Quilt complex

The quilt on the left is a Broken Star Variation. As Escher knew so well, there's no such thing as plain geometry. However it's possible to take this too far (ok, I don't really mean that!) Here are some amazing mathematical quilts by Diana Venters and Elaine Ellison, who have written a book, Mathematical Quilts: No Sewing Required.

Well, now it's time for me to quilt while I'm ahead. (I'll let you know how it turns out...)

Santa Claus does too exist!

Contrary to the assertion of the scandalous propaganda on the left, Santa Claus does indeed exist. And I can prove it. Start with proposition A:

A. If A is true, then Santa Claus exists.

Now, suppose A were true. Then it would follow that if A is true, then Santa Claus exists, and again since we're supposing A is true, it would follow that Santa Claus exists. So we've shown that if A is true, then Santa Claus exists. But that is proposition A, so we've proven that proposition A is true. So that means that Santa Claus exists! (A remarkable conclusion given my recent post on things that probably don't exist.)

The only trouble is that the reasoning above lets you prove anything (e.g. that penguins rule the universe). It's an example of Curry's paradox, which can't be easily explained away, and is the subject of ongoing research by logicians.

Bah humbug!

Still not convinced? Thomas Aquinas to the rescue! Well, actually, his modern admirers. Aquinas came up with 5 ways of proving the existence of God. Dr. Joseph Magee, a Thomistic scholar, has used similar methods to develop 5 ways of proving the existence of Santa Claus. For example:

The fourth way is taken from the grades which are found in Christmas spirit. Indeed, in this world, among men there are some of more and some of less Christmas spirit. But "more" and "less" is said of diverse things according as they resemble in their diverse ways something which is the "maximum." Therefore there must be something which has the most Christmas spirit, and this we call Santa Claus.

I would question, however, the implicit assumption that it's a man who fits the bill.

Visions of sugarplums

If you think I'm just trying to flatter the jolly old elf so as to garner more loot this Christmas, well ... keep quiet about it, would ya?

log base 2

It turns out that the number 1 reason people visit this blog is to calculate log base 2 of an integer. So here is log₂ of 1 through 10, to 16 digits precision:

log₂(1) = 0
log₂(2) = 1
log₂(3) = 1.584962500721156
log₂(4) = 2
log₂(5) = 2.321928094887362
log₂(6) = 2.584962500721156
log₂(7) = 2.807354922057604
log₂(8) = 3
log₂(9) = 3.169925001442312
log₂(10) = 3.321928094887362

Note that log₂(x) is defined for any x greater than zero. If you have a calculator than computes the natural logarithm (often denoted ln), then you can calculate log₂(x) = ln(x)/ln(2). The same thing works with log base 10, i.e. log₂(x) = log₁₀(x)/log₁₀(2).

But what does it mean?
log₂(x) means the power you have to raise 2 in order to get x. For example, 2² = 4, so log₂(4) is 2. Similarly, 2³ = 8, so log₂(8) = 3. It turns out that 2^1.58496 is very nearly 3, so log₂(3) is roughly 1.58496.

Some cases deserve special mention. log₂(2) = 1 because 2¹ is 2. log₂(1) = 0 because by mathematical convention 2⁰ = 1 (this holds not just for 2, but for any base). Finally, note that log₂(0) is undefined, although some software will return -Infinity (which is the limit of log₂(x) as x approaches zero).

What is it used for?

The logarithm is useful for a variety of purposes. One of the more common is when describing exponential growth or decay. For example, the time for a radioactive substance to decay to half its mass is called the half life. Similarly we can describe accelerating growth in terms of the doubling time. I previously applied this to the number of blogs tracked by Technorati.

In computing, log₂ is often used. One reason is that the number of bits needed to represent an integer n is given by rounding down log₂(n) and then adding 1. For example log₂(100) is about 6.643856. Rounding this down and then adding 1, we see that we need 7 bits to represent 100. Similarly, in order to have 100 leaves, a binary tree needs log₂(100) levels. In the game where you have to guess a number between 1 and 100 based on whether it's higher or lower than your current guess, the average number of guesses required is log₂(100) if you use a halving strategy to bracket the answer.

Two much of nothing

Although I can't provide additional help to people with logarrhythmias, I hope this note is of some assistance.

In memoriam: Ram Myers

Goodbye, dear friend.

I learned of Ram's death on Tuesday evening, from the mother of Ram's wife (Rita, who is also a dear friend).

She added that Rita has requested that people not try to contact her.

It's now early Wednesday morning, and I am going to bed.