By: Sydney Krantz

Math in Art

Show Me the Math

Although you may not see it, math is in everything you do. And, art is no exception. All famous masterpieces are filled with lines, shapes, patterns, tessellations, and formulas. Math doesn’t just apply to boring textbook equations. Sure, maybe it’s not the type of math you’re used to seeing, but I guarantee you it’s still math.

About Tessellations

• A tessellation is a pattern made of identical shapes that must fit together without gaps or overlaps.

• The word originated from the Latin word tessella which were the small, square stones or tiles used in ancient Roman mosaics.

• Tessellations are commonly on fabric used for apparel and rugs, wallpapers, tiles, carpeting, paintings, and other types of art.

• Tessellations are forms of mathematics because they are patterns.

• All patterns can be expressed algebraically.

The Math Behind Tessellations

• There are 17 possible combinations of regular polygons around a vertex.

• This makes for 12 different arrangements.• The smallest possible angle in any regular

polygon is 60°.• The number of polygons at a vertex must be

anywhere from 3 to 6.• The measure of any angle of a regular polygon

with n sides and n angles is represented by the expression (n-2) 180°

n

• The sum of the angles around a vertex is 360°.

• The following are mathematical expressions for a vertex with 3, 4, 5, and 6 sides. – 3 polygons:

• Simplified:

– 4 polygons:

– 5 polygons:

– 6 polygons:

The Wonderful World of Geometry

• Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.

• However, geometry is usually just referred to by people as the math of shapes which is also correct.

• A shape is defined as ANY enclosed space made by a closed line.

Shapes?

• For instance, these are shapes.

• These are not.

Oh, Shapes

• It is pretty self explanatory how the idea of shapes and geometry contribute to art.

• But if you still don’t follow, let me help you out.

• EVERY single thing in this world is made up of shapes

• Art is just a huge conglomeration of different shapes, and the laws of geometry help us define these shapes.

How Basic Can It Get?

• A shape is technically any enclosed area, however, all shapes are made up of basic shapes.

• Basic shapes are the most common as well as simple shapes there are.

• Some examples would be triangles, squares, rectangles, circles, ovals, and hexagons.

• How about an example?

•Identifying different basic shapes in a painting

such as this or this is

pretty easy.

•Doing so in something like this might be a bit harder.

•However, I have outlined them

for you.

•But this isn’t only for two dimensional works of art.

•Do you see what I mean?

Symmetry

• Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis.

• There are a few different types, but we are going to be focusing on reflectional symmetry which is otherwise referred to as line symmetry.

• Reflectional symmetry is a type of symmetry where one half is the reflection of the other half.

• The imaginary line down the center of a symmetrical image is called the Line of symmetry.

How Symmetrical!• It is a tested fact that images that are

symmetrical are more aesthetically pleasing than those that are not symmetrical.

• This also applies to faces.

• For example, who do you think looks better, me or my evil twin?

Me Evil Twin

• If your answer is me, that is perfectly understandable.

• And, I am flattered.• If we look at the headshots like this, we can see

the symmetry in the two faces.

Me Evil Twin

• The line going vertically was the line of symmetry (both of our heads were tilted), and the other two went through the centers of our eyes and mouths.

• At least, they were supposed to.• If you looked closely, you could see that my eyes and

mouth were more symmetrically aligned than my twin’s.• That is why if you answered “me”, it was perfectly

understandable.

My Mouth Twin’s Mouth

My Eyes Twin’s Eyes

Connection Please?• If you are looking for a connection between what we just

discussed and art, it is right here.• Most famous paintings, drawings, and photographs

(especially of people) are symmetrical.• As concluded by Dr. Christopher Tyler, it is a fact that

many famous works such as the Mona Lisa are symmetrical, but the artists and photographers did not do this consciously.

• There is proof of symmetry in these pieces, but their creators’ did not keep symmetry in mind while creating them.

• Tyler’s original theory about a different topic was not correct, but he did discover that the brains of all people can unconsciously detect symmetry.

• It also backs up the idea that symmetrical things are more pleasing to look at.

Ratios

• Lastly, we will be discussing ratios and their place in art.

• Ratios are the quantitative relations between two amounts showing the number of times one value contains or is contained within the other.

• Ratios are used to express many things such as batting averages in baseball, the number of students to teachers at schools and colleges, and the amount of miles per hour you travel.

A Rainbow of Ratios

•Ratios can also be applied to art.•They are very important to color mixing.•For instance, everybody knows that mixing red with yellow makes orange.•However, there are many different shades of orange such as: amber, burnt orange, carrot orange, orange, international orange, peach, Princeton orange, and many more than can be named in one slide.•That means that for each type of orange, there is a different ratio of red to yellow.

•When mixing paints, being just one drop off could turn into a terrible mess.

•For each specific color, there is a ratio to be used in order to make it.

•If I am trying to make yellow orange the red to yellow ratio I would use is 4

•In this ratio, the 4 represents the number of yellow paint drops, and the 1 represents the number of red paint drops.

• This also means that you can use an equivalent ratio for making the color yellow orange.

• You can use 8 yellow for every 2 red, 12 yellow for every 3 red, or even 16 yellow for every 4 red.

No Math = No art• Without math, there would be no art.• If there weren’t any tessellations, beautiful wallpapers such as

this and stunning tiles such as this could

not exist. • Because there are no algebraic patterns or equations, there

would be no way of knowing how to make them.• Without geometry and shapes, there would be no art at all.• All art would have to look like this

• Do you see any meaning or identifiable shapes in there? I don’t.

• And without symmetry, nothing would look right, and no artwork would be aesthetically pleasing.

• Also, people with Obsessive

Compulsive Disorders would

not be able to contain themselves. • Lastly, without ratios, mixing colors would be impossible!

• Without a way to know how much of a color you are supposed to use you would end up making a completely different color each time.

• Conclusively, there is no

art without math.

Works Cited

Roth, Elizabeth. "Perception, symmetry of art discussed at brain lecture.“ mc.vanderbilt.edu. Vanderbilt University Medical Center , 2007. Web. 10 Dec. 2011.<http://www.mc.vanderbilt.edu/reporter/index.html?ID=2007>.

Roza, Greg. An Optical Artist. New York City: Rosen Publishing Group, Inc., 2005. Print.

Seymour, Dale, and Jill Britton. Introduction to Tessellations. Paloalto: Dale Seymour Publications, 1989. Print.

The End