Earlier today I posted a comment about searching through the decimal expansion of Pi for a given sequence. And, I gave an example of the sequence that the expansion contains.
Later in the day I thought:
What is the probability that the given length of the expansion of Pi contains a specified (albeit randomly chosen) sequence?
Consider the expansion of Pi to be a random sequence in that for a location the probability that the digit is d is 1/10. That's not really true, of course, but let's assume it anyway.
So, the first problem is to find the probability that the expansion of Pi contains any given random sequence. Let the length of the given sequence be variable. The length of the expansion is fixed.
The second part is to plot the probability of the expansion containing a random sequence of digits for an increasing length of the chosen sequence. Thus, the probability that the expansion contains a single digit (somewhere in the expansion) is 1. But what about the sequence "16" then, what about a sequence like "598" and continue on for longer sequences.
I don't have the answer, but I'm hoping you do.
Tuesday, November 29, 2005
Are you in Pi?
Search the decimal expansion of pi for a given string of digits.
Here's a neat site that finds a string of digits in the first 200-million digits of pi.
Example: A search for the string "4444" gives back:
"The string 4444 was found at position 54,525 counting from the first digit after the decimal point. The 3. is not counted.
The string and surrounding digits:
21855877513127211793 4444 82014404257450830639 "
Here's a neat site that finds a string of digits in the first 200-million digits of pi.
Example: A search for the string "4444" gives back:
"The string 4444 was found at position 54,525 counting from the first digit after the decimal point. The 3. is not counted.
The string and surrounding digits:
21855877513127211793 4444 82014404257450830639 "
Thursday, November 17, 2005
NSA: For the kids
The National Security Agency, the one that seeks to decode various secrets, has a website for children. It's actually pretty neat and fun to try.
This is an excellent idea, by the way. The agency will need cryptographers in the future and this is one way to get children interested in this field. Children should start early if they can be hooked to do it.
Kudos to NSA for a good idea.
(Hat tip: Bruce Schneir's Cryptogram)
This is an excellent idea, by the way. The agency will need cryptographers in the future and this is one way to get children interested in this field. Children should start early if they can be hooked to do it.
Kudos to NSA for a good idea.
(Hat tip: Bruce Schneir's Cryptogram)
Sparklines Implemented
In an earlier post I mentioned the Edward Tufte idea of Sparklines.
Sparklines are small, compact graphics that show the change of a variable over time (or some other parameter). Their beauty and utility stems from their small size and ability to relate information easily and intuitively. I encourage you to click to Edward Tufte's site for more information.
There is now a site that makes the sparkline plots for you, at the link above. I hope you'll give them a try and make them part of your own writings. The more they're used the more people will come to understand them. And because Sparklines convey information so readily, the more people will understand the information you want them to understand.
Sparklines are small, compact graphics that show the change of a variable over time (or some other parameter). Their beauty and utility stems from their small size and ability to relate information easily and intuitively. I encourage you to click to Edward Tufte's site for more information.
There is now a site that makes the sparkline plots for you, at the link above. I hope you'll give them a try and make them part of your own writings. The more they're used the more people will come to understand them. And because Sparklines convey information so readily, the more people will understand the information you want them to understand.
Tuesday, November 15, 2005
Powers of 10
If you haven't seen this before, it's worth a click. This site shows you the universe through various resolutions and detail.
Hubble picture
Explore distance galaxies back to just a few hundred millions years after the Big Bang!
Sunday, November 13, 2005
Tin foil hats: You're not crazy....but...
"Among a fringe community of paranoids, aluminum helmets serve as the protective measure of choice against invasive radio signals. We investigate the efficacy of three aluminum helmet designs on a sample group of four individuals. Using a $250,000 network analyser, we find that although on average all helmets attenuate invasive radio frequencies in either directions (either emanating from an outside source, or emanating from the cranium of the subject), certain frequencies are in fact greatly amplified. These amplified frequencies coincide with radio bands reserved for government use according to the Federal Communication Commission (FCC). Statistical evidence suggests the use of helmets may in fact enhance the government's invasive abilities. We speculate that the government may in fact have started the helmet craze for this reason."
(Hat tip: Julius)
Wednesday, November 09, 2005
Serious Lego
Now I've seen Lego projects my 12-year old builds, but this site is for the serious Lego enthusiast.
Beware going to it. You'll want to do your own.
Beware going to it. You'll want to do your own.
Tuesday, November 08, 2005
Good quote
From Edward Tufte's discussion site:
"I was also pleased to learn that 'the messy desk is not necessarily a sign of disorganization. It may be a sign of complexity: those who deal with many unsolved ideas simultaneously cannot sort and file the papers on their desk, because they haven't yet sorted and filed the ideas in their head.' "
Says it all.
"I was also pleased to learn that 'the messy desk is not necessarily a sign of disorganization. It may be a sign of complexity: those who deal with many unsolved ideas simultaneously cannot sort and file the papers on their desk, because they haven't yet sorted and filed the ideas in their head.' "
Says it all.
Coding Theory: No more to do
Erica Klarreich from Science News reports on how coding has almost reached the Shannon limit.
The Shannon limit (for Claude Shannon who discovered it) tells how efficient a code can be so that the original message can be recovered from a digitized version of the message. Take the message, put it into bits, and transmit it. When you receive the message some of the bits may have (likely will have!) errors. Coding theory has devised many clever ways to send the data bits with extra bits to correct the errors, at some efficiency. Not long ago, that efficiency was much less than what was theoretically possible.
Now, two engineers have discovered Turbo Codes that get close to the Shannon limit. What's more, an older discovery called low-density parity-check (LDPC) coding could get even closer.
So, much of what engineers spent careers trying to do can be done. But, most of that previous work is obsolete.
" 'The ideas behind turbo codes and LDPC codes have rendered much of the preceding 50 years of coding theory obsolete', says David MacKay of Cambridge University in England, one of the coding theorists who rediscovered LDPC codes. 'Future generations won't have to learn any of the stuff that has been the standard in textbooks,' he says."
The Shannon limit (for Claude Shannon who discovered it) tells how efficient a code can be so that the original message can be recovered from a digitized version of the message. Take the message, put it into bits, and transmit it. When you receive the message some of the bits may have (likely will have!) errors. Coding theory has devised many clever ways to send the data bits with extra bits to correct the errors, at some efficiency. Not long ago, that efficiency was much less than what was theoretically possible.
Now, two engineers have discovered Turbo Codes that get close to the Shannon limit. What's more, an older discovery called low-density parity-check (LDPC) coding could get even closer.
So, much of what engineers spent careers trying to do can be done. But, most of that previous work is obsolete.
" 'The ideas behind turbo codes and LDPC codes have rendered much of the preceding 50 years of coding theory obsolete', says David MacKay of Cambridge University in England, one of the coding theorists who rediscovered LDPC codes. 'Future generations won't have to learn any of the stuff that has been the standard in textbooks,' he says."
Confusions: Probability
A current posting at Mathematical Association of America is about probability theory and how we often misunderstand it.
Keith Devlin makes the point that people have problems when they want to assign a probability to knowledge:
"In my experience, it's when probabilities are attached to information that most people run into problems.
The concept of probability you get from looking at coin tossing, dice rolling, and so forth is generally referred to as "frequentist probability". It applies when there is an action, having a fixed number of possible outcomes, that can be repeated indefinitely. It is an empirical notion, that you can check by carrying out experiments.
The numerical measure people assign to their knowledge of some event is often referred to as "subjective probability". It quantifies your knowledge of the event, not the event itself. Different people can assign different probabilities to their individual knowledge of the same event. The probability you assign to an event depends on your prior knowledge of the event, and can change when you acquire new information about it."
Devlin gives a beautiful example of subjective probability. I have my own though. Suppose you have ten beans covered by cups. Nine beans are black, one is red. What is the probability that when you turn over a cup the bean is red? Intuitively it's 1/10. Now, suppose I turn over nine cups so you see the covered beans. What's the probability that the last cup has a red bean? At this point it's no longer a probability but rather a statement of known fact. You can see the other beans. If one of those is red, the cup has a black bean. If none of the nine exposed beans is red, the covered bean is red. There's no uncertainty at all.
I've often wondered just what probability meant and how we can use it consistently and properly. Devlin makes a good start to doing just that.
Keith Devlin makes the point that people have problems when they want to assign a probability to knowledge:
"In my experience, it's when probabilities are attached to information that most people run into problems.
The concept of probability you get from looking at coin tossing, dice rolling, and so forth is generally referred to as "frequentist probability". It applies when there is an action, having a fixed number of possible outcomes, that can be repeated indefinitely. It is an empirical notion, that you can check by carrying out experiments.
The numerical measure people assign to their knowledge of some event is often referred to as "subjective probability". It quantifies your knowledge of the event, not the event itself. Different people can assign different probabilities to their individual knowledge of the same event. The probability you assign to an event depends on your prior knowledge of the event, and can change when you acquire new information about it."
Devlin gives a beautiful example of subjective probability. I have my own though. Suppose you have ten beans covered by cups. Nine beans are black, one is red. What is the probability that when you turn over a cup the bean is red? Intuitively it's 1/10. Now, suppose I turn over nine cups so you see the covered beans. What's the probability that the last cup has a red bean? At this point it's no longer a probability but rather a statement of known fact. You can see the other beans. If one of those is red, the cup has a black bean. If none of the nine exposed beans is red, the covered bean is red. There's no uncertainty at all.
I've often wondered just what probability meant and how we can use it consistently and properly. Devlin makes a good start to doing just that.
Friday, November 04, 2005
The fourth dimension: You can play with it
The University of Mass. has a lab that allows students to play (or study) shapes from the fourth dimension.
Thursday, November 03, 2005
Google Print
If you haven't heard yet, Google wants to digitize books and make them available on the web. Great idea! I think it'll increase sales of books and make books that are out of print available to everyone.
Stephen Hawkins: God Created the Integers
I bought this book on my way tonight. Looks excellent. I'll review it when I've gone through it.
Tufte Bulletin Board
Tufte also has a bulletin board for discussions on data presentations with links to other sites of interest. This is worth spending a lot of time looking at the links, reading the comments, and digesting the information.
Edward Tufte: Sparklines
Edward Tufte is an expert on how data should be presented and he's written three excellent books on this subject. I recommend all his books and especially his advice.
My friend Andy (hattip: Ilachina) sent me this link to a chapter of Tufte's next book. This chapter, entitled "Sparklines" tells how data can presented within text to show a trend and allow one to quickly see trends. This is not totally new, Euclid did it as did others. What's new is the advocacy of it for papers today. While data is presented so hap-hazardly, it refreshing and needed to see Sparklines.
My friend Andy (hattip: Ilachina) sent me this link to a chapter of Tufte's next book. This chapter, entitled "Sparklines" tells how data can presented within text to show a trend and allow one to quickly see trends. This is not totally new, Euclid did it as did others. What's new is the advocacy of it for papers today. While data is presented so hap-hazardly, it refreshing and needed to see Sparklines.
Photographing flying insects
The man shows you how he designed and built and used! a camera, plus electronics to photograph flying insects. It's not simple and it's amazing how much work this man did.
Science Toys
Make your own toys at home with a science theme. Neat site and neat toys.
Christmas is coming, maybe you don't have to shop far to find something fun?!
Christmas is coming, maybe you don't have to shop far to find something fun?!
Who is the greatest mathematician?
Cast your vote for greatest mathematician. Naturally, Gauss is a competitor but I think Euler should be it.
What do you think?
What do you think?
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