Math & Money

The mathematics of sidewalk illusions

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.

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