## Golden Rectangle

October 20th, 2009 by Math Tricks | No Comments | Filed in Golden Rectangle, Mathematics Concepts

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## The Golden Rectangle

The golden rectangle is a mathematical concept that goes back to antiquity. As a definition, the golden rectangle can be described as a rectangle which has a height to base proportion of 1: 1.6180339 (approximately). This ratio also as a special name – the golden ratio. It is also know by several other names, including the divine proportion, the golden mean, and the golden number. In mathematics, it is denoted by the Greek letter phi (φ).

So what is so special about the golden rectangle? Why does it have such a divine proportions? Why am I planning on writing several posts on this seemingly mundane rectangle?

First, let me say that this special rectangle is found in many places. It is found in art, architecture, and nature as well. Look at the Mona Lisa, and you will see that the subjects face is bounded by a golden rectangle. The Parthenon, built in ancient Greece, has several golden rectangles. In nature, the logarithmic growth of nautilus shells is at a rate of phi (φ).

In this section, I will be adding more posts on this subject. I’ll show you how to construct this rectangle, and how to derive phi several ways. This really is a fascinating mathematical concept, and I hope you come back for the post to follow.

## Math Patterns

October 12th, 2009 by Math Tricks | 5 Comments | Filed in Math Patterns

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## Math Patterns

This is the first post to the new category Math Patterns.  Here in this category, I will post many interesting math patterns; some of these are very well known, and some are obscure.  For many of these mathematical patterns, one can derive general formulas very easily by just carefully observing how the math pattern develops.  Other numeric patterns will have you pounding your head on your keyboard while trying to come up with the next number in the series.

So, I give you here the first in the series (pun intended!) of math patterns:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379 . . .

Can you see the next number in the pattern?

Here is a subset of this math pattern:

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379 . . .

The subset above is simply all of the prime numbers in the first set.  So then, what is the pattern of the first set?  This is the set of happy numbers up to and including 379.

So what the heck is a happy number?  A happy number is happy if you take each digit in the number, square them, add the squares together, and then repeat the process with the result and then eventually get the value 1.

For example, take the number 338.  First square each digit and add them together:

9 + 9 + 64 = 82

Now repeat the process with the result (82):

64 + 4 = 68

Continue the process with 68:

36 + 64 = 100

And then with 100:

1 + 0 + 0 = 1

Here the result is 1, and thus 338 is a happy number!

So what if a number is not happy?  Interestingly, an unhappy number ends up in a cyclic loop with the pattern:

4, 16, 37, 58, 89, 145, 42, 20, 4 . . .

So is there a limit to the number of happy numbers?  Happily, there are an infinite number of happy numbers – and an infinite number of unhappy numbers as well.