[math-fun] antikythera device: an ancient computer
a while back, someone posted a link to this youtube video of the antikythera device made out of legos: http://www.youtube.com/watch?v=RLPVCJjTNgk here is an interesting (58 minute) bbc documentary that explains in more detail how the device actually works, as far as we know today: http://www.youtube.com/watch?v=-rUsgL2VGeU many of the gears had a prime number of teeth. this was very helpful in figuring out what the device was used for, and how it worked. here are several shorter videos that might also be of interest: The Antikythera Mechanism - 2D (8 minutes) http://www.youtube.com/watch?v=UpLcnAIpVRA The Antikythera Mechanism - 3D (4 minutes) http://www.youtube.com/watch?v=L1CuR29OajI Antikythera mechanism working model.mov (4 minutes) http://www.youtube.com/watch?v=4eUibFQKJqI in case anyone passes through bozeman, montana, there is a great computer museum that has a replica of the antikythera device. the museum also has an old ibm 1620 computer, a curta calculator, and many other neat old things: http://www.compustory.com/ note: their rather primitive website doesn't do the place justice! bob baillie
At 08:48 AM 11/25/2012, Robert Baillie wrote:
here is an interesting (58 minute) bbc documentary that explains in more detail how the device actually works, as far as we know today: http://www.youtube.com/watch?v=-rUsgL2VGeU
many of the gears had a prime number of teeth. this was very helpful in figuring out what the device was used for, and how it worked.
Fabulous video; highly recommended!! Ok, all you algebra geeks: these gears with prime numbers of teeth had to be built, so which of the prime numbers in this device are "constructible" using a straight-edge and compass ? For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
If I were doing it, I'd wrap a wire tightly around the circumference of the gear blank and mark the circumference on the wire. Then I would use a grid of N equally spaced parallel lines to put N equally spaced marks on the wire; rewrapping the wire would let me transfer the marks to the gear blank. I'm sure there are other equally practical contructions. On Fri, Nov 30, 2012 at 2:42 PM, Henry Baker <hbaker1@pipeline.com> wrote:
At 08:48 AM 11/25/2012, Robert Baillie wrote:
here is an interesting (58 minute) bbc documentary that explains in more detail how the device actually works, as far as we know today: http://www.youtube.com/watch?v=-rUsgL2VGeU
many of the gears had a prime number of teeth. this was very helpful in figuring out what the device was used for, and how it worked.
Fabulous video; highly recommended!!
Ok, all you algebra geeks: these gears with prime numbers of teeth had to be built, so which of the prime numbers in this device are "constructible" using a straight-edge and compass ?
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The only constructable primes are 2 and the Fermat primes, probably just 3, 5, 17, 257, 65537. It doesn't seem like any explicit geometric construction was used for this device, probably just measuring out the circumference and tooling the gaps as appropriate. They did have techniques (mechanical linkages, I think, and certainly neusis) which allowed the construction of many more, though in neither case is 223 constructable. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Nov 30, 2012 at 2:42 PM, Henry Baker <hbaker1@pipeline.com> wrote:
At 08:48 AM 11/25/2012, Robert Baillie wrote:
here is an interesting (58 minute) bbc documentary that explains in more detail how the device actually works, as far as we know today: http://www.youtube.com/watch?v=-rUsgL2VGeU
many of the gears had a prime number of teeth. this was very helpful in figuring out what the device was used for, and how it worked.
Fabulous video; highly recommended!!
Ok, all you algebra geeks: these gears with prime numbers of teeth had to be built, so which of the prime numbers in this device are "constructible" using a straight-edge and compass ?
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 11/30/2012 11:42 AM, Henry Baker wrote:
At 08:48 AM 11/25/2012, Robert Baillie wrote:
here is an interesting (58 minute) bbc documentary that explains in more detail how the device actually works, as far as we know today: http://www.youtube.com/watch?v=-rUsgL2VGeU
many of the gears had a prime number of teeth. this was very helpful in figuring out what the device was used for, and how it worked. Fabulous video; highly recommended!!
Ok, all you algebra geeks: these gears with prime numbers of teeth had to be built, so which of the prime numbers in this device are "constructible" using a straight-edge and compass ?
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
The way uniform machines were first constructed was by using symmetry. Polishing three metal blocks against one another produces a plane surface. If you run two chased screws against one another you get a uniform helical pitch. I imagine the Greeks did the same. If you rough out the teeth on two gears that are relatively prime and then run them against one another with some abrasive compound you get uniform spacing. Brent
Does anyone here know what the fundamental _period_ of the antikythera device is? I.e., by all reports, this device is a "planetarium in a box". So how many years of simulated time does it take for this device to return to its initial configuration ? If we start the antikythera device in 70 B.C., which appears to be approximately the date of its demise, how far into the future can this device "see" ? Also, if the antikythera device really is a "planetarium in a box", and if it really does do a good job of modeling the known universe at the time, how come we can't tell _the exact date_ of its demise ? Let's assume that the device was used on a daily, weekly or monthly basis. Presumably, the position of the wheels would be indicative of the last time it was used. This is essentially equivalent to being able to determine the exact time of an earthquake from the position of the hands of clocks which stopped as a result of the earthquake.
http://www.scientificamerican.com/article.cfm?id=antikythera-mechanism-eclip... -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Gear wheels have teeth that have to engage, which implies that they have the same pitch. It is easy to construct gear wheels with 2^n teeth. Since gear wheels are made from circular blanks, and since the circumference is proportional to the diameter (we don't even have to know the value of pi), we can produce a circular blank whose diameter is any integral ratio k/(2^n) to the diameter of a 2^n circular blank. All we have to do now is to mark off gear teeth of the same pitch in the new circular blank. If we measured everything carefully, the circumference on the new non-2^k blank should be an integral number of teeth. It's too bad that the antikythera mechanism is so corroded; it would be very interesting to study an uncorroded ancient gear train to see exactly what the shape of the gears was. I'd be willing to bet that the Greeks probably figured out a very close approximation to the correct shape for the gear teeth, as well, so that the gear train would have very little "play" which might throw off the accuracy of the device. At 11:42 AM 11/30/2012, Henry Baker wrote:
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
At 04:07 PM 11/30/2012, you wrote:
Gear wheels have teeth that have to engage, which implies that they have the same pitch.
It is easy to construct gear wheels with 2^n teeth.
Since gear wheels are made from circular blanks, and since the circumference is proportional to the diameter (we don't even have to know the value of pi), we can produce a circular blank whose diameter is any integral ratio k/(2^n) to the diameter of a 2^n circular blank.
All we have to do now is to mark off gear teeth of the same pitch in the new circular blank. If we measured everything carefully, the circumference on the new non-2^k blank should be an integral number of teeth.
Nutz! That should be ^^^^^^^ non-2^n
It's too bad that the antikythera mechanism is so corroded; it would be very interesting to study an uncorroded ancient gear train to see exactly what the shape of the gears was. I'd be willing to bet that the Greeks probably figured out a very close approximation to the correct shape for the gear teeth, as well, so that the gear train would have very little "play" which might throw off the accuracy of the device.
At 11:42 AM 11/30/2012, Henry Baker wrote:
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?
participants (6)
-
Allan Wechsler -
Charles Greathouse -
Henry Baker -
meekerdb -
Mike Stay -
Robert Baillie