World War I officially ended in 1918 at the 11th hour of the 11th day of the 11th month. Remembrance Day is observed in Commonwealth countries to recall the end of the war and to remember the members of the armed forces who gave their lives for the cause. This year we observe Remembrance Day on 11/11/11. Beyond the somberness of this memorial day, those of us who are mathematically inclined consider the surprising ways we can combine 1s to achieve beautiful results. Some of these combinations involve rather unpleasant calculations, so we'll let Mathematica do the heavy lifting while we marvel at the results! Today I'll share 11 interesting places in which 1 appears. Let's jump in. 1. Can anything of interest come from combining the humble digit 1 with the square root and plus symbol? As the demonstration below suggests, given the right nesting, you get an infinite series that converges to φ = 1.61803..., the golden ratio.

The art of pumpkin carving is hard to master, yet once a year parents in many countries are asked to perform this traditional and messy form of art. It's time for a change in this old tradition. In fact, our colleague Jon McLoone already made a significant advance in pumpkin carving, mainly using implicit functions and RegionPlot3D. This year, I decided to make a contribution of my own that is more interactive and easier to use, with Mathematica or Mathematica Home Edition, of course. Let's start with a list. These are the things you need for traditional pumpkin carving. A nice looking pumpkin Carving tools of your choice: from a spoon and knife (if you are a true professional) to an industrial 36,000 rpm power rotary tool (seriously, I know someone who uses one) A bunch of candles to be placed inside the pumpkin A pattern for the carving on paper For industrial Mathematica pumpkin carving, you need these tools. B-spline curve, surface, and function Color processing functions Morphological image processing functions ParametricPlot3D with Texture A pattern for the carving as a bitmap Intrigued? Let us begin.

What could be a better way to celebrate the Fourth of July than beautiful fireworks in the dark sky? And what could be a better way to create fireworks on your screen than using Mathematica? There are a few different ways to create firework "effects" on computers, but it would be a shame if we chose to use just graphical effects with Mathematica. Yes, we are going for the full-scale particle simulation. Here is the synopsis. We create a firework simulation. With a mouse click, we seed a number of particles on the screen. Each particle has a different initial velocity, and it will follow the projectile motion. The particles spend a limited time on the screen, in which their opacity will diminish gradually. There will be a few customizable effects—colors and trails.

When I first learned about π, I was told that a good approximation was 22/7. Even when I was 12 years old, I thought this was utterly pointless. 22/7 agrees with π to two decimal places (so three matching digits): Since there are three digits to remember in 22 and 7, what have you gained? You have just as much to remember, but have lost the notion that π is “just over 3”. Is there a better rational approximation where we actually get out more digits than we put in? Here is a brief and rather low-brow investigation (and the chance to win something if you can do better).