Re: Emergency Sundial
> Edley, Steve, Andrew, et al, > > Approximate 'trig.' from memory is all very well, but > why not simple geometry of equality and bisection? > Because nobody ever pointed it out to me before !! > With a string, twig, or weed-stem part as arbitrary > unit-length b, swing a circular arc. Cut successive > chords of length b, which if connected to the center > form 60° equiangular triangles. Then successively > bisect ... Yep, it's nifty. S.
RE: Emergency Sundial
Edley, Steve, Andrew, et al, Approximate 'trig.' from memory is all very well, but why not simple geometry of equality and bisection? With a string, twig, or weed-stem part as arbitrary unit-length b, swing a circular arc. Cut successive chords of length b, which if connected to the center form 60 equiangular triangles. Then successively bisect with any fixed length object longer than the chords of the sub-angles in question, to get 30, 15, 7 1/2, 3 3/4... etc. as wanted. Binary math is also great for graduating lineal scales, as for example, the traditional English inch. ( for Andrew) Sciagraphically, Bill Maddux > Hi Andrew, > > A closer and somewhat easier method would be to continue dividing the > string in half til it had been divided into 16 equal parts. Measure > out 15 of those parts and then at a right angle go 4 parts. As I > mentioned to Steve LeLievre this approximates the tangent of 15 > degrees with better than 1/2 percent accuracy. [ tan(15) roughly is > equal to 0.267949 and 4/15 is about equal to 0.27, arctan(4/15) > is about 14.93 degrees ]. Of course any object marked out in equal > divisions would do as well to mark out the 15 and 4. It just so > happens that my fist is right at 4 inches wide and the length > measured from my elbow to the end of my fist is right at 15 inches, > so, I'm lucky, I carry this measure about with me! > > How many of the rest of you are so lucky? How many find your fist > width to elbow-fist length ratio to be even closer to the tangent of > 15 degrees? > > Are there even better measurements taken from our self dimensions? > > Edley. > > You wrote: > > Steve Lelievre suggested using arc tan 1/4 to find 15 degrees "in > > emergency". > > > > A slightly better approximation is to go via > > sin(15) = 0.2588 and arcsin(0.25) = 14.4775 deg > > cf. tan(15) = 0.2679 and arctan(0.25) = 14.0362 degrees > > > > So, take your string and find a quarter of its length as before by two > > halvings. Start from the centre point and mark a point on a first line at > > the string's length away. From that point draw an arc (or mark a few > > points to indicate it) using the quarter string as radius. Go back to the > > centre and take the tangent to that arc as the second line. > > > > The angle subtended is arcsin 0.25 or about 14 degrees 29 minutes, close > > enough for practical purposes I would think. > > > > If you want to get still closer, then do it both ways and add the > > difference between them to the larger of the two angles to get 14 degrees > > 55 minutes. (And if you dislike estimating the addition to make rather > > than constructing it, then take two of the sine angles one after the other > > i.e. added together and go back one of the tan angles to subtract it.) > > > > Or of course you could just use the same process, with half the string > > length instead of a quarter, to go via arcsin 1/2 and obtain 30 degrees, > > then take the string and mark two points equidistant from the origin on > > the two lines 30 degrees apart, and use doubling the string to find the > > point midway between those two - hence bisecting the angle into two angles > > of exactly 15 degrees with no error at all. But we're getting close to > > terrestrial origami I fear ... > >
RE: Emergency Sundial
Hi Andrew, A closer and somewhat easier method would be to continue dividing the string in half til it had been divided into 16 equal parts. Measure out 15 of those parts and then at a right angle go 4 parts. As I mentioned to Steve LeLievre this approximates the tangent of 15 degrees with better than 1/2 percent accuracy. [ tan(15) roughly is equal to 0.267949 and 4/15 is about equal to 0.27, arctan(4/15) is about 14.93 degrees ]. Of course any object marked out in equal divisions would do as well to mark out the 15 and 4. It just so happens that my fist is right at 4 inches wide and the length measured from my elbow to the end of my fist is right at 15 inches, so, I'm lucky, I carry this measure about with me! How many of the rest of you are so lucky? How many find your fist width to elbow-fist length ratio to be even closer to the tangent of 15 degrees? Are there even better measurements taken from our self dimensions? Edley. You wrote: > Steve Lelievre suggested using arc tan 1/4 to find 15 degrees "in > emergency". > > A slightly better approximation is to go via > sin(15) = 0.2588 and arcsin(0.25) = 14.4775 deg > cf. tan(15) = 0.2679 and arctan(0.25) = 14.0362 degrees > > So, take your string and find a quarter of its length as before by two > halvings. Start from the centre point and mark a point on a first line at > the string's length away. From that point draw an arc (or mark a few > points to indicate it) using the quarter string as radius. Go back to the > centre and take the tangent to that arc as the second line. > > The angle subtended is arcsin 0.25 or about 14 degrees 29 minutes, close > enough for practical purposes I would think. > > If you want to get still closer, then do it both ways and add the > difference between them to the larger of the two angles to get 14 degrees > 55 minutes. (And if you dislike estimating the addition to make rather > than constructing it, then take two of the sine angles one after the other > i.e. added together and go back one of the tan angles to subtract it.) > > Or of course you could just use the same process, with half the string > length instead of a quarter, to go via arcsin 1/2 and obtain 30 degrees, > then take the string and mark two points equidistant from the origin on > the two lines 30 degrees apart, and use doubling the string to find the > point midway between those two - hence bisecting the angle into two angles > of exactly 15 degrees with no error at all. But we're getting close to > terrestrial origami I fear ...
RE: Emergency Sundial
Steve Lelievre suggested using arc tan 1/4 to find 15 degrees "in emergency". A slightly better approximation is to go via sin(15) = 0.2588 and arcsin(0.25) = 14.4775 deg cf. tan(15) = 0.2679 and arctan(0.25) = 14.0362 degrees So, take your string and find a quarter of its length as before by two halvings. Start from the centre point and mark a point on a first line at the string's length away. From that point draw an arc (or mark a few points to indicate it) using the quarter string as radius. Go back to the centre and take the tangent to that arc as the second line. The angle subtended is arcsin 0.25 or about 14 degrees 29 minutes, close enough for practical purposes I would think. If you want to get still closer, then do it both ways and add the difference between them to the larger of the two angles to get 14 degrees 55 minutes. (And if you dislike estimating the addition to make rather than constructing it, then take two of the sine angles one after the other i.e. added together and go back one of the tan angles to subtract it.) Or of course you could just use the same process, with half the string length instead of a quarter, to go via arcsin 1/2 and obtain 30 degrees, then take the string and mark two points equidistant from the origin on the two lines 30 degrees apart, and use doubling the string to find the point midway between those two - hence bisecting the angle into two angles of exactly 15 degrees with no error at all. But we're getting close to terrestrial origami I fear ... Regards Andrew James 51 04 W 01 18 N
Re: Emergency Sundial
Hi Steve, You Wrote: > Edley, John, > > Another easy way and reliable to approximate 15 degrees - > > Use the fact that tan(15) is very close to 1/4 (only 7% error) > > Take a length of string, bring the ends together and pull out the loop > until tight. This gives you the half length of the string. Repeat the > process with the doubled string, to get the quarter length. Mark off where > the quarters are. > > Now place one end of the string at the centre of your intended wedge and > stretch it in the direction required for one side of the wedge. Make a > mark on the surface at the end of the piece of string. Judging by eye what > is the perpendicular to this, now position the string to be at right > angles to your first line and make your second mark at the quarter length. > > Et voila, you have a triangle with the two long sides separated by 15 > degrees (closer to 14 actually, but still a good enough approximation for > a campsite sundial) > > Steve That is a good idea! , so I started looking for easy to remember ratios that were close to the tangent of 15 degrees and discovered that 4/15 is accurate to better than 1/2 percent and might be easy to remember because of the 15 in the denominator. Thanks! Edley.
RE: Emergency Sundial
Dear John, Here is another, easier Emergency Sundial. Hope this satisfies as well as the last one. Our research has finally paid off. As you may recall our last quick fix sundial required shoestrings and writing empliments. This newly developed method can now supply an emergency sundial with only a properly facing surface, a few rocks or other lumps and proper use of your body. No longer must you suffer countless hours without a sundial! Find a surface that is parallel to the equator, tilt a table, find a leaning rock, pile up some sand, whatever. For many people, including myself, the angle subtended by a tight fist, looking from the elbow toward it, is very close to 15 degrees. So, stick the elbow down where the gnomon will be and mark off 15 degree marks with rocks, using your fist as a guide, rotating your arm about the spot your elbow is on. ( The back of the fist down is the classical method ). These mark the hour lines. Pop your arm up vertical to the surface and there you are, an equatorial sundial. With this basic dial you can tell pretty close how long it has been since you built it, but still have a large error in where 12 o'clock might be. This error will be composed of meridianal error ( wrong guess where north is ), Longitudinal error ( not right on prime meridian) and the equation of time. Using some other time instrument you can calibrate out this main error by discovering, for existence that 12 on your sundial is really 5:15 PM or some such. A large rock or twig might be used in place of your arm sticking up if you have other things to do than just watch the sundial all the time. You can also make a shadow plane dial instead, using a similar approach, but starting with your elbow away from the sun and telling time by the shadow of the rock that hits your elbow. If further research improves on these latest findings we will happily report them. :-) Edley McKnight [43.126N 123.357W]
