Figure 4 LMa Step two you will find the midpoint between C and D and form point

# Figure 4 lma step two you will find the midpoint

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Figure 4. [LMa] Step two, you will find the midpoint between C and D and form point E. Figure 5. [LMa]
Step three, requires you to extend the line from point C (on D, C segment) out so you can complete the rectangle. Place your point (stabby side) of the compass on point E and your pencil side on point B. Draw an arc from point B until you intersect the line extension from point C, this intersection forms point F. Figure 6. [LMa] Step four requires you to draw a horizontal line extending up towards point B starting at point F. Figure 7. [LMa] Step five is the final step in completing the triangle as you will extend out your line from point B (on the AB segment) to intersect with the horizontal line that started at point F. This intersection will form point G. Figure 8.
[LMa] Points A, G, F and D form a golden rectangle. If you were to measure the length and the width of the rectangle and divide the length of segment A, G by the width of segment A, D you should come up with a ratio close to the golden ratio. Keep in mind that the size of the rectangle does not matter, what does matter is the ratio between the two sides. The golden rectangle also possesses a rather unique property. The golden rectangle is comprised of smaller golden rectangles. Just like Russian nesting dolls, you open one and there is another inside of it, and another, and another. If you keep dividing the golden rectangle using the same method we used above you will find several more golden rectangles inside of your first rectangle. This particular rectangle has six golden rectangles inside of it. Once you have found all of the golden rectangles inside of the first something else appears. Figure 9.
[LMa] If you look closely at the golden rectangle that has been reduced to many different golden rectangles something else appears. By drawing a quarter of a circle inside of each square you will get a golden spiral. The spiral approximates the logarithmic spiral, occurs in nature in various forms, such as the nautilus sea shell [EBB]. The golden spiral grows from the center as it expands outward, however, the growth factor is equal to the golden ratio (1.618…). The golden spiral is also found in nature and is directly related to the golden rectangle. Figure 10. [LMa] As we already know the Fibonacci numbers and golden rectangle are related. There is also a Fibonacci spiral. To create the Fibonacci spiral with to small squares that are equal in size. Then we will draw a square on top of the first two and that will be square two. Square two is equal in size to the addition of the two size one squares. Square 3 will cover the area of all five
previous units. This continues on with each new square having a side that is the sum of the last two squares side (See figure eleven below), notice that the two square you are adding together are all Fibonacci numbers. If you look at Figure twelve you will notice that a quarter circle was drawn in each of the square to form a spiral just like the spiral very much like the golden spiral that was demonstrated above.

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• Winter '15
• Andrew Martino
• The Da Vinci Code, Golden ratio, golden rectangle

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