By Nami Youn

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**Some Golden Geometry**

**1. Golden
Rectangle**

A Golden Rectangle is a rectangle with proportions that are two consecutive
numbers from the Fibonacci sequence.

The Golden Rectangle has been said to be one of the most visually satisfying
of all

geometric forms. We can find many examples in art masterpieces
such as in edifices of ancient Greece.

GSP file

**2. Golden
Triangle**

If we rotate the shorter side through the base angle until it touches
one of the legs, and then, from the endpoint, we draw a segment down to
the opposite base vertex, the original isosceles triangle is split into
two golden triangles. Aslo, we can find that the ratio of the area of the
taller triangle to that of the smaller triangle is also 1.618…. (=Phi)

If the golden rectangle is split into two triangles, they are called golden
triangles suing the Pythagorean theorem, we can find the hypotenuse of
the triangle.

**3. Golden
Spiral**

The Golden Spiral above is created by making adjacent squares of Fibonacci
dimensions and is based on the pattern of squares that can be constructed
with the golden rectangle.

If you take one point, and then a second point one-quarter of a turn
away from it, the second point is Phi times farther from the center than
the first point. The spiral increases by a factor of Phi.

This shape is found in many shells, particularly the nautilus.

**4. Penrose
Tililngs**

The British physicist and mathematician, Roger Penrose, has developed
an aperiodic tiling which incorporates the golden section. The tiling is
comprised of two rhombi, one with angles of 36 and 144 degrees (figure
A, which is two Golden Triangles, base to base) and one with angles of
72 and 108 degrees (figure B).

When a plane is tiled according to Penrose's directions, the ratio of
tile A to tile B is the Golden Ratio.

In addition to the unusual symmetry, Penrose tilings reveal a pattern
of overlapping decagons. Each tile within the pattern is contained within
one of two types of decagons, and the ratio of the decagon populations
is, of course, the ratio of the Golden Mean.

**5. Pentagon
and Pentagram**

We can see there are lots of lines divided in the golden ratio. Such
lines appear in the pentagon and the relationship between its sides and
the diagonals.

We can get an approximate pentagon and pentagram using the Fibonacci
numbers as lengths of lines. In above figure, there are the Fibonacci numbers;
2, 3, 5, 8. The ratio of these three pairs of consecutive Fibonacci numbers
is roughly equal to the golden ratio.

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