I'm PROUD to live in a universe where
Yes, proud indeed. The equation above is an all time favorite, but what can it possibly mean? It is related to the three formulas below: ix = ln(cos x + i sin x) e±ix = cos x ± i sin x (cos x + i sin x)n = cos nx + i sin nx The first was published (posthumously) by the English mathematician Roger Cotes (1682-1716), of whom Newton said, "If he had lived, we might have known something." The second two equations were published (not necessarily discovered) by Leonard Euler in 1748, though the final equation is more commonly known as De Moivre's Theorem. The first two are, of course, equivalent. Cool, hunh? Setting x = pi in the first two equations gives the simple little identity that includes four of the all time great constants. If you like constants, and you should, MathSoft has all Jake's favorites.

• The Complete Law O' Sines, by yours truly, apparently many countries (other than the US) routinely prove and use this version.
• The one, the only Fractint! Still the greatest fractal generating program, just stroll through the par files (F6), no need to know anything!
• Dr. Matrix's World
• Play with Penrose tiles. 
• The world of math from Eric W. Weisstein.
• The (fairly) complete elements of Euclid.
• Finding the line of best fit visually, pretty neat.
• More to come...

Another great constant is phi, about 1.618, also known as the golden ratio. It's everywhere... Though not really in nature, or art, or music all that much, despite attempts to find it.

The Irrational Acre

    An acre is based on (drum roll please), ten square chains. A chain? Four rods. A rod? Sixteen and one/half feet! The rod was an Anglo-Saxon unit equal to roughly twenty (human) feet, converted by the Norman conquerers to exactly 16 1/2 feet. So one chain is 66 feet, and one acre is 66 by 660 feet, or One Acre = 43,560 square feet. The thing is, this results in a unit of measurement for area which can't be made into a square!

There is no such thing as a square acre.

The acre is based on a small narrow Anglo-Saxon field which can be plowed in one day. This was one furlong (40 rods) long by 4 rods wide. You can make a lot of rectangular acres, but a perfectly square one has sides of irrational length, or more specifically, sides of a length incommensurate with a foot or any fraction of a foot. On the other hand, you can get as close as you like, so it's not really that big a deal... Also on the other hand, while one square acre is tough, ten square acres is a square ten chains to a side, a very convenient size to old time surveyors carrying their 66 foot chains. Note: Dayton's streets are famously wide, and they are ... one chain wide.

Infinite Series