Da Vinci's Vitruvian Man of math - James Earle

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What's so special about Leonardo da Vinci's Vitruvian Man? With arms outstretched, the man fills the irreconcilable spaces of a circle and a square -- symbolizing the Renaissance-era belief in the mutable nature of humankind. James Earle explains the geometric, religious and philosophical significance of this deceptively simple drawing.

Lesson by James Earle, animation by TED-Ed.

This image of the Vitruvian Man,

taken from Leonardo's sketches,

has become one of the most recognizable

symbols of the Renaissance.

But why?

It's a simple pen and ink drawing, right?

Wrong!

Let's start to answer this question

with a math problem.

I know how to calculate the area of a circle.

I take the value for pi

and multiply it by the radius squared.

I also know how to take the area of a square.

I multiply the base by itself.

But, how can I take the area of a circle

and create a square with an equal area?

This is a problem often called "squaring a circle"

that was first proposed in the ancient world.

And like many of ideas of the ancient world,

it was given new life during the Renaissance.

As it turns out,

this problem is impossible to solve

because of the nature of pi,

but that's another story.

Leonardo's sketch,

which is influenced by the writings

of a Roman architect, Vitruvius,

places a man firmly at the center

of a circle and a square.

Vitruvius claimed the navel

is the center of the human body

and that if one takes a compass

and places the fixed point on the navel,

a circle can be drawn perfectly around the body.

Additionally, Vitruvius recognized

that arm span and height

have a nearly perfect correspondence in the human body,

thus placing the body perfectly inside a square as well.

Leonardo used the ideas of Vitruvius

to solve the problem of squaring a circle metaphorically

using mankind as the area for both shapes.

Leonardo wasn't just thinking about Vitruvius, though.

There was an intellectual movement

in Italy at the time

called Neoplatonism.

This movement took an old concept

from the 4th century developed by Plato and Aristotle,

called "The Great Chain of Being".

This belief holds that the universe is a hierarchy

resembling a chain,

and that chain starts at the top with God,

then travels down through the angels,

planets,

stars,

and all lifeforms

before ending with demons and devils.

Early in this philosophic movement,

it was thought that mankind's place in this chain

was exactly in the center.

Because humans have a mortal body

accompanied by an immortal soul,

we divide the universe nicely in half.

Around the time Leonardo sketched

the Vitruvian Man, however,

a Neoplatonist named Pico della Mirandola

had a different idea.

He pried mankind off the chain

and claimed that humans have a unique ability

to take any position they want.

Pico claimed that God desired

a being capable of comprehending

the beautiful and complicated universe he had created.

This led to the creation of mankind,

which he placed at the center of the universe

with the ability to take whatever form he pleases.

Mankind, according to Pico,

can crawl down the chain and behave like an animal

or crawl up the chain and behave like a god,

it's our choice.

Looking back at the sketch,

we can see that by changing the position of the man,

he can fill the irreconcillible areas

of a circle and a square.

If geometry is the language the universe is written in,

then this sketch seems to say

we can exist within all its elements.

Mankind can fill whatever shape

he pleases geometrically

and philosophically as well.

In this one sketch,

Leondardo was able to combine

the mathematics,

religion,

philosophy,

architecture,

and artistic skill of his age.

No wonder it has become such an icon

for the entire time period.