Cosmos Computed

Some people pay hundreds to be able to photograph tiny subjects such as snowflakes, but now you can do it for $5. Taken with a jeweler's loupe and a digital camera I recently read an article about a man who took photos of snowflakes by taping an extra lens on the front of his camera. Naturally, I was interested. I had always enjoyed taking close-up photos of things, but I hadn't been able to take very many high-quality photos of anything as small as a snowflake. After reading the details of his method, I had an idea: instead of using an expensive lens like he was using, I could use one of the cheap plastic jeweler's loupes I already have ( available on Amazon for about $5). I had tried using loupes to take close-up images before, but without much success. After reading the article, though, I realized that there was a trick I had been missing: after taping the lens on the front of the camera, I should have zoomed in all the way before taking the photo. At first, it seems to...

Glass cellphones... brilliant About a month and a half ago, an Orbital Sciences rocket exploded during liftoff. Although nobody was injured from the explosion, hundreds of millions of dollars of supplies and equipment were destroyed in a matter of seconds. The fireball, which could be seen for miles, stunned observers; many hadn't even seen a lift-off before, much less an exploding one. Only three days later, another space-related accident occurred: during a test flight of Virgin Galactic's SpaceShipTwo spacecraft, a premature wing rotation due to pilot error resulted in the destruction of the spacecraft, killing one of the two occupants and severely injuring the other. These technological accidents may seem out of the ordinary, but they are actually quite common. All throughout history, humans have suffered accident after accident related to their own technology. Even if they do not make the news, small accidents happen every day: a car breaks down, somebody drops a light bulb...

Here's a program I wrote earlier this year in July. It's a two-player pong game that involves communicating over the internet using Firebase. As I wrote this program, I also wrote a JavaScript library to take care of graphics and data communication over the internet. The library took the most work; the pong program just required some math and some creativity to help solve the problems I ran into... such as how the two computers would figure out who gets the top paddle and who gets the bottom paddle. To play the game, you'll need to send the URL to one of your friends (otherwise you won't have an opponent). Move your paddle (the white one) using the arrow keys, and click "Reset" to start over. The ball will bounce to the left or to the right depending on where it hits the paddle – you can use this to your advantage. And forgive the glitches; the computers are communicating over the internet, so it's not going to be perfect. Have fun! New posts every month ...

About a week ago, I was finally able to observe the second solar eclipse of my life – a partial eclipse with a maximum coverage of about 50%. The only other eclipse I have observed was another  partial solar eclipse in May 2012 . A solar eclipse occurs when the moon casts its shadow on Earth's surface. From the point of view of an observer in the shadow, the moon covers the sun. It's not always obvious when a solar eclipse is happening; if the observer is in near the edge of the shadow, so the sun is only partly covered, it might not seem like anything unusual is happening. This was my situation during the eclipse last week. The moon only covered part of the sun, so when I told one of my friends that an eclipse was occurring, he hadn't even realized it. And I don't blame him. Actually, a few minutes earlier, I hadn't realized it either. I got the time mixed up, and didn't think the eclipse had begun yet. The sunlight was weaker than usual, but I assumed that was...

If time were stopped, would this falling pen look blurry to us? Time can be a very confusing subject. It has a past, a present, and a future, but the future keeps changing and the present is always moving, while the past grows larger. Although most people think they understand time, they often get confused when they try to describe it. To help visualize and understand time, many people compare it to a very familiar concept: a line. This line, called a timeline, represents time as a whole. The current moment, or the present, is a single point which moves smoothly along the timeline. The part of the timeline behind the point is the past, and the part ahead of the point represents the future. The whole timeline, including the moving point that represents the present, is considered exactly the same for everyone. Although this whole approach may be a useful way to represent time, it is actually inaccurate in describing time. First of all, the present does not have to be experienced in the s...

It's that time of the year when I get the most hummingbirds at my feeder. It's hot, and the fields are brown, but the little whirring birds make summer fun the whole season long. With their buzzing wings and long beaks, hummingbirds can seem almost like giant mosquitoes. Hummingbirds usually beat their wings around 50 or 60 times per second. That's very fast, and it takes a lot of energy. That energy has to come from somewhere. Do you ever drink energy drinks before exercising? It's like that for hummingbirds, except that they need even more energy than you do. Hummingbirds drink more than their weight in nectar every day. Predictably, hummingbird feeders help hummingbirds a lot. The hummingbirds can easily get as much nectar as they need, and they can get it whenever they need it (unless the feeder runs out of nectar). Feeders are also nice for people, because they get to see the hummingbirds that visit it. One thing about hummingbirds is that they try to dominate the ...

What is a vitamin? I don't mean the different types of vitamins or the shops where they can be bought. I mean, what makes something a vitamin? Perhaps a vitamin is only a useless chemical that serves no purpose, and is therefore called a vitamin to make it seem useful when it really isn't. I prefer to think of a vitamin, however, as an essential chemical that our body needs to grow and survive. Unfortunately, the former definition is sometimes the more accurate one; in fact, some of the vitamins being sold today do nothing good for the human body. One example of this is vitamin B 17 . Laetrile was named a vitamin in the 1900's when Ernst T. Krebs, Jr. wanted to get around the restrictions on drugs by selling laetrile as a nutritional supplement. Some say that Krebs was a medical doctor, but he never even got a doctoral degree, except for an honorary degree from a school not accredited to award advanced degrees. In reality, Krebs was just a chemist that wanted to make money....

I recently added a "Did You Know" app to the sidebar of my blog. The app randomly picks a math/science/technology fact from a list, and displays it. I had to write the app myself (with JavaScript and HTML), because Blogger doesn't have an app like that built-in, and even if it did, I like to have control over my things. To get technical: the data is stored on Firebase. The app reads the data from Firebase, and adds it all to an array. There is no way to tell how many items there are without reading all of them from Firebase. The app then looks at how many elements are in the array (call that number  a ) and then randomly chooses a number  b from 1 to a . It then takes the b th element from the array, and displays it. (I know, I didn't have to store them all in an array, but it's easier that way.) Because of the data storage method, I can add new facts very easily: I just have to go to Firebase and add the new fact to the database. Of course, my app updates as soo...

Here's a simple JavaScript + HTML program I wrote. It will graph a given number of terms of a Fourier series. To graph a Fourier series, type the appropriate expressions into the text fields and click "Update." The text fields support JavaScript, so you can write a whole function in there if you need to. I also added some functions. even(x)  - returns true if x is even, false otherwise. odd(x)  - returns true if x is odd, false otherwise. power(x, a)  - returns x a . Some examples of input to try: odd(k)?(2/k):0 even(k)?(4/k/PI):0 odd(k)?(4/(k+PI)/k):0 The default Fourier series is equal to sin(2.5x) on (-π, π). Enjoy! Fourier Series Grapher Min x: Max x: Min y: Max y: *sin(k*x) *cos(k*x) Terms: '> To receive updates, subscribe now!

I was reading a calculus textbook when I noticed it said that cos( x 2 ) doesn't have an elementary antiderivative. Elementary antiderivative? Clearly, they were hiding something. They didn't say it didn't have an antiderivative; they said it didn't have an elementary  antiderivative. Of course, I wanted to know what the antiderivative was. If it wasn't elementary, it had to be really awesome. I looked up the integral of sin( x 2 ). Turns out, the integral cannot be expressed as anything other than itself. It's known as the Fresnel S integral, is written as S( x ), and is defined as the integral of sin( x 2 ). There's another Fresnel integral known as the Fresnel C integral which is written as C( x ) and defined as the integral of cos( x 2 ). I also saw some graphs of the integrals. One really cool graph involved the parametric equations x  = C( t ) and y  = S( t ), and was called the "Euler spiral." It had a cool spirally shape, and I immediately...

Happy π-day! And happy π-month! Today's month and day - that is, March 14 or 3.14 - includes the first 3 digits of π. And today's month and year - March 2014 or 3.14 - also includes the first 3 digits of π. We won't have another double-day for π for the next 100 years, so enjoy this one! For the special occasion, I'm posting two π-related posts, one for π-month and the other for π-day. In both posts, I'm setting the font size to 16.1527897 pixels, which is approximately π * π + π + π. This is the second post, for π-day; for the first, go to  http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-1.html . In this post, I will list 5 common myths about π, and explain why they're wrong. 1. Pi is wrong Actually, π is defined as the ratio between the circumference of a circle and its diameter. There really isn't anything about this ratio that could possibly be wrong, because a ratio is a ratio - just like an orange is an orange or a piano is a piano. You can...

The digits of π, organized in a very new way Happy π-day! And happy π-month! Today's month and day - that is, March 14 or 3.14 - includes the first 3 digits of π. And today's month and year - March 2014 or 3.14 - also includes the first 3 digits of π. We won't have another double-day for π for the next 100 years, so enjoy this one! For the special occasion, I'm posting two π-related posts - one for π-month, and the other for π-day. In both posts, I'm setting the font size to approximately π * π + π + π. This is the first post, for π-month; to see the second, go to http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-2-5-common-pi-myths.html . In this post, I am including an essay I wrote about whether π or τ is the more superior constant. This was written for people who know very little about math, so the basic idea should be easy to understand even for people who are not mathematically inclined. Should Tau Replace Pi? A constant is any number or value that ne...

I know it's already February, but I wanted to post some of my favorite photos I took last year. This post will include those photos, along with some commentary under each one. None of the photos have been posted in previous posts, so you won't have to see any repeats. I took all of the photos in this post using the same camera. Winter This is a waning gibbous moon rising above distant city lights. I took this photo late at night in early January. Unfortunately, my camera isn't the best for taking astronomical photos, but the photo is still pretty good considering that fact. I saw these rays of light beaming down through the clouds when I was skiing at Keystone. I had no choice but to take a picture. This is a black-billed magpie. I see these a lot in the winter where I live. For a blog post about identifying birds, go to  http://greatmst.blogspot.com/2012/07/bird-watching.html . These two photos are from my trip to Arizona . The thing on the left is a really cool butte...