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>> prehistoric village of stone houses with connecte 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 primitive compass, > as a pre-school rainy-day amusement. > > Yours truly, Dave Webster, Kentville > > > ----- Original Message ----- From: "Stephen Shaw" <srshaw@Dal.Ca> > To: <naturens@chebucto.ns.ca> > Sent: Friday, August 22, 2014 3:48 PM > Subject: RE: [NatureNS] Neolithic stone rings - encore. > > >> Hi Eleanor, >> Many years ago I recall reading that the neolithic denizens of >> Skara Brae used to cache the bones of their forebears in an >> ossuary, on stone ledges somewhere in their dwellings. One of the >> memorable findings was that experts analyzed these bones as to time >> of death, revealing that practically nobody at Skara Brae had lived >> beyond the age of 30, apparently testifying to the hard life there. >> I couldn't find any mention of this latterly, searching a couple >> of recent sources e.g. Wikipedia. Did you come across any such >> information when you were there: is it still believed that they had >> nearly all died by an age that we would consider a very young? I'm >> not sure that this is reflected in other early societies -- not the >> contemporary Egyptians, I think, who however were presumably much >> better fed. >> >> Hi Dave: Maybe this flogging a dead horse, but I think you ha