John    I saw a reference last night to the grid square EL fx and wanted to figure out where that is There are many programs that will do this for you but I wanted to do it by hand I wrote about how grid squares work a year ago but I was rusty on the details so The post Decoding a grid square

John    Draw a line in the complex plane What is the image of that line when you apply the exponential function A line through w with direction z is the set of points w tz where w and z are complex and t ranges over the real numbers The image of this line is exp w The post Exponential of a line

John    If you ve taken calculus and someone asks you what the derivative of x is you can say without hesitation that it s x Now suppose they come back and say I m sorry I forgot to give you any context Here x is a polynomial in the field of elements It turns out that this additional

John    I ve wondered occasionally about the patterns in how chemical elements are abbreviated If you don t know the abbreviation for an element is there a simple algorithm that would let you narrow the range of possibilities or improve your odds at guessing Here s a survey of how the elements are

John    A few years ago I wrote about Manual Blum s proposed method for mentally computing a secure hash function He proposed using this method as a password manager using the hash of a web site s name as the password for the site I first wrote about Blum s method on the Heidelberg Laureate Forum blog

John    Suppose you have a standard deck of cards You pull out a card put it back in the deck shuffle and pull out another card How long would you expect to do this until you ve seen every card Here s a variation on the same problem Suppose you re a park ranger keeping data on tagged The post

John    Literature about Lisp rarely resists that narcissistic pleasure of describing Lisp in Lisp Christian Queinnec Lisp in Small Pieces Applying software development tools to themselves has a dark side and a light side There s a danger of becoming obsessed with one s tools and never getting

John    I started a new Twitter account this morning ElementFact I m thinking the account will post things like scientific facts about each element but also some history around how the element was discovered and named and other lore associated with the element We ll see how this goes I ve started many

John    The previous post included this image of a logarithm spiral passing through the corners of squares in a sequence of golden rectangles The portion of the spiral in each square looks like a quarter of a circle How well would circular arcs approximate the spiral Very well Here s a plot The

John    I ve seen an image similar to the following many times but I don t recall any source going into detail regarding how the spiral is constructed This post will do just that The previous post constructed iterated golden rectangles We start with a golden rectangle and imagine chopping of first the

John    I ve seen the illustration of nesting golden rectangles many times but I ve never seen a presentation go into much detail This post will go into more detail than usual including Python code Start with a golden rectangle in landscape mode We ll plot our rectangle with the lower left corner at

John    In my previous post I mentioned John Conway s Doomsday rule for calculating the day of the week for any date This method starts off very simple but gets more complicated when you actually use it This post will present an alternative method that s easier to use in practice and can be described

John    John Horton Conway came up with an algorithm in for mentally calculating what day of the week a date falls on His method which he called the Doomsday rule starts from the observation that every year the dates and fall on the

John    Intervals of a fourth such as the interval from C to F are common in western music but consecutive intervals of this size are not Quartal harmony is based on intervals of fourths and quartal melodies use a lot of fourths particularly consecutive fourths Maybe the most famous quartal melody

John    I m listening to a podcast interviewing Neil Richards the author of Why Privacy Matters Richards makes a couple interesting points about the infamous example of Target figuring out which women were pregnant based on their purchase history First pregnancy is a point at which women are open to

John    As I ve written about here and elsewhere the following simple approximations are fairly accurate log x x x loge x x x log x x x It s a little surprising that each is as accurate as it is but it s also surprising that the approximations for The post

John    A couple days ago I wrote about homogeneous coordinates projective planes I said that the lines y and y intersect in a point at infinity In projective geometry any two distinct lines intersect in exactly one point and you can compute that intersection point the same way whether the

John    A Blaschke product is a function that is the product of Blaschke factors functions of the form b z a a a z a a z where the complex number a lies inside the unit circle and a is the complex conjugate of a I wanted to plot Blaschke products with random The post Random

John    The previous post explained how to define a projective plane over a field F Now let s look at how we do geometry in a projective plane Definitions We have a definition of points from the other post a point is a triple a b c of elements of F with not all elements equal to The post

John    Given a field F finite or infinite you can construct a projective plane over F by starting with pairs of elements of F and adding points at infinity one point for each direction Motivation Bézout s theorem A few days ago I mentioned Bézout s theorem as an example of a simple theorem that

John    Introduction I was puzzled the first time I saw bilinear transformations also known as Möbius transformations I was in a class where everything had been abstract and general and suddenly thing got very concrete and specific I wondered why we had changed gears and I wondered how there could be

John    This post looks at how to partition complexity between definitions and theorems and why it s useful to be able to partition things more than one way Quadratic equations Imagine the following dialog in an algebra class Quadratic equations always have two roots But what about x

John    I got an email from a student in France who asked about a French counterpart to my post on Morse code palindromes and this post is a response to that email Palindromes A palindrome is a word that remains the same when the letters are reversed like kayak A Morse code palindrome is a word The

John    Blaschke factors are complex functions with specified zeros inside the unit disk Given a complex number a with a the Blaschke factor associated with a is the function Notice the semicolon in b z a This is a convention that a few authors follow and that I wish more would adopt From a

John    Inversion in the unit circle is a way of turning the circle inside out Everything that was inside the circle goes outside the circle and everything that was outside the circle comes in Not only is the disk turned inside out the same thing happens along each ray going out from the origin Points

John    Male and female heights both have a standard deviation of about inches with means of inches and inches That s a good first pass model using round numbers If you ask what the height of an average adult is not specifying male or female you get a mixture of two normal distributions If we

John    A few days ago I wrote about the Hilbert transform and gave as an example that the Hilbert transform of sine is cosine We ll bootstrap that example to find the Hilbert transform of any periodic function from its Fourier series The Hilbert transform of a function f t is a function fH x defined

John    I got an evaluation copy of The Best Writing on Mathematics yesterday One article jumped out as I was skimming the table of contents A Zeroth Power Is Often a Logarithm Yearning to Be Free by Sanjoy Mahajan Great title There are quite a few theorems involving powers that have an

John    I explained the basics of how a slide rule works in the previous post But how does a circular slide rule work Apparently the prop Mr Spock is holding is an E B aircraft slide rule It includes a circular slide rule and more functionality Start with an ordinary straight slide rule with each bar

John    Suppose you have two sticks The length of one is log x and the length of the other is log y If you put the two sticks end to end the combined length is log x log y log xy That s the basic idea behind a slide rule The simplest slide rule consists The post Why a slide rule works first