Geometric Constructions Involving Phi

This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.


(Previous Section: The Golden Ratio)

Master Fibonacci
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Rectangles with sides the lengths of Fibonacci numbers maintain a constant ratio (dividing the long side by the short side) no matter how large the rectangle is. Rectangles made with the Golden Ratio are called “Golden Rectangles” because many people believe them to have the most aesthetically pleasing proportions. A rectangle with sides of 8 × 5, for example, has a ratio of 1.6.

Golden Rectangle

Golden RectangleGolden Rectangles can be “cloned” by partitioning a square of side length equal to the length of the short side of the rectangle (Seewald), as shown previously in the image of The Golden Spiral.

Golden Triangle

Golden Triangle

The Golden Ratio
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Another geometric variation is the golden triangle, also known as the sublime triangle, which is an isosceles triangle in which the ratio of a side to the base is Phi. In a golden triangle, a base angle of 72° can be bisected to create two additional, self-similar triangles (the internal angles and the ratios between the sides are identical no matter the length of the sides).

Golden Gnomon

Golden Gnomon

The result is an acute isosceles triangle of the same dimensions as the original, and an obtuse isosceles triangle in which the length of the equal (shorter) sides to the length of the third side creates a ratio that is the reciprocal of the Golden Ratio. This obtuse isosceles triangle is known as a “gnomon.” Pentagons, pentagrams, and decagons can be generated this way, as well (Seewald).

Golden Spiral

Golden triangles

Scientists and mathematicians have studied these logarithmic spirals (named for the way the radius of the spiral grows when moving around it in a clockwise direction) because they recognize the patterns in objects of biology and nature, from animals and plants to vast galaxies (Seewald). They seem evident in the harmony and proportion of art and architecture, as well. The golden spiral has been utilized in the design of some modern art and architecture, but whether ancient artists and architects (such as Phidias, architect of the Parthenon) deliberately incorporated the Golden Ratio is still being debated.






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