# How Mathematical Models are Used in Art

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn how math helps artists create realistic work that looks three-dimensional. You'll also learn how math is involved in photography.

## Art

Leonardo da Vinci painted the Mona Lisa, a painting that became famous.

M.C. Escher made some visually intriguing drawings.

Both of these artists used mathematical principles to get these results. Now, don't think that the math involved had to be difficult. It's actually not all that complicated. Let's take a look.

## The Mona Lisa

Let's start with the Mona Lisa. Leonardo da Vinci used what is known as the golden ratio. The golden ratio is approximately 1.618 and is represented by the Greek letter phi. This ratio is achieved when you divide a line so that the longer part divided by the smaller part is equal to the whole line divided by the longer part.

If you divided your line into a longer portion a and a shorter portion b, then the golden ratio will have your a / b equal to a + b / a. This number will be the Greek letter phi representing 1.618.

The Mona Lisa uses the golden ratio throughout the painting. If you take a ruler and begin measuring different parts, you'll see that you'll get the golden ratio of 1.618 when you divide the longer measurement by the shorter measurement. For example, dividing Mona Lisa's head by the distance from the top of her head to her nose will give you the golden ratio. The placement of her arms also fit the golden ratio.

The golden ratio is used elsewhere too. Photographers for example, try to use the golden ratio when composing their photography. Instead of having the skyline right in the middle, the skyline will be at one of the golden ratios within the photograph.

## M.C. Escher

Artist M.C. Escher wanted to draw things with multiple perspectives. When you look at his drawings, everything looks normal at first glance. But when you take a closer look, you see that things are not as they seem. In fact, things shouldn't be functioning as they are because they are physically impossible. And some of those things are mathematically impossible too. Escher didn't use any fancy mathematical calculations to do his work either. All he used was his eye and pen.

Escher also drew tessellations by hand. A tessellation is a pattern of shapes that fit together perfectly and repeat in a pattern. It might not seem like it has anything to do with math, but tessellations are all about math as they can be tiled to form a continuous pattern on an infinite plane.

Tessellations aren't always simple grids. They can also be very complex like this one.

Notice how there are no overlaps and no gaps between each individual component.

## Three-Dimensional Art

Another art form that relies heavily on math is creating a three-dimensional illusion from a two-dimensional painting. It requires distorting proportions so the flat two-dimensional image looks three-dimensional from a certain perspective. This is called anamorphosis or anamorphic art. A three-dimensional cube seen from a certain point of reference will look like a flat distorted rectangle from a different point of reference. Once a point of reference is chosen, then points are mapped from the three-dimensional object to the flat two-dimensional plane so that it will give the illusion of being three-dimensional when seen from the point of reference.

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