Projective Illusion

By Xah Lee. Date: . Last updated: .
3D painted room 3D painted room
Two views of a room. The room is painted in such a way so that from a particular view point it becomes meaningful. [image source ]

Basically, the ideas is that we perceive things by projection. (or rather, physically speaking light get reflected into our retina) Now, forget that we have two eyes and the processing to get into a stereoscopic imagery in our head. But suppose we just have one eye for simplicity of this account. Now, we see things by projection. So, essentially what we get is a planar representation of a 3D image. Besides having two eyes to get different perspective to reconstruct the original 3D image, we also heavily relies on past experiences of reality of likely objects. So, given a drawing on paper (which is 2D), we are able to interpret its original 3D scene as the drawing is of. The simplest example is that a drawing of a cube, which on paper is a hexagonal thing. (and if you view the cube from its corner perspective, it's a perfect hexagon, which SGI exploited for their logo design. See: SGI Logo Visual Illusion.)

Now, however, given a piece of drawing (which is a 2D image), there are really a myriad ways to interpret it. Put it technically, there are infinite 3D scenes that can project into a given 2D image. Our brain simply automatically choose the simplest, possible, earthly interpretation. For most images this works out fine. But for some drawings, our brain are kinda indecisive and ambiguous on how to interpret it. This is why, given a line drawing of a cube, sometimes you see it as this side out, while in a blink of an eye you see it as the other side out. To recap, this is because a 2D image cannot define a 3D object.

Now, it becomes really interesting if you were to exploit this fact into applications. Given a 2D image, you can create several 3D objects whose projection is this 2D image. Of course, almost all 3D scenes will in reality be absurd objects and shapes, but still they are real 3D objects. One simple application of this exploit is for example in creating decorative visual displays for the entertainment industry. (For example, screen-savers)

In particular, given a line drawing, say, a cube. In your program, you find a few real 3D object of which this line drawing is a projection. Then, you start to rotate this 3D object. (say, in OpenGL) So, the effect is that when a viewer sees a drawing on screen, then the thing on screen suddenly starts to rotate like a real 3D object, and it came to the surprise and realization of the viewer the real nature and shape of the object. Now, when the 3D object has rotated a 360°, in your program you switch to another 3D interpretation, so that, the effect being the object as if thru a 2D-space-hole suddenly and smoothly transformed into another, rather different, 3D object.

Likewise, you can start with some 3D object in rotation and thru this 2D screen presentation turn it into another 3D object. This understand can be used to explore unexplored fields in illusion, visualization, and computer animations.

Some Artworks Related to Dimensionality and Projection

Artist M C Escher did several artworks based on the idea of 3D objects and its 2D projection. He exploited a aspect so that seemingly realistic drawing are actually “impossible” objects. Note, however, these the vast potential of dimensionality and projection remain unexplored.

M C Escher's Waterfall. Note here that in fact all planar images can be interpreted as the projection of some 3D objects onto a plane. So, in a sense it is not possible to draw something that's a “impossible object”. However, when we say it being “impossible”, it means realistically unlikely.

For example, this site shows the actual 3D objects of many Escher's “impossible” 2D drawings.

[Escher for Real By Gershon Elber. At , accessed on 2016-12-13 ]

a folded dragon drawing
M C Escher's Dragon. Here, two types of drawings are intermixed in one, to create an odd effect. One type of drawing is the flat etching. Such is the embroidery on clothing or wallpaper patterns. This type of drawing is used on the dragon's neck, body, tail, wings. The other type of drawing, is depiction of real 3D object using primarily technique of shading. This type of drawing is used to create the box in the center, the dragon's legs, and the crystals. Escher skillfully mixes the two to create a very odd effect that makes the viewer become aware of the interpretation process.
M C Escher's Belvedere. This is another employment of the 2D/3D projection illusion. When happened here is that a rectangular 3D object with 8 legs can be positioned in more than one way, such that their legs will have the same position in 2D projections. So, in Belvedere, the top block is positioned in one way, while the base block is positioned in another way, but both in a way such that their columns positions in 2D projection matches exactly. So, Escher just connected them in the drawing.

Math and Illustrations