Fibonacci Sequence

In a book completed in the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on?

The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each new term is the sum of the previous two terms. This set of numbers is now called the Fibonacci sequence.

Fibonacci numbers come up surprisingly often in nature, from the number of petals in various flowers to the number of scales along a spiral row in a pine cone. They also arise in computer science, especially in sorting or organizing data.

Amazingly, the ratios of successive terms of the Fibonacci sequence get closer and closer to a specific number, often called the golden ratio. It can be calculated as $\frac(1 + \sqrt5)(2)$, or 1.6180339887…. For instance, the ratio 55/34 is 1.617647…, and the next ratio, 89/55, is 1.6181818….


Fibonacci's immortal rabbit problem. A red rectangle designates a newborn pair, which doesn't produce offspring until the second month.

Proof Without Words

1³ + 2³ + … + n³ = (1 + 2 … + n)².
Mathematically, $\sum_{i=1}^{n} i^3 = (\sum_{i=1}^{n} i)^2$.

Proof Without Words

Area(Parallelogram) = ad - bc.

Proof Without Words

Law of Cosines: $\a^2 +b^2 - 2ab\cos\theta = c^2$.

Proof Without Words

1 + 4 + 9 + … + n² = n(n + 1)(2n + 1)/6.
Mathematically, $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$.

Scaling and Counting Concepts

Secret Worlds: The Universe Within (Powers of 10) shows you the scale of the universe and the microscopic world.

The MegaPenny Project helps you to visualize large numbers.

A Dot for Every Second in the Day

Very interesting website that shows you how things are scaled. Similar to Starship Dimensions.

Famous Unsolved Codes and Ciphers

This site contains an unofficial list of well-known unsolved codes and ciphers. A couple of the better-known unsolved ancient historical scripts are also thrown in, since they tend to come up during any discussion of unsolved codes.

Proof Without Words

$AB + AX = AD$.

XML MAIDEN Project

More Cascading Style Sheets (CSS) to help you create better looking math sites.

Indian American Mathematician is Awarded the Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2007 to Srinivasa S.R. Varadhan, Courant Institute of Mathematical Sciences, New York. He receives the prize “for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation”.

Click here for more information.

MIT OpenCourseWare

MIT recently started offering its entire learning curriculum for free to anyone. Click here for their math portal.

Proof Without Words

Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c,
$a^2 + b^2 = c^2$.

Proof Without Words

The area of a regular dodecagon is 3R².

Proof Without Words

1/4 + (1/4)² + (1/4)³ + … = 1/3.
Mathematically, $\sum_{i=1}^{\infty} \frac{1}{4^i} = \frac{1}{3}$.

Proof Without Words

1 + 2 + 3 + … n = n(n + 1)/2.
Mathematically, $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.

Islamic Quasicrystalline Architecture

Medieval Muslims Made Stunning Math Breakthrough
Magnificently sophisticated geometric patterns in medieval Islamic architecture indicate their designers achieved a mathematical breakthrough 500 years earlier than Western scholars, scientists said on Thursday.

By the 15th century, decorative tile patterns on these masterpieces of Islamic architecture reached such complexity that a small number boasted what seem to be "quasicrystalline" designs, Harvard University's Peter Lu and Princeton University's Paul Steinhardt wrote in the journal Science.

Click here for more information.

Lie Groups - E8

Mathematicians Map E₈
Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E₈. This achievement is significant both as an advance in basic knowledge and because of the many connections between E₈ and other areas, including string theory and geometry.

Click here for more information.

Proof Without Words

The sum of the first n odd numbers is n².
Mathematically, $\sum_{i=1}^{n} (2i - 1) = n^2$.

Starship Dimensions

Okay, this isn't a mathematical post per se, but it does make use of dimensions and exponents.

Note: The images on that site are all exactly to scale, 10 pixels to a meter. Internet Explorer users may Click and Drag the starships to compare them as you like. At the bottom of the image are some contemporary vehicles and buildings for reference. Have fun!

GNU MP Bignum Library

The GNU Multiple Precision (MP) Bignum Library is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on. GMP has a rich set of functions, and the functions have a regular interface.

Compare it to MIRACL for functionality and speed.

MathIWYG

MathIWYG allows you to enter math directly into your website using a Flash-based WYSIWYG (what-you-see-is-what-you-get) editor.

One: A Poem: A Raven

Here's an interesting poem by Mike Keith that counts the digits of π as well.