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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)
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---- ---------------------- --------------------------------------------------
-1.9 BAYES_00 BODY: Bayes spam probability is 0 to 1%
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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