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>> Hi Dave: Maybe this flogging a dead horse, but I think you ha Content preview: Thanks Stephen, We have drawn hexagons inside of circles, bisected angles, drawn equilateral triangles, perpendiculars, kids enjoy these investigations and can't believe such simple tools (two pencils and a paper clip) can show the relationships among angles, segments, circles and be able to draw such "perfect" shapes! [...] Content analysis details: (-1.9 points, 5.0 required) pts rule name description ---- ---------------------- -------------------------------------------------- -1.9 BAYES_00 BODY: Bayes spam probability is 0 to 1% [score: 0.0000] Thanks Stephen, We have drawn hexagons inside of circles, bisected angles, drawn equilateral triangles, perpendiculars, kids enjoy these investigations and can't believe such simple tools (two pencils and a paper clip) can show the relationships among angles, segments, circles and be able to draw such "perfect" shapes! George Quoting Stephen Shaw <srshaw@dal.ca>: > Hi George, Dave, others: > I haven't seen the National Geographic article Dave cited: did they > use a straightedge to incise the lines? The idea raised by both of > you is that interesting and even useful constructions could have > been discovered accidentally, operationally by 'pre-geometrical' > people 'doodling' casually with implements like primitive rulers and > compasses. Obviously this is impossible to deny directly, so the > follow-up question is whether there is any evidence that any early > 'pre-geometrical' cultures actually might have done this, and > whether any such discoveries were passed on, to become part of the > local culture. I don't remember ever seeing evidence of this and > couldn't find any in a cursory search. > > All the remarkable, artistic palaeolithic inscriptions on cave walls > seem to have been inscribed freehand, and this seems true also in > the later spiral megalithic incisions on rocks. In Lascaux type > caves, you don't find straight-ish lines like spears drawn with a > straightedge and roundish images constructed in a way that suggests > a compass was used. By contrast, in some later Egyptian > inscriptions (Book of the Dead, papyrus versions starting 1550 BC) > it is difficult to see how vertical lines separating hieroglyphs > that straight could have been drawn without a straightedge for > guidance, but that seems to be about the first indication of this. > Round things like images of the sun still didn't seem to be drawn > with a compass in a few images that I looked at, but perhaps someone > has better information. It would be surprising if Egyptian temple > architects didn't have compasses as well as rulers. > > George, if you don't already know it, I came across > 'Compass-and-straightedge_construction' on-line, which gives a > repeating animation of constructing a hexagon inside a circle that > might supplement your students' efforts. It also discusses/solves > the classical problems of trisecting a line segment and trisecting > an angle. The related link to the 'Neusis construction' used widely > by the Greeks, is interesting but rather opaque as to particular > usage. > Steve > ________________________________________ > From: naturens-owner@chebucto.ns.ca [naturens-owner@chebucto.ns.ca] > on behalf of George E. Forsyth [g4syth@nspes.ca] > Sent: Friday, August 29, 2014 12:49 AM > To: naturens@chebucto.ns.ca > Subject: Re: [NatureNS] Neolithic stone rings - encore. > > Hi, > > I teach this same process in grade seven math! We use a primitive > compass, a paper clip and two pencils. We also look at the use of this > symbol in historic terms, a hex. The students all associate "hex" with > a bad spell used by a witch or sorcerer, but soon find that it was > used in northern European history as sign or symbol of good luck and > fortune. The Pennsylvania "Dutch" use it as a protection on their > barns, as a bearer of protection. > > Interesting wondering how so many discoveries could have been made by > "primitive" people without the computers and communication of our world. > > Cheers, George Forsyth > > > > Quoting David & Alison Webster <dwebster@glinx.com>: > >> Hi Steve & All, >> We appear to be in essential agreement on this. Practical >> geometric insights would likely all have come by accident in the >> course of small scale and perhaps perishable decorative art >> exercises; and once recognized and learned perhaps incorporated as a >> part of practical culture long before any attempt theoretical >> analysis. The latter requires leisure. >> >> That same article provides a good example of this process on page >> 33. where parallel evenly spaced straight lines engraved in stone >> cross a sequence of other straight lines to produce a double row of, >> what we would call isosceles triangles. And then secondary patterns >> are inscribed within these triangles; some messy and some >> attractive. The two long sides of one of these original triangles is >> neatly bisected and the points joined to form a triangle of >> identical shape but half as high. Then the base of the original >> triangle is bisected and the points joined to form a total of four >> identical triangles all within the original triangle that was twice >> as high. >> If that rather attractive pattern were to become widely used >> then someone would eventually notice that when the height of a >> figure like this is doubled the area will be four times as great. >> And if this became understood then someone might notice that the >> same applies to squares and rectangles. And those experienced in >> dividing fields for various purposes would say "Well duh". >> Decorative arts would also likely have revealed the circle >> hexagon connection. If drawing careful circles using a forked stick >> with one side sharpened and the other charred >> had come into common usage at some point then someone would >> eventually have noticed that by placing the pointed arm anywhere on >> a circle the charred end would pass through the center. And someone >> would have noticed that this can be repeated 5 more times to yield >> an attractive flower-like pattern with six-fold symmetry. Drop the >> arcs that extend beyond the original circle, join the adjacent >> points of the 6 petals and you have a hexagon just fitting a circle. >> Perhaps more than one person on naturens will recall attempting >> to draw this figure exactly, using an even more pr