## Saturday, 28 November 2009

### Numbers, Big Numbers.

A googolplex is the number 10googol, which can also be written as the number 1 followed by a googol of zeros (i.e., 10100 zeros).
1 googolplex
= 10googol
= 10(10100)
= 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Writing it in numerals would be physically impossible, since doing so would require more space than there is in the known universe.
An average book of 60 cubic inches can be printed with 5 x 105 '0's (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3 × 103 '0's per cubic inch. The observable universe contains 6 × 1083 cubic inches (1.3 × π × (14 × 109 light year in inches)3).
This implies that if the universe is stuffed with paper printed with '0's, it could contain only 5.3 × 1087 '0's, far short of a googol of '0's. In fact there are only about 2.5 × 1089 elementary particles in the observable universe so even if you could write a zero on each elementary particle, you would still have to make the universe's mass about a trillion times larger to complete the number.
The time it would take to write such a number also renders the task impossible: if a person writes two digits per second, it would take around about 1.51 × 1092 years, which is 1.1 × 1082 times the age of the universe, to write a googolplex.
That's pretty fucking big, but a bright chap called Graham has an even bigger one named after him, notated here as G. $\left. \begin{matrix} G &=&3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots\cdots \uparrow}3 \\ & &3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots \uparrow}3 \\ & &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\ & &3\uparrow \uparrow \uparrow \uparrow3 \end{matrix} \right \} \text{64 layers}$
The number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,
$G = g_{64},\text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3,\ g_n = 3\uparrow^{g_{n-1}}3,$
G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3's; the second step is to calculate g2 with g1 up-arrows between 3's; the third step is to calculate g3 with g2 up-arrows between 3's; and so on, until finally calculating G = g64 with g63 up-arrows between 3's.
$g_1 = 3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3) = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots ))$
(3^^^3 = 3^^(3^^3) is 3^^7,625,597,484,987 = 3^(7,625,597,484,987^7,625,597,484,987), which makes a tower of exponents 7,625,597,484,987 layers high.)
Even n, the mere number of towers in this formula for g1, is far greater than the number of Planck volumes (roughly 10^185 of them) into which one can imagine subdividing the observable universe. And after this first term, there are still another 63 terms in the g sequence before Graham's number G = g64 is reached.
That's really fucking big, but size isn't everything. It's not perfect, unlike these few examples of the Perfect Number, starting with our humble 6 (no-one knows how many more there are):

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128,2658455991569831744654692615953842176,191561942608236107294793378084303638130997321548169216,13164036458569648337239753460458722910223472318386943117783728128,14474011154664524427946373126085988481573677491474835889066354349131199152128,23562723457267347065789548996709904988477547858392600710143027597506337283178622239730365539602600561360255566462503270175052892578043215543382498428777152427010394496918664028644534128033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646976,141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128, 54162526284365847412654465374391316140856490539031695784603920818387206994158534859198999921056719921919057390080263646159280013827605439746262788903057303445505827028395139475207769044924431494861729435113126280837904930462740681717960465867348720992572190569465545299629919823431031092624244463547789635441481391719816441605586788092147886677321398756661624714551726964302217554281784254817319611951659855553573937788923405146222324506715979193757372820860878214322052227584537552897476256179395176624426314480313446935085203657584798247536021172880403783048602873621259313789994900336673941503747224966984028240806042108690077670395259231894666273615212775603535764707952250173858305171028603021234896647851363949928904973292145107505979911456221519899345764984291328.

And then, of course, you've got infinity a.k.a. Aleph Null. Sort of.
It can't be annotated but Albert Einstein, who knew a thing or two about numbers, said
"Only two things are infinite, the Universe and human stupidity, and I'm not sure about the former."

## Friday, 27 November 2009

### Go on....String a sentence together.

Sarah Palin's 'book' has sold c.700,000 copies. Instaputz.blogspot.com reviewed "Going Rogue", they weren't terribly impressed:

"If you are in a hurry, here is the succinct version of this review: Going Rogue is shit. It is groundbreaking in its banality and disregard for facts. If you are sentient, it will pain you to read it. Imagine watching your parents 69 one another while John Madden sits behind you and bellows out color commentary and you will have some idea of how excruciating and profoundly scarring it is to plow through each page of this wholly fictional monument to self-aggrandized mediocrity. Going Rogue is to the art of writing what the Holocaust is to the concept of a just God – the piece of disconfirming evidence so overwhelming that we are left questioning whether it can exist at all.
Going Rogue is not without merit. It certainly delivers what its intended audience wants. Readers who already like Palin will love it, much as America’s pedophiles will find the latest Jonas Brothers DVD to their liking. The authors’ talent for communicating the ex-Governor’s unique rhetorical style in print is remarkable – the Sesame Street cadence of her delivery and the intermittent Tourette’s-like winks leap off the page. The book, recession priced at just \$9, is also an ideal gift for the Aunt or Uncle who assaults your email inbox with a dozen weekly communiqués on the President’s Kenyan birth and the constitutionality of income taxes.
Unfortunately that is an exhaustive list of its strengths."

## Wednesday, 25 November 2009

### So you think nobody listens to You?

For 23 years (that's about 200 million breaths) Rom Houben was imprisoned in his own body. He saw his doctors and nurses as they visited him during their daily rounds; he listened to the conversations of his carers; he heard his mother deliver the news to him that his father had died. But he could do nothing. He was unable to communicate with his doctors or family. He could not move his head or weep, he could only listen.
Doctors presumed he was in a vegetative state following a near-fatal car crash in 1983. They believed he could feel nothing and hear nothing. For 23 years.
Then a neurologist, Steven Laureys, who decided to take a radical look at the state of diagnosed coma patients, released him from his torture. Using a state-of-the-art scanning system, Laureys found to his amazement that his brain was functioning almost normally.

"I had dreamed myself away," said Houben, now 46, whose real "state" was discovered three years ago, according to a report in the German magazine Der Spiegel this week.
Laureys, a neurologist at the University of Liege in Belgium, published a study in BMC Neurology earlier this year saying Houben could be one of many cases of falsely diagnosed comas around the world. He discovered that although Houben was completely paralysed, he was also completely conscious — it was just that he was unable to communicate the fact.
He realised when he came round after his accident, which had caused his heart to stop and his brain to be starved of oxygen for several minutes, that his body was paralysed. Although he could hear every word his doctors spoke, he could not communicate with them.

"I screamed, but there was nothing to hear," he said, via his keyboard.

The Belgian former engineering student, who speaks four languages, said he coped with being effectively trapped in his own body by meditating. He told doctors he had "travelled with my thoughts into the past, or into another existence altogether". Sometimes, he said, "I was only my consciousness and nothing else".
The moment it was discovered he was not in a vegetative state, said Houben, was like being born again. "I'll never forget the day that they discovered me," he said. "It was my second birth". Houben now communicates with one finger and a special touchscreen on his wheelchair – he has developed some movement with the help of intense physiotherapy over the last three years.
Experts say Laureys' findings are likely to reopen the debate over when the decision should be made to terminate the lives of those in comas who appear to be unconscious but may have almost fully-functioning brains. Belgian doctors used an internationally-accepted scale to monitor Houben's state over the years. Known as the Glasgow Coma Scale, it requires assessment of the eyes, verbal and motor responses. But they failed to assess him correctly and missed signs that his brain was still functioning.

Last night Houben's mother, Fina, said in an interview with Belgian RTBF that they had taken him to the US five times for reexamination. The breakthrough came when it became clear that Houben could indicate yes and no with his foot.
"Powerlessness. Utter powerlessness. At first I was angry, then I learned to live with it," he tapped out on to the screen during an interview with the Belgian network last night, AP reported.
Laureys, who is head of the Coma Science Group and department of neurology at Liege University hospital, has advised on several prominent coma cases, such as the American Terri Schiavo, whose life support was withdrawn in 2005 after 15 years in a coma. He concluded that coma patients are misdiagnosed "on a disturbingly regular basis". He examined 44 patients believed to be in a vegetative state, and found that 18 of them responded to communication.
"Once someone is labelled as being without consciousness, it is very hard to get rid of that," he told Der Spiegel. He said patients suspected of being in a non-reversible coma should be "tested 10 times" and that comas, like sleep, have different stages and need to be monitored.

Houben hopes to write a book detailing his trauma and his "rebirth".

## Tuesday, 3 November 2009

### R.I.P.Claude Levi-Strauss

Tricked by the Eternal Trickster.