10-Minute Math

Jorge Luis Borges‘ short story Library of Babel conceives of a library comprised of all possible books.  Specifically, all possible permutations of 25 characters (22 letters, comma, period, space) within books with 410 pages.  (Each page has 40 lines; each line has 80 characters).  This set of books would contain all books written so far (spanning multiple volumes if needed), along with all books that could possibly be written in the future.  The majority of the books in the library would be gibberish, of course.

A few questions, ordered from easy to hard:

• How many books are in The Library of Babel?
• How much space would this set of books consume (let’s ignore the rooms and shelves, and just stack the books, which measure, say, 6 in. x 4 in. x 1.5 in. )?
• What proportion of the books would ready coherently from start to finish?

In this library, there would be a book explaining the solutions to each of these questions!

Here is a long-term project with the intent of making all systematic knowledge immediately searchable.  Try some of the example queries or browse by topic.

I’ve been searching for online programs (not requiring downloads, easier to use at school) to explore the Mandelbrot set.  The programs I have found all have strengths and weaknesses, but here are the top ones.  If you know of one that should be on the list, let me know.

1. Neal Ziring’s Mandelbrot Applet

Pros: Easy navigation, anti-aliasing, control over the color gradient.
Cons: I couldn’t get it to save the image (had to PrtScn), and it aborted computation when I selected a large image size.

2. Mandelbrot Fractal Explorer

Pros: Good navigation controls (double-click or shift+drag to zoom, drag to pan), lots of preset color options.  Nice anti-aliasing.  Image can be downloaded.
Cons: Cannot specify image size (you need larger images for printing)

3. Julia and Mandelbrot Set Explorer

Pros: Allows you to set the image size in pixels.  Shows corresponding Julia set.
Cons: Navigation isn’t easy to use.  No anti-aliasing.

Courtesy of Fail Blog.

Lately my precalculus and calculus classes have been watching episodes of this series, exploring the concept of spacial dimensions. Distributed under a Creative Commons License, the series is free to download and show in class, and gives an excellent overview of projective geometry in two, three, and four dimensions! Incidentally, the animations were created using POV-Ray, a free ray-tracing program that generates photo-realistic images using precise mathematical modeling.

This problem comes from Vaughn:

A light bulb begins to burn in the middle of outer space.  It burns for one year.

a) The “sphere of light” now has a certain volume.  If the light bulb keeps burning, how long must you wait for this volume to double?

b) If, after the year is up, the light bulb stops burning, how long will it take the “sphere of light” to double its volume?

For this exercise, the “sphere of light” is a rough term for the set of locations in outerspace from which an observer can see the light.