For some reason, The Four-Color Theorem went viral* today. Maybe it was a slow news day, as they say on SlashDot.
io9 (“We come from the future”) featured a slightly puzzling post, The first math problem that we needed a computer to solve.
In 1976, a computer was used to solve a math problem… the first one that would probably remain unsolved if it weren’t for computers. Computers were routinely used to solve things… But prior to 1976, they weren’t required to prove any math problem. They just made things easier… until Kenneth Appel and Wolfgang Haken used a computer to prove a 124-year-old conjecture…
specifically, The Four-Color Theorem introduced by mathematician Francis Guthrie in 1852.
Better yet was the response that the io9 entry elicited in one of our own, Flaneur in Pajamas, here on Tumblr. The pleasingly colorful image above is from the Flaneur.
Via flaneurinpajamas:
Correct me if I’m wrong, guys… Wouldn’t the Four-Color Theorem fall apart if confronted with a country traversing four concentric countries, or traversing two sets of two concentric countries? Consider Lesotho, San Marino, or the Vatican… surely it’s possible that a set of neighborhoods could evolve this way?
EDIT: And the internet responds! Anon writes:
“In the two examples you provide, the red and yellow don’t touch, nor do the blue and green. Which means that any of those could (and would) be changed to the other color and the blank space would be the color it changed from. Thus making it work.”
So simple now that it’s pointed out to me! Turns out that Wikipedia has an explanation [click on image] that applies here but even having read that I wasn’t able to generalize it to my own attempt at disproof. Oh well, my dreams of inserting myself into the world of higher math by abrupt genius will have to wait till another time!
UPDATE: Credit where credit is due
io9 does reference a January, 2005 post from Devlin’s Angle, a proto-blog on the Mathematical Association of America (MAA) website. It is a good, fun read, explaining how a network approach was successfully used as the starting point by Appel and Haken. And it clarifies the significance of computing to mathematical proofs, particularly with respect to The Four Color Theorem:
Appel and Haken started their computer-assisted investigation in 1972 and four years later they had their answer. It took 1,200 hours of computer time, during which the computer had to carry out billions of calculations. The two mathematicians themselves had to analyze by hand some 10,000 portions of networks.
With the Appel-Haken result, something had happened that mathematicians had wondered about since computers had first appeared in the 1950s: Machines had finally taken over some of the task of proving theorems.
* Maybe I am exaggerating ever so slightly. I don’t truly know the quantitative particulars for determining viral. Yet I feel confident in saying that The Four Color Theorem hasn’t received anything close to 11,500 page views in a 12-hour interval prior to now. Of course, it doesn’t come close to the 28,000+ page views snagged by today’s other popular io9 post, Researchers quantify just how badly your favorite celebrity is photo-shopped, but that shouldn’t be a surprise to any one. It is the way of the world.
![prostheticknowledge:
10 PRINT CHR$(205.5+RND(1);: GOTO 10
Piece of code in the Commodore 64 manual, used in a small digital art show called ‘Codings’ at the Pace Digital Gallery, New York, which looks at code, text and digital art:
A 3-line version of this program appeared in the original Commodore 64 User’s guide:
10 PRINT “{CLR/HOME}20 PRINT CHR$(205.5+RND(1)30 GOTO 20
This is one of many short basic programs, for this and other computers, that have been entered by users seeking to puzzle their friends, to learn more about computing, and to see aesthetically pleasing output.
You can find out more about the show here
[sad confession - I never knew there was an easy bit of code to do this on a Commodore … I discovered the effect myself, but did it manually, typing the two characters myself at random …]
Nothing to be ashamed of, ProstheticKnowledge. I see some old friends there: GOTO and RND.](http://25.media.tumblr.com/tumblr_m02ynrdYiJ1qav3uso1_500.png)
![For some reason, The Four-Color Theorem went viral* today. Maybe it was a slow news day, as they say on SlashDot.
io9 (“We come from the future”) featured a slightly puzzling post, The first math problem that we needed a computer to solve.
In 1976, a computer was used to solve a math problem… the first one that would probably remain unsolved if it weren’t for computers. Computers were routinely used to solve things… But prior to 1976, they weren’t required to prove any math problem. They just made things easier… until Kenneth Appel and Wolfgang Haken used a computer to prove a 124-year-old conjecture…
specifically, The Four-Color Theorem introduced by mathematician Francis Guthrie in 1852.
Better yet was the response that the io9 entry elicited in one of our own, Flaneur in Pajamas, here on Tumblr. The pleasingly colorful image above is from the Flaneur.
Via flaneurinpajamas:
Correct me if I’m wrong, guys… Wouldn’t the Four-Color Theorem fall apart if confronted with a country traversing four concentric countries, or traversing two sets of two concentric countries? Consider Lesotho, San Marino, or the Vatican… surely it’s possible that a set of neighborhoods could evolve this way?
EDIT: And the internet responds! Anon writes:
“In the two examples you provide, the red and yellow don’t touch, nor do the blue and green. Which means that any of those could (and would) be changed to the other color and the blank space would be the color it changed from. Thus making it work.”
So simple now that it’s pointed out to me! Turns out that Wikipedia has an explanation [click on image] that applies here but even having read that I wasn’t able to generalize it to my own attempt at disproof. Oh well, my dreams of inserting myself into the world of higher math by abrupt genius will have to wait till another time!
UPDATE: Credit where credit is due
io9 does reference a January, 2005 post from Devlin’s Angle, a proto-blog on the Mathematical Association of America (MAA) website. It is a good, fun read, explaining how a network approach was successfully used as the starting point by Appel and Haken. And it clarifies the significance of computing to mathematical proofs, particularly with respect to The Four Color Theorem:
Appel and Haken started their computer-assisted investigation in 1972 and four years later they had their answer. It took 1,200 hours of computer time, during which the computer had to carry out billions of calculations. The two mathematicians themselves had to analyze by hand some 10,000 portions of networks.
With the Appel-Haken result, something had happened that mathematicians had wondered about since computers had first appeared in the 1950s: Machines had finally taken over some of the task of proving theorems.
* Maybe I am exaggerating ever so slightly. I don’t truly know the quantitative particulars for determining viral. Yet I feel confident in saying that The Four Color Theorem hasn’t received anything close to 11,500 page views in a 12-hour interval prior to now. Of course, it doesn’t come close to the 28,000+ page views snagged by today’s other popular io9 post, Researchers quantify just how badly your favorite celebrity is photo-shopped, but that shouldn’t be a surprise to any one. It is the way of the world.](http://24.media.tumblr.com/tumblr_lvlbx6p63q1qeaf2mo1_500.jpg)
![engineering:
This is a VT220 serial console (circa 1983) set up as a terminal for my Mac Pro (circa 2010), a nerdy dream I’ve had for a long time that I finally made a reality yesterday.
Some quick history: in the early days of office computers, it was rare that you would actually have one on your desk. Instead there might be a central mainframe (running Unix) and everyone would have a terminal that connected to it over a long serial cable or modem connection. One computer, many users.
The terminal has a keyboard and monitor, but it’s not a full computer and worthless without the mainframe. It’s more like a teletype machine, all it can do is display the text sent to it (like a paperless printer) and send text back. It doesn’t have any knowledge of pixels or colors or graphics of any kind.
In modern times we don’t have mainframes anymore, but Unix is more prevalent than ever. It runs on the servers delivering this page and the iPhone in your pocket. For developers and power users the command line has never gone away, but instead of a dedicated hardware serial console we have Terminal.app (with translucent backgrounds and anti-aliased fonts). The software is just emulating the old hardware, though. The protocols haven’t changed much in 30 years. The Unix underpinnings of OS X still have all the stuff required to use a real serial terminal, it’s just no one actually does it (well, almost no one).
I’ve always thought those old terminals were beautiful, and I’m not the only one—there’s a Mac app called Cathode that does a convincingly wonderful job simulating vintage terminals, using OpenGL to degrade things into a nice analog haze. But it’s not quite the same as the real thing…
Hardware terminals regularly crop up on eBay for around $100. They’re actually still used in a lot of places (old warehouse systems, supermarkets, banks) and there are still companies that support and refurbish them. Back at Vimeo we discovered one abandoned in a server closet when we moved into the office. Finding one isn’t a problem, the main challenge is stringing together the right adapters to use an ancient serial port with modern USB.
[see content source for details omitted here]
Eventually I found this page, which explains the problem and how to fix it. After adding a line in /etc/gettytab to manually set the terminal type to vt220-8bit everything works perfectly! A real hardware terminal directly connected the old fashioned way, with no emulation. Awesome.
If this is something you want to attempt yourself please drop me a line; I learned a lot about how terminals work over the last couple weeks and the final result is quite satisfying, a soft amber glow and one less window on my desktop. It’s also a nice reminder that we didn’t get to where we are overnight, user interfaces and software development have been evolving in an unbroken chain for a long time and some of the old ideas are so solid that they persist 30 years later. Why not use the proper hardware?](http://24.media.tumblr.com/tumblr_lpo5dfqyIz1qz4mo8o1_r1_500.jpg)
