Cosmos Computed

For some reason I felt like this was relevant Happy Pi Day! The digits of the date - 3.14 - contain the first couple digits of π. But that's not all. If you add the year to get 3.14.16, you have the first 5 rounded digits of π. This is better than last year's version (3.14.15), which contained a truncated version of π. Today's date contains the more accurate rounded version. In this post, I'll describe the process of finding an algorithm to approximate the number π. There are three main types of algorithms. The first type converges very quickly to the correct solution; we use algorithms of this type in computers and calculators. The second type converges slowly, but is interesting to think about; we take this type of algorithm to mathematics conferences. The third type is slow to converge, and is boring to think about; these are pretty much useless so we get rid of them. This may be interesting to know, but I wasn't too worried about it when I wrote my Pi Day post l...

Here's a simple physics puzzle: suppose you're launching a model rocket. You want the rocket to go as high as possible. Is it better to design the rocket to: A. Use all of its fuel very quickly, and then use its momentum to travel high B. Use the fuel gradually over the whole duration of the flight In other words, what is the fuel consumption rate that will maximize the height of the flight path? For this problem, we can assume that the thrust is proportional to the fuel consumption rate. I'll try to post a solution to the problem in the future. New posts every month - subscribe for free !

First off, apologies for this long-overdue post. I've had a busy month, and although I would have liked to post this within a few days of the beginning of 2016, I wasn't able to get it done until now. 2015 was a great year, and I think the images in this post will reflect that. In the previous year's image collection ( click here to view ), the images were only photographs. The image editing was restricted to cropping; I didn't need anything more. In this collection, though, I have drawings, CGI, and microphotography. These new types of images required more processing to prepare them for viewing. Of course, I used GIMP. I can't afford to pay for software. Instead of listing tons of images, I picked out the most important ones. I also tried not to add the images from previous blog posts; the majority of what follows is new stuff. I also attempt to analyze the images from both a scientific and artistic standpoint. Most likely, by the time most of you read this, you...

Planet original Last month, I drew a couple astronomy-related pictures using a graphite pencil on a sheet of white paper. The problem with pencil sketching is that it's hard to draw white glowing objects... so I sketched a negative, and then uploaded the images to my computer and inverted them. Then I increased the contrast to improve the black background and enhance the white. The image at the top of this post is a ringed planet, possibly Saturn. I included three moons and some lighting on the backside from the rings. The whole image was smaller than a dime. Black hole original The image below is an imagined view of a black hole. I included high-energy jets of ejected material and an accretion disk. Most black holes are not actively consuming material, and do not have jets. But those are boring to draw. New posts every month - subscribe for free !

Imagine the universe is filled with water. Instead of empty space, every inch of it contains pure water. No planets, no stars, only water. What happens? And what would happen if an air bubble formed? The answer to this question requires a basic understanding of gravity. Gravity is very important. It helps hold matter together, bends light, and distorts space-time (which, incidentally, is how it bends light). It also makes it possible to play football, and as Americans are big football fans, they would certainly agree that we couldn't live without it. Unfortunately, many Americans don't understand how gravity works. Admittedly, scientists haven't figured out a lot of things, but we do understand it well enough to make predictions and model physical events. One of the important things about gravity is that its strength is proportional to the inverse of the square of the distance. In other words, it gets weaker as you get farther away, based on the equation: F = c/d 2 where c ...

Keystone on opening day Keystone Ski Resort just opened for the ski season on Friday. I went up to the resort that same day. There was only one run open (not including the beginner area at the top of the mountain), but it was awesome nevertheless! One of the main downsides to skiing on opening day is the number of people. There must have been about 2500 people on the mountain at the same time as me (not including the people snacking in the lodge). To keep the lines moving, the lift attendants made sure that the lift was completely full, with 4 people per chair. Even so, the wait to get on the lift took a long time. At one point when I was standing in line, a thought came to mind: the line was constantly being filled with more and more people, but it never got longer because the chair lift was carrying the people away at the same rate. So what would happen if the lift attendants only put 3 people on each chair, instead of 4? This would disrupt the balance: the inflow of skiers would be ...

Over my fall break (which lasted 1 day), I wrote a JavaScript program that would render a sphere. I thought it would be a fun way to practice math and programming skills simultaneously. As usual, I used my FireTools.js library to help with graphics as well as some other functions. To render the sphere, I wanted to take every pixel on the screen, and calculate the color of the pixel. The color would vary depending on which part of the sphere the pixel was displaying. If the part of the sphere was facing the light source, it would need to be brighter than a part of the sphere facing away from the light source. To find the brightness of each point on the surface, I first calculated the XYZ coordinates of that part of the sphere. This is also equal to the normal vector at that point. I took the dot product between the light source's direction and the normal vector. This is a mathematical operation that tells me whether the surface is aligned with the light source, as well as the percen...