Leonardo da Vinci

The Golden Rectangle is proposed to be the most aesthetically pleasing of all possible rectangles. For this reason, it and the Golden Ratio have been used extensively in art and architecture for thousands of years. The most prominent and well known uses of the Golden Rectangle in art were created by the great Italian artist, inventor, and mathematician, Leonardo da Vinci.

 Move your mouse over the image to see the Golden Rectangles. (scan by Mark Harden)
"The Mona Lisa," undisputably Leonardo's most famous painting, is full of Golden Rectangles. If you draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head, you will have a Golden Rectangle. Then, if you draw squares inside this Golden Rectangle, as was shown in the Golden Spiral in Action page, you will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. It is believed that Leonardo, as a mathematician, purposefully made this painting line up with Golden Rectangles in this fashion in order to further the incorporation of mathematics into art. In the spirit of mathematics in art, it is also worth mentioning that the overall shape of the woman is a triangle with her arms as the base and her head as the tip. This is meant to draw attention to the face of the woman in the portrait.

 Move your mouse over the image to see the Golden Rectangles. (scan by Mark Harden)
Leonardo's famous study of the proportions of man, "The Vetruvian Man" (The Man in Action), is also full of Golden Rectangles. Unlike the Mona Lisa, where all the lines of the Golden Rectangle are assumed by the mathematician, in "The Vetruvian Man", many of the lines of the rectangles are actually drawn into the image, at least in part. There are three distinct sets of Golden Rectangles in this painting: one set for the head area, one for the torso, and one for the legs.

To find the first set of rectangles, the one for the head, draw a rectangle whose base goes along the man's neck from shoulder to shoulder (stop at the shoulder lines provided by Leonardo). The top of the rectangle should meet the top of the man's head. This creates the first Golden Rectangle. Once you have that, inscribe a square in the left side of the rectangle, creating a smaller Golden Rectangle on the right side of the man's head. Then do the same with the right side of the original rectangle, creating a long, thin rectangle that runs vertically through the center of the man's head. Note that the smaller Golden Rectangles intersect with the focal points of the head: the eyes.

The second set of rectangles is found in a similar way. This time, all the lines are provided by Leonardo in the painting. Draw a rectangle which runs from elbow to elbow and from neck to waist. This creates a Golden Rectangle. Then, in a similar fashion to the first set, inscribe a square in each side of the rectangle, creating two more Golden Rectangles. Note this time that these new smaller Golden Rectangles intersect with the innermost portion of the man's torso.

For the third set, draw a rectangle whose lower two vertices are at the places where the man's outermost toes touch the outlying circle. The rectangle should extend vertically to the man's waist. This creates yet another Golden Rectangle. Now inscribe squares in the sides of the rectangle as you did before. This time it seems that the two smaller rectangles come to where the man's legs would be if they weren't turned out in that fashion.

Random Fibonacci number from the 1st through the 200th:

All contents, unless otherwise specified, are © 1999 by Matt Anderson, Jeffrey Frazier, and Kris Popendorf.
Created by Team 27890 for the 1999 competition.