Numerical Analysis Better Left Unsaid (and other musings of interest)
A gyroid is a fascinating geometric structure. It's a three-dimensionally-tileable unit that creates an infinitely connected surface. The surface is triply-periodic, meaning it repeats in all three dimensions. Gyroids also occur naturally in polymer science and biology. For 3D printing, a gyroid is a useful infill pattern. Not only does it fill volume without using much material, but it also…
Have a look at this centipede walking. You can clearly see waves of legs propagating forward along the body, from back to front, as the centipede walks forward. The motion of the legs is known as a metachronal rhythm , appearing as traveling waves caused by actions happening in squence. It's even more obvious with a millipede. In each case, a leg in back leads the one in front. Is this universal?…
I recently joined a Toastmasters club local to me, to improve my public speaking skills. In the educational pathways available, the first formal speech one gives is the "icebreaker", in which you speak for 4–6 minutes about a comfortable personal topic. Here is my icebreaker speech, which I accompanied with a slideshow of pictures as I spoke. Something that mattered Thank you, Mr…
You don't see aircraft with elliptical wings anymore. The most famous aircfaft with such wings is probably the Supermarine Spitfire fighter aircraft from World War II. Elliptical wings have the most uniform theoretical distribution of lift and therefore the least induced drag. In the case of the Spitfire, the gentle taper of the ellipse near the wing root also provided more room to mount weapons…
A recent project to create an ergonomic handle for 3D printing led me down a path that introduced me to anthropometric measurements of the human hand, which in turn revealed some interesting facts about hand sizes. I had no idea that hand size is dependent on nationality, but I found a number of research articles on this topic. For those who don't want to read further, the answer to the title…
Back in July 2005 I became intrigued with crafting weird and improbable magical items for the amusement of my Dungeons and Dragons gaming group. At the time, I had also been employed in the defense industry for about half my life, working around various weapons systems. Naturally, these things came together in creating a couple of D&D weapons, one of which became a meme in both D&D and in…
I learned this card trick in the fourth grade, decades ago, before the World Wide Web existed. I have never seen it written about, and anyone to whom I have shown it has never seen it either. This is surprising given how long I've known this trick. Did a brilliant classmate (or a parent) invent it? I'd love to know the origin. This is a mathematical card trick. Meaning, there is no sleight of…
A game master / dungeon master (DM) should never allow a player to bring 3D printed dice to a game. However, a DM could supply them to players, for certain purposes. In this article I examine the characteristics of "most fair" and "most unfair" designs of d20 dice, which I made in a CAD program and 3D printed for experiments. A fair and balanced d20 (white), a d20 biased to 20 (bronze, with the ☺…
I decided to create a procedurally-generated 3D landscape in a CAD program, and wrap it around a globe, which led me to an investigation of erosion algorithms. I include a JavaScript erosion demo at the end of this article for you to play with. As an aside, let me say that I like OpenSCAD in spite of its idiosyncrasies. Other CAD programs can do many things that OpenSCAD can't, but OpenSCAD is the…
The Bollinger Bands plot on a price bar chart is a common component of technical analysis among financial traders. Bollinger Bands display a graphical envelope around the moving average of price, with the width of the envelope representing volatility of the financial instrument charted. Exchange-traded fund SPY showing 20-day moving average (cyan centerline) and 2-standard-deviation Bollinger…
Some years ago I visited the Exploratorium in San Francisco, where I saw an impressive work of kinetic art, a large mechanical contraption with many articulated legs made from PVC pipe. It walked easily when gently pushed. Kids would take turns in groups pushing it back and forth in a parking lot. This was my first exposure to a strandbeest (Dutch for "beach beast"), a name given by the artist…
When my employer switched over from the Google office suite to Microsoft's Office 365, it caused unexpected disruption in my family. I had been sharing my work calendar with my wife (sharing no details, just busy times). It became an integral part of her calendar, to help her coordinate (between her job and my job) who picks up and drops off my son from school, and to let her know when I was free…
This is an intellectual exercise with a practical end-result. There are better approaches to this problem, but the approach I describe here is interesting. A screw thread from a stack of discs Imagine a stack of 50 coins, stacked so that each coin is slightly offset from the one below it, with the offset moving around in a circle with each new coin. Here is what it looks like for fifty coins 20 mm…
There are stories on the internet about "the fourth little pig", but before I ever saw those, a decade ago I made up my own story called The four little pigs as a bedtime story for my toddler. I've always liked the perspective I put on the original tale. Once upon a time... ...there were four little pigs: Curley, Hurley, Turley, and Xerxes, all brothers. These four pigs knew they had to build a…
Practitioners of agile project management, particularly the scrum agile framework for software development, might be familiar with with this problem: When the project starts, there are many stories and tasks in the project backlog. There aren't any bugs yet. When planning each sprint, the stories / tasks / improvements are assigned story points — not time units, but dimensionless numbers…
This entry has nothing to do with numerical methods, mathematics, logic, gaming, or anything else I've written about. Well, there may be a few numbers. Maybe. One day in January this year, while spending precious minutes of my life browsing YouTube as people tend to do on occasion, my clicks led from one thing to another ranging from music videos to Maker projects to magic acts, and then I landed…
My son and I began playing Minecraft together, me on my Windows laptop and him on his iPad. In my opinion, Minecraft is probably the only game in which a parent and child can play together in the same virtual world and both of them enjoy it. Quality time with your child in a video game? Who would have thought? The game is easy enough for a child to understand and complex enough for an adult to…
Thrust from air So now we're ready to put all the calculations together: launch tube, water thrust, air thrust, and ballistic flight. The launch tube calculation is more like an initialization for the water thrust phase; we just have one time step of the duration of launch tube traversal. Get the initial pressure \(P_0\) for water thrust from (L1), and the velocity \(v_0\) from (L4), and use these…
Thrust from water Numerical calculations After the water is expended, the tank (bottle) still contains substantial pressure that releases quickly through the nozzle. This final burst of air can impart a significant boost to velocity (at least 30% depending on the mass of the rocket), so we shouldn’t ignore this contribution to thrust. For convenience, however, we ignore any further affects on air…
Launch tube thrust Thrust from air Once the rocket has left the launch tube, the water thrust phase of the flight begins. But before we begin those calculations, we need to have equations for ballistic flight, which is the last step described in the introduction . Why? Because during ballistic flight (coasting through the air), the only forces acting on the rocket are gravity and wind drag —…
The water rocket: Introduction Thrust from water A launch tube inside the bottle does two important things: A tube with its open end above the water level keeps the water from spilling into the launcher pipe-work, leaving it inside the bottle for the rocket to use as reaction mass. It allows the bottle to gain velocity as the nozzle slides along the length of the tube, with negligible leakage of…
Launch tube thrust I recently became interested in water rocketry after realizing how complicated the physics actually can be, and the fact that the equations are best solved with numerical methods because so many interacting variables change rapidly in a non-linear fashion. For those unfamiliar, a water rocket is a rocket that uses water as its reaction mass, powered by air pressure. The cool…
While I'm on the subject of Dungeons & Dragons (see my previous post on ability score probabilities ), I recall something I did involving a tesseract way back in 2007 and posted on the community forum of Wizards of the Coast. WOTC took down their forum in 2015, but fortunately the Wayback Machine has an archived copy. Imagine a cubical room. It has four walls, a ceiling, and a floor (six faces).…
There's an element to the game Dungeons & Dragons that lends itself to a numerical analysis: the initial array of ability scores assigned to a character. Some background: A character in the game has scores assigned to each of six abilities (strength, dexterity, constitution, intelligence, wisdom, and charisma). The first step in creating a character is to generate an array of six numbers by…
A Koch snowflake. CC-BY-SA Wikimedia Commons While thinking about the Koch snowflake it occurred to me that a recursive algorithm to generate one side of the curve would be trivial, but a non -recursive algorithm is pretty simple as well, although it took a lot of thought on my part to get it to work. I don't claim to be an expert at algorithms but I'm pleased when I can figure something out.…