Cuyler,  It's a good thing geometry was invented by the Greeks, because
it's all Greek to me.  I did, however, recognize the only bit of geometry I
ever learned, namely, the square of the hypotenuse of a right angle
triangle is equal to the sum of the squares of the other 2 sides.  I think
Pythygoras must have been unemployed for a large part of his life to have
time to think this stuff up, and all just to torment poor high school
students.

However, I asked 16 year old son and he knew all about the Golden
Rectangle, however, he sounded as though it was not his favorite part of
Algebra II.  Ruth





At 10:27 AM -0800 2/1/05, Cuyler Page wrote:
OOPS !

Not meaning to tarnish your reputation and clever e-drawing in front of
friends, but, hey jc., your Golden Rectangle is a little tarnished.
Sounds more like a Bronze Rectangle, close, but not quite.

Golden Rectangle arrived at by geometry:

1) draw a square

2) from the mid-point of one side, let's call it the bottom of the
square, swing an arc using that point as centre and an opposite (top)
corner of the square as the end of the radius.   Swing the arc to the base
line (the bottom line with the centre point) extended (right or left)beyond
the square to allow the arc to intersect with it.

3) from the intersection of the arc and the extended bottom side of the
square, erect a line at right angles (vertical line).

4) extend the top line of the square to intersect with the new vertical.

5) the new large rectangle, including the square and the small new
rectangle just created, is a Golden Rectangle.

Golden Ratio Proportions are approximately 1 : 1.618...  or 1 : 0.618... ,
reciprocal and an irrational number (Ralph will like that).
You can prove it with the resulted rectangle constructed as above.   AxA +
BxB = CxC  (my e-mail doesn't have the symbol for "square", but that is
nice because even those without Rich Text, like Ruth, can read it, "In a
right triangle with short sides a and b, and long side c,  a-square plus
b-square equals c-square.")

6) if you use the long side of the new large rectangle as the length of the
side of a square to add beside that rectangle, then the new very large
rectangle will be another Golden Rectangle, etc., etc., etc.   Connecting
similar points on all the new-to-infinity rectangles will describe a
logarithmic curve, but that is another story.

jc., you describe beautifully the benefit of looking for the proportion
systems used by the original designers, not always Golden, as a means to
discovering missing parts or creating new work that blends harmoniously
with old.   Have experienced this frequently with delightful success while
working with heritage home owners in revitalizing their houses, finding old
clues to everything from missing walls to the original position of picture
or plate rails and the height of wainscoting.

The Golden Ratio is a mathematical or geometrical relationship that
describes how many living things grow their form in nature, has its own
wonderful curiosity as a piece of mathematical gymnastics, describes sonic
and musical relationships, and seems to be hard-wired into our
visual perception system, so it has a powerful intrigue from many points of
view.   Wordsworth, a serious geometer, constructed poems based on its
proportions, and the story goes on and on.   It is not universal, but it
sure can be interesting.

cp in proportional bc
(Kamloops, not Golden. That town is near the Rockies)



--
Ruth Barton
[log in to unmask]
Dummerston, VT

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