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The Golden Rectangle is all around us -- whether we're aware of it
or not. After learning more about this curious shape, you might spot
it in buildings, hurricanes, and art. But what exactly *is* the
Golden Rectangle? Why is it so important? What can it teach us about
math? Let's find out:

Many people believe that the Golden Rectangle is the most visually pleasing shape in the entire universe. Considering how many beautiful paintings and buildings contain this shape, this argument is quite convincing. For example, the Golden Rectangle is present in the Mona Lisa, the Parthenon of Ancient Greece, and many other works of art.

Let's build our own Golden Rectangle. Start with a square, then draw a line from the corner of one side to the midpoint of another adjacent side. We then use this line as the radius of a circle to draw an arc down to the level of the side we found the midpoint of. Then add on a rectangle on the side of our square using that point we found with the arc and voilà! We have ourselves a Golden Rectangle.

Now let's have some fun with the Golden Rectangle. Turn one side
into a perfect square, and you'll have another Golden Rectangle on
the other side. Turn one side of your second rectangle into a
square, and you'll be left with a *third* Golden Rectangle. You
can continue this pattern forever (although you might run out of
space eventually).

If we trace an arc through each perfect square within our Golden Rectangle, we're left with a very distinct spiral. This is the exact same spiral you'll find in seashells, flowers, and even spinning galaxies. Take a trip into your backyard, and you'll find plenty of examples of this pattern -- also known as the Fibonacci Sequence.

Now take a look at a picture of the Greek Parthenon from the front. Give the short side of this architectural rectangle a value of "1" and solve the equation:

$\frac{1}{2}+\frac{\sqrt{5}}{2}$

You should be left with:

$\frac{1}{2}+\frac{2.236068}{2}=1.618034$

Another way to calculate the golden ratio with a continued fraction as follows:

$\varphi =1+\frac{1}{(1+\frac{1}{(1+\frac{1}{(1+\frac{1}{(1+\cdots )})})})}$

Common Core: High School - Geometry Flashcards

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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