In Golden Rectangle & Golden Ratio page, I
mentioned that: Return to my Mathematics pages
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© Copyright 1999, Jim Loy
In Golden Rectangle & Golden Ratio page, I
mentioned that:
The golden ratio shows up (as an exact fit) in mathematics in many unexpected places. The ratio shows up everywhere in the pentagram (five pointed star) and its circumscribed pentagon (shown on the left). a/b=(a+b)/a=(a+b+a)/(a+b)=phi (the golden ratio). Here we also see the two kinds (acute and obtuse) of golden triangles (I've painted two of them green). They are any of the isosceles triangles in this diagram. These are triangles which exhibit the golden ratio.
There are several ways to construct a
pentagon (using compasses and straightedge) in a circle. They normally start
with: (1) draw a diameter, (2) draw a diameter perpendicular to the first
diameter, (3) bisect one of the radii (this step produces the
Golden Ratio in several forms), (4) draw a line from
the middle of this radius to an endpoint of the other diameter. After that
step, there are several ways to continue. The way I do it is to (5) draw an arc
(with its center at the midpoint of the radius) from this last endpoint of a
diameter to the other diameter. This produces the length marked with two
vertical marks in the diagram. If the radius is 1 then that length is 1/phi or
1-phi. Alternatively, if this length is 1, then the radius is phi. So that
length is the side of a decagon. In the diagram, (6) I drew two sides of a
decagon. (7) Connect two vertices of the decagon (skipping a vertex), and you
have one side of a pentagon. (8) Draw the other four sides. I discovered this
method on my own. But, I suspect that it is one of the standard methods.
Another method was discovered by Richmond
in 1893. The first steps are as above. Then (5) bisect the angle that is at the
bisection of the radius. (6) Where this angle bisector crosses the other
diameter, draw a line parallel to the bisected radius. Now (7) draw one side of
the pentagon, from where that parallel line crosses the circle to the endpoint
of the diameter that is perpendicular to the bisected radius, as shown in the
diagram. (8) Draw the other four sides.
Tie an overhand knot in a strip of paper, being
careful to tighten it slowly, and flatten it as it gets tight. Many books claim
that this produces a regular pentagon. See the diagram. It is fairly obvious
that this is at least close to being a regular pentagon, as each side is
slightly longer than the width of the paper strip. And the angles are roughly
the same. But it is not obvious that it is perfect, rather than close.
By symmetry (the knot is the same from either side of the paper), or in other ways, we can show that angles E (of the pentagon) and B are congruent. Also angles C and D are congruent. It is not difficult to show that angle A is congruent to B (by the way angle A is folded, which makes the exterior angles congruent).
The diagram on the right shows the pentagon with the
diagonals. Each of the diagonals is parallel to a side, because the strip of
paper has parallel edges. We can show that all kinds of angles are congruent,
by symmetry and by parallel lines. In particular, we can show that the blue
triangle is isosceles. From there, we can show that the two red triangles are
congruent. And from there, we can show that at each vertex of the pentagon, all
three acute angles are congruent. From that we show that the five vertices of
the pentagon are congruent. And we can easily show, by congruent triangles,
that the five sides of the pentagon are congruent.
So the pentagon is indeed regular. I did not show each step of the proof. Some of the steps may take some thought.
Also see Home Plate, which is an irregular pentagon.
Addendum #1:
It is fairly easy to figure out the angles of a
pentagon. Draw a central angle, as in the diagram at the left. This central
angle is 1/5 of 360 degrees, or 72 degrees. The sum of the angles of a triangle
is 180 degrees. And so the three angles of the isosceles triangle are 72
degrees, 54 degrees and 54 degrees. That makes each angle of the pentagon 108
degrees. The sum of the five angles is 108x5=540 degrees.
This method works for any regular
polygon. But we can deduce a general formula for all polygons. At the right, we
see a pentagon with three inscribed triangles. An n-gon (polygon with n sides)
would contain n-2 triangles in a similar way. The sum of the angles of each of
these triangles is 180 degrees. Each angle of the n-gon is the sum of angles of
these triangles, and there are no angles of the triangles that are not part of
angles of the n-gon. And so the sum of the angles of a regular n-gon is
180(n-2) degrees. This formula works for irregular polygons (convex or
not); but dividing a nonconvex polygon (with some angles greater than 180
degrees) into triangles may not be quite as easy. If the n-gon is regular, then
each angle is 180(n-2)/n degrees. You can check that these two formula work for
triangles and squares and pentagons.
The diagonals split each angle of a regular pentagon
into three identical angles. In this diagram, we see that quadrilateral AEDF is
a rhombus (parallel sides and consecutive sides are congruent (equal)). And so,
the diagonal AD bisects the angle EAF. And, by symmetry, all three angles are
congruent (36 degrees).
Using the same diagram, I will now show that the golden ratio (See Golden Rectangle and Golden Ratio) shows up in a regular pentagon. Let's say that each side of our pentagon is 1. AF=1, because it is the side of our rhombus. Triangles ADF and BCF are similar. So, x/1=1/y. But y = x+1. So x = 1/(x+1). x(x+1) = 1. So x^2+x-1=0 [where x^2 is x squared]. Solving for x, using the Quadratic Formula, x = (-1 +/- sqr(5) )/2 [where sqr() is the positive square root function, and +/- means plus or minus]. One of these lengths is positive: x = (-1+sqr(5))/2. And y = (1+sqr(5))/2. And that is the golden ratio.
Addendum #2:
The great book Amazing Origami by Kunihiko
Kasahara describes the following method as "the American method" for making a
pentagon by paper folding. We start with a square piece of paper (which can be
created from any odd shaped piece of paper, by folding). We fold it diagonally
in half (the horizontal line in the diagram), then crease one side in half at
point A. We then fold point B over to point A, forming line CD. The idea is
that angle ACD (and its image BCD) is about 72 degrees. It is actually about
71.56 degrees, almost half a degree off. This is close enough for most
purposes. The book goes on to fold and cut a pentagon with center at C. I won't
do that here, as the pentagon is not exact.
The same book gives this method (on the left) as "the
Japanese method" for folding a pentagon. We fold the square in half, as above.
Then we crease the long diagonal of the square at C. We crease the upper right
side twice, in half and then in fourth at A. We fold line BC over to point A,
producing line CD (point B will coincide with point A). Then we bisect angle
CDE, producing line CF. And angle DCF (and its image ECF) is supposed to be
near 72 degrees. Actually it is 72.11 degrees, which is closer than the
American method.
The same book produces a "regular pentagon template" by folding a regular hexagon, cutting along one radius, and overlapping one sector over its neighbor. This produces a short pentagonal pyramid. We can use a similar method to fold any regular polygon that we want, theoretically.
The book says, "Many people in the past have tried to fold a regular pentagon." Apparently we are not going to do it by making a pyramid, or by tying an overhand knot. I'm sure that people have actually succeeded, as it is fairly easy to fold a golden rectangle. And, it has actually been proved that anything that can be done with compasses and straightedge can be done by folding paper, and more. I'm trying to discover a method. If our paper were cut into a circle, we could use one of the construction methods shown above (which used compasses and straightedge).
OK, here is a method that seems to work, based upon our first
construction (which produced a decagon). We crease our paper into four smaller
squares, as shown. We crease the right center horizontal line in half, and fold
the diagonal AB. We bisect angle BAC to find point D. CD is the length of our
decagon in our first construction. We fold the point C onto point D producing
the perpendicular bisector EF of CD. We fold point B onto line EF to find point
E. EG is now one side of our pentagon. From there we can reproduce every side
of the pentagon. All vertices will be within our original square.
In practice, this may not be any more accurate as our American and Japanese approximations, as we don't get very precise angles when we fold paper. But in theory, this method is exact.
That method suggests an improvement to our
construction of a pentagon. See the diagram on the right. We start as before,
constructing a golden ratio. Then we bisect the segment which I have marked
with the little double vertical lines. Then we draw the perpendicular bisector
from there, and we can draw one side of our pentagon (in red) without bothering
with a decagon. Then we can draw the other four sides.
In Vietnam, I sometimes painted white stars on green vehicles, as the officers found out that I was an artist. Nobody could ever find a template. Not remembering how to construct a pentagon, I just experimented until I had a roughly regular pentagon, and then drew my star. Once I just painted the star freehand, starting slightly small, and adjusting it over and over until it looked regular. The person watching me do this was amazed that I could do that.
Addendum #3:
Here is a nice construction of a pentagon, as shown in
Mathographics by Robert Dixon. It creates the same lengths as the
previous construction, but oriented in different directions.
There are many ways to construct a regular
pentagon. On the right we see the construction of a golden triangle, from which
we can make pentagons, decagons, and pentagrams in many ways. We draw the
square ABCD, bisect AB at E, make EF=ED, then find G by making AF=AG and BA=BG.
Then we draw FG and AF. Triangle BFG is also a golden triangle. And ABG is the
obtuse kind of golden triangle.
Here is another construction that I received by email. It is similar
to most of the above constructions.