect various recent changes to the language and standard libraries (such as the AMP and BBP/FTP). Along the way I also added more links to related reading as well as more exercises. How you can help I am always on the lookout for more exercises to add and for more links to interesting further rea
I had fun this past December solving Advent of Code problems in Haskell. I was particularly proud of my solution to one particular problem involving generating and processing large bitstrings, which I’d like to share here. I think it really shows off the power of an algebraic, DSL-based approach to problem solving. This post is literate Haskell—download it and play along! > {-# LANGUAGE GADTs #-}
Summary: The species package now has support for bracelets, i.e. equivalence classes of lists up to rotation and reversal. I show some examples of their use and then explain the (very interesting!) mathematics behind their implementation. I recently released a new version of my species package which adds support for the species of bracelets. A bracelet is a (nonempty) sequence of items which is co
Summary: given an injective function , it is possible to constructively “divide by ” to obtain an injection , as shown recently by Peter Doyle and Cecil Qiu and expounded by Richard Schwartz. Their algorithm is nontrivial to come up with—this had been a longstanding open question—but it’s not too difficult to explain. I exhibit some Haskell code implementing the algorithm, and show some examples.
Introduction In an attempt to solidify and extend my knowledge of category theory, I have been working my way through the excellent series of category theory lectures posted on Youtube by Eugenia Cheng and Simon Willerton, aka the Catsters. Edsko de Vries used to have a listing of the videos, but it is no longer available. After wresting a copy from a Google cache, I began working my way through t
Many parser combinator libraries in Haskell (such as parsec) have both a Monad interface as well as an Applicative interface. (Actually, to be really useful, you also need MonadPlus along with Monad, or Alternative along with Applicative, in order to encode choice; from now on when I talk about Monad and Applicative note that I really have MonadPlus or Alternative in mind as well.) The Applicative
While working on an article for the Monad.Reader, I’ve had the opportunity to think about how people learn and gain intuition for abstraction, and the implications for pedagogy. The heart of the matter is that people begin with the concrete, and move to the abstract. Humans are very good at pattern recognition, so this is a natural progression. By examining concrete objects in detail, one begins t
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