Euler Spiral

I was reading a calculus textbook when I noticed it said that cos(x²) 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(x²). 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(x²). There's another Fresnel integral known as the Fresnel C integral which is written as C(x) and defined as the integral of cos(x²).

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 knew that I had to graph it myself. I ended up writing an interactive JavaScript program to graph the parametric equations. Here it is; enjoy!

x = C(t)
y = S(t)

Max t:
t step size: * .001

Zoom X: %
Zoom Y: %

To receive updates, subscribe now!

5 Common Pi Myths

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.

Should Tau Replace Pi?

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.

2013 in Photos

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.

Google Earth

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 blackness around the planet is filled with stars that correspond to the actual positions of stars in the sky. The dark side of the planet is filled with city lights, just as the real planet would have, and both the light and dark sides of the planet contain clouds in the actual positions of clouds on the real planet. It is amazing.

The lighting of the planet, as well as the position of the sun relative to the other stars, is beautifully accurate. I was very impressed that Google even included the planets in the starry sky.

When I start to zoom in on the planet (using the mouse scroll-wheel), all this changes. The clouds are still there, but the stars disappear and the planet no longer has a dark side. This allows for easier navigation of the globe. The clouds fade away after zooming in further.

At closer zoom levels, finer details are visible. In most places, only the terrain is 3D, and the houses are steam-rolled flat to the ground. In other places, however, Google used a special algorithm using visual data to detect 3D structures such as houses or trees. In these areas, the scenery is much more impressive. It may look like a cheap video game, but when you realize that the houses and trees were built using computer software and photographs, it is very amazing.

Racine, WI, near Lake Michigan; notice the 3D structures

The main problems I found are the following:
1. Sometimes, adjacent satellite images are tinted differently, resulting in a striped landscape
2. The bumps on the ocean floor are treated as mountains, so when viewing the ocean at an angle, it looks bumpy and unrealistic
3. There's no way to hide the labels
4. Countries look the same in Summer as in Winter, even when zoomed out completely

So that's what I have to say about Google Earth. Have anything you'd like to add? Leave a comment!