Cosmos Computed

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...

Recently, Google released a new map system to allow for a 3D Earth view. The new maps depict Earth as a 3D planet, and allow users to explore mountains and cities in 3D.  At first, Google Earth could be downloaded and installed on a computer, but now it is available at maps.google.com for Google users to use without downloading it. To try the new maps, sign in to your Google account and go to maps.google.com. In the tab on the left, click "Get it now" where it describes the new Google maps. If you do not see the button, try  typing "new maps" into the search bar and pressing Enter. The "Get it now" button should appear above the results. My photograph of the Grand Canyon on left, Google Earth's Grand Canyon on right (same location) I really like the new maps. Besides being able to explore the Grand Canyon or Mount Everest, I especially like what happens when I zoom out all the way: Earth becomes visible with a lit side and a shadowed side, and the bl...