Golden Ratio - The Math Book

Fra Luca Bartolomeo de Pacioli (1445 - 1517) - In 1509, Italian mathematician Luca Pacioli, a close friend of Leonardo da Vinci, published Divina Proportione, a treatise on a number that is now widely known as the "Golden Ratio." This ratio, symbolized by , appears with amazing frequency in mathematics and nature. We can understand the proportion most easily by dividing a line into two segments so that the ratio of the whole segment to the longest part is the same as the ratio of the longer part to the shorter part, or (a+b)/b = b/a = 1.61803 ...

If the lengths of the sides of a rectangle are in the golden ratio, then the rectangle is a "golden rectangle." It's possible to divide a golden rectangle into a square and a golden rectangle. Next, we can cut the smaller golden rectangle into a smaller square and golden rectangle. We may continue this process indefinitely, producing smaller and smaller golden rectangles.

If we draw a diagonal from the top right of the original rectangle to the bottom left, then from the bottom right of the baby (that is, the next smaller) golden rectangle to the top left, the intersection point shows the point to which all the baby golden rectangles converge. Moreover, the lengths of the diagonals are in golden ratio to each other. The point to which all the golden rectangles converge is sometimes called the "Eye of God."

The golden rectangle is the only rectangle from which a square can be cut so that the remaining rectangle will always be similar to the original rectangle. If we connect the vertices in the diagram, we approximate a logarithmic spiral that "envelops" the Eye of God. Logarithmic spirals are everywhere - seashells, animal horns, the cochlea of the ear - anywhere that nature needs to fill space economically and regularly. A spiral is strong and uses a minimum of materials. While expanding, it alters its size but never its shape.