Out of the bosom of the Air, Out of the cloud-folds of her garments shaken, Over the woodlands brown and bare, Over the harvest-fields forsaken, Silent, and soft, and slow Descends the snow. "Snow-flakes" by Henry Wadsworth Longfellow Happy holidays to all my readers! In my last post I showed

## Tag: **Just for Fun**

I have a fondness for fractals. In previous articles, I've used SAS to create some of my favorite fractals, including a fractal Christmas tree and the "devil's staircase" (Cantor ) function. Because winter is almost here, I think it is time to construct the Koch snowflake fractal in SAS. A

Last week I blogged about how to draw the Cantor function in SAS. The Cantor function is used in mathematics as a pathological example of a function that is constant almost everywhere yet somehow manages to "climb upwards," thus earning the nickname "the devil's staircase." The Cantor function has three

I was a freshman in college the first time I saw the Cantor middle-thirds set and the related Cantor "Devil's staircase" function. (Shown at left.) These constructions expanded my mind and led me to study fractals, real analysis, topology, and other mathematical areas. The Cantor function and the Cantor middle-thirds

Lo how a rose e'er blooming From tender stem hath sprung As I write this blog post, a radio station is playing Chrismas music. One of my favorite Christmas songs is the old German hymn that many of us know as "Lo, How a Rose E're Blooming." I was humming

In my previous blog post, I showed how you can use SAS to program a "weaving" algorithm that takes an image, cuts it into strips, and weaves the strips together to create mathematical art. I used matrices and heat maps for the computations and visualization. At the end of the

An artist friend of mine recently created a beautiful abstract image and described the process on her blog. She says that "after painting my initial square, I cut it into strips and split them down the middle, then wove them together.... I had no idea when I started piecing these

Last week I was chatting with some mathematicians and I mentioned the blog post that I wrote last year on the distribution of Pythagorean triples. In my previous article, I showed that there is an algorithm that uses matrix multiplication to generate every primitive Pythagorean triple by starting with the

Saturday, March 14, 2015, is Pi Day, and this year is a super-special Pi Day! This is your once-in-a-lifetime chance to celebrate the first 10 digits of pi (π) by doing something special on 3/14/15 at 9:26:53. Apologies to my European friends, but Pi Day requires that you represent dates

The xkcd comic often makes me think and laugh. The comic features physics, math, and statistics among its topics. Many years ago, the comic showed a "binary heart": a grid of binary (0/1) numbers with the certain numbers colored red so that they formed a heart. Some years later, I

O Christmas tree, O Christmas tree, One year a fractal made thee! O Christmas tree, O Christmas tree, A heat map can display thee! From Pascal's matrix we define! Reflect across, divide by nine. O Christmas tree, O Christmas tree, Self-similar and so divine! Eventually I will run out of

Pascal's triangle is the name given to the triangular array of binomial coefficients. The nth row is the set of coefficients in the expansion of the binomial expression (1 + x)n. Complicated stuff, right? Well, yes and no. Pascal's triangle is known to many school children who have never heard of polynomials

When I studied high school geometry, I noticed that many homework problems involved right triangles whose side lengths were integers. The canonical example is the 3-4-5 right triangle, which has legs of length 3 and 4 and a hypotenuse of length 5. The triple (3, 4, 5) is called a

Many people enjoy solving word games such as the daily Cryptoquote puzzle, which uses a simple substitution cipher to disguise a witty or wise quote by a famous person. A common way to attack the puzzle is frequency analysis. In frequency analysis you identify letters and pairs of letters (bigrams)

My previous blog post describes how to implement Conway's Game of Life by using the dynamically linked graphics in SAS/IML Studio. But the Game of Life is not the only kind of cellular automata. This article describes a system of cellular automata that is known as Wolfram's Rule 30. In

A colleague jokingly teases me whenever I write a blog that demonstrates how to write fun and exciting programs by using SAS software. "Why do you get to have all the fun?" he mock-chides. Today I'm ready to face his ribbing, because this article is about Conway's Game of Life

Last week Chris Hemedinger posted an article about spam that is sent to SAS blogs and discussed how anti-spam software helps to block spam. No algorithm can be 100% accurate at distinguishing spam from valid comments because of the inherent trade-off between specificity and sensitivity in any statistical test. Therefore,

Yesterday I blogged about the Hilbert matrix. The (i,j)th element of the Hilbert matrix has the value 1 / (i+j-1), which is the reciprocal of an integer. However, the printed Hilbert matrix did not look exactly like the formula because the elements print as finite-precision decimals. For example, the last

Many geeky mathematical people celebrate "pi day" on March 14, because the date is written 3/14 in the US, which is evocative of the decimal representation of π = 3.14.... Most people are familiar with the decimal representation of π. The media occasionally reports on a new computational tour-de-force that

Although I currently work as a statistician, my original training was in mathematics. In many mathematical fields there is a result that is so profound that it earns the name "The Fundamental Theorem of [Topic Area]." A fundamental theorem is a deep (often surprising) result that connects two or more

Prime numbers are strange beasts. They exhibit properties of both randomness and regularity. Recently I watched an excellent nine-minute video on the Numberphile video blog that shows that if you write the natural numbers in a spiral pattern (called the Ulam spiral), then there are certain lines in the pattern

O Christmas tree, O Christmas tree, Last year a fractal made thee! O Christmas tree, O Christmas tree, A heat map can display thee! O tree of green, adorned with lights! A trunk of brown, the rest is white. O Christmas tree, O Christmas tree, A heat map can display

Each year my siblings choose names for a Christmas gift exchange. It is not unusual for a sibling to pick her own name, whereupon the name is replaced into the hat and a new name is drawn. In fact, that "glitch" in the drawing process was a motivation for me

This article is about rotating matrices. No, I don't mean "rotation matrices," I mean rotating matrices. As in turning a matrix 90 degrees in a clockwise or counterclockwise direction. I was reading a program written in MATLAB in which the programmer used a MATLAB function called ROT90, which rotates a

While walking in the woods, a statistician named Goldilocks wanders into a cottage and discovers three bears. The bears, being hungry, threaten to eat the young lady, but Goldilocks begs them to give her a chance to win her freedom. The bears agree. While Mama Bear and Papa Bear block

Editor's Note: My 8th grade son, David, created a poster that he submitted to the 2013 ASA Poster Competition. The competition encourages students to display "two or more related graphics that summarize a set of data, look at the data from different points of view, and answer specific questions about

In my previous post, I described how to implement an iterated function system (IFS) in the SAS/IML language to draw fractals. I used the famous Barnsley fern example to illustrate the technique. At the end of the article I issued a challenge: can you construct an IFS whose fractal attractor

Fractals. If you grew up in the 1980s or '90s and were interested in math and computers, chances are you played with computer generation of fractals. Who knows how many hours of computer time was spent computing Mandelbrot sets and Julia sets to ever-increasing resolutions? When I was a kid,

Last week I wrote a SAS/IML program that computes the odds of winning the game of craps. I noted that the program remains valid even if the dice are not fair. For convenience, here is a SAS/IML function that computes the probability of winning at craps, given the probability vector

Gambling games that use dice, such as the game of "craps," are often used to demonstrate the laws of probability. For two dice, the possible rolls and probability of each roll are usually represented by a matrix. Consequently, the SAS/IML language makes it easy to compute the probabilities of various