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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 have it
> backwards. In fact I suggested that the neolithic farmers could well
> have 'solved' what would later be called the "inscribed regular hexagon
> conjecture" by a simple practical-knowledge construction procedure of the
> sort that you advocate, without any foundation in theoretical geometry
> that would not arrive until much later, usually associated with the
> Greeks. At the same time, it's not clear why stone circle-makers would
> have been sequentially pegging out the boundary of a large circle by trial
> and error to make any such discovery (if that's how they did it), if they
> didn't have some informal geometrical insight in the first place. But I
> doubt that they could get that simply from looking at snowflakes, without
> a magnifying glass and ruler, though we did see plenty of snowflakes when
> we lived in Scotland.
> Steve
> ________________________________________
> From: naturens-owner@chebucto.ns.ca [naturens-owner@chebucto.ns.ca] on
> behalf of Eleanor Lindsay [kelindsay135@gmail.com]
> Sent: Tuesday, August 19, 2014 11:45 AM
> To: naturens@chebucto.ns.ca
> Subject: Re: [NatureNS] Neolithic stone rings etd.
>
> On a completely different aspect of this topic, I spent time in Orkney
> in the '70s during the early displays of the first discovery of ancient
> dwellings which became exposed at Skara Brae after a major storm tore
> masses of turf off the nearby shoreline, uncovering an entire
> prehistoric village of stone houses with connected walkways. It was not
> hard to understand why this site had been chosen as the nearby cliffs
> around the bay consisted of a type of rock that, to this day still
> appears to shelve off in long slim slabs; these slabs were evident in
> every house and what, for me, remains so memorable was their use for
> everyday needs which were identical to ours today - small horizontal
> slabs inserted at various levels in the walls to provide shelves and,
> most striking of all, rectangular bed frames on the ground consisting of
> narrow strips of the stone slabs for the sides, tall upright slabs for
> the head and slightly smaller ones for the foot of the bed - exactly how
> we still do it today!! And what I saw at that time is only a mere
> fraction of what has been discovered since then...
> The other site there that made a deep impression was the standing stones
> circle at the Moor of Brodgar; seeing it there in its (at least at that
> time) splendidly isolated setting looking no different than the day it
> was completed made a very powerful impression that left poor beleaguered
> Stonehenge, with all the traffic whizzing by, way behind.
>
> Orkney is a totally fascinating place to visit, not so much for its
> scenery, but for its spectacularly rich endowment of an amazing variety
> of prehistoric to second world war history.
>
> Eleanor Lindsay
>
>
>
> On 18/08/2014 9:07 PM, David & Alison Webster wrote:
>> Hi Steve & All,
>> I think you are confusing theoretical logic with practical know how
>> and these northern folk had an impressive amount of know how.
>>
>> For example, the walls of the Knop of Howar (occupied 3700
>> BC-1800 BC) are still standing. How many of our structures will still
>> be around 4000 years from now ? They lived on islands so likely knew
>> how to build boats that could actually be steered ( able to go out,
>> turn around and come back) and which cost less than a king's ransom.
>>
>> You don't need to be a Greek Philoso