The mathematics of sidewalk illusions - Fumiko Futamura

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Have you ever come across an oddly stretched image on the sidewalk, only to find that it looks remarkably realistic if you stand in exactly the right spot? These sidewalk illusions employ a technique called anamorphosis — a special case of perspective art where artists represent 3D views on 2D surfaces. So how is it done? Fumiko Futamura traces the history and mathematics of perspective.

Lesson by Fumiko Futamura, animation by TED-Ed.

If you're ever walking down the street

and come across an oddly stretched out
image, like this,

you'll have an opportunity
to see something remarkable,

but only if you stand in exactly
the right spot.

That happens because these works
employ a technique called anamorphosis.

Anamorphosis is a special case
of perspective art,

where artists represent realistic
three-dimensional views

on two-dimensional surfaces.

Though it's common today,

this kind of perspective drawing has only
been around since the Italian Renaissance.

Ancient art often showed all figures
on the same plane,

varying in size by symbolic importance.

Classical Greek and Roman artists realized
they could make objects seem further

by drawing them smaller,

but many early attempts at perspective
were inconsistent or incorrect.

In 15th century Florence,

artists realized the illusion
of perspective

could be achieved with higher degrees
of sophistication

by applying mathematical principles.

In 1485, Leonardo da Vinci
manipulated the mathematics

to create the first known
anamorphic drawing.

A number of other artists later
picked up the technique,

including Hans Holbein
in "The Ambassadors."

This painting features a distorted
shape that forms into a skull

as the viewer approaches from the side.

In order to understand how artists
achieve that effect,

we first have to understand how
perspective drawings work in general.

Imagine looking out a window.

Light bounces off objects
and into your eye,

intersecting the window along the way.

Now, imagine you could paint the image
you see directly onto the window

while standing still and keeping
only one eye open.

The result would be nearly
indistinguishable from the actual view

with your brain adding depth
to the 2-D picture,

but only from that one spot.

Standing even just a bit off
to the side

would make the drawing
lose its 3-D effect.

Artists understand that
a perspective drawing

is just a projection
onto a 2-D plane.

This allows them to use math to come up
with basic rules of perspective

that allow them to draw without a window.

One is that parallel lines, like these,

can only be drawn as parallel if they're
parallel to the plane of the canvas.

Otherwise, they need to be drawn
converging to a common point

known as the vanishing point.

So that's a standard perspective drawing.

With an anamorphic drawing,
like "The Ambassadors,"

directly facing the canvas makes the image
look stretched and distorted,

but put your eye in exactly the right spot
way off to the side,

and the skull materializes.

Going back to the window analogy,

it's as if the artist painted
onto a window positioned at an angle

instead of straight on,

though that's not how Renaissance artists
actually created anamorphic drawings.

Typically, they draw a normal image
onto one surface,

then use a light,

a grid,

or even strings to project it
onto a canvas at an angle.

Now let's say you want to make
an anamorphic sidewalk drawing.

In this case, you want to create
the illusion

that a 3-D image has been added
seamlessly into an existing scene.

You can first put a window
in front of the sidewalk

and draw what you want to add
onto the window.

It should be in the same perspective
as the rest of the scene,

which might require the use of those
basic rules of perspective.

Once the drawing's complete,

you can use a projector placed
where your eye was

to project your drawing down
onto the sidewalk,

then chalk over it.

The sidewalk drawing
and the drawing on the window

will be nearly indistinguishable
from that point of view,

so viewers' brains will again be tricked

into believing that the drawing
on the ground is three-dimensional.

And you don't have to project onto
a flat surface to create this illusion.

You can project onto multiple surfaces,

or assemble a jumble of objects,
that from the right point of view,

appears to be something else entirely.

All over the planet, you can find
solid surfaces

giving way to strange, wonderful,
or terrifying visions.

From your sidewalk
to your computer screen,

these are just some of the ways
that math and perspective

can open up whole new worlds.