Far-Out Corners
At first glance, this activity might look like a bunch of boxes. But there are no boxes at all. A closer look reveals that what you're actually seeing is a cluster of corners lit from below. Move your head around the assembly with one eye closed, and the cubes seem to turn mysteriously to follow your movement.
- Three clean milk cartons (quart size works well) or a piece of white poster board large enough to make nine 4 x 4-inch (10 x 10-cm) squares —see Assembly
- Scissors
- Ruler
- Transparent tape
- A bright light source that can be used to illuminate the final assembly from below
- One or more partners to hold the completed assembly and the light source (alternatively, you can devise a stand or hang the assembly to achieve the desired effect)
- You will need three partial cubes, or corners, for this activity. You can use the inside corners of white, square-cornered containers such as clean milk cartons or make your own corners from poster board.
To make corners out of milk cartons:- Cut a milk carton as shown in the diagram below. You should keep the lower three-sided corner that includes the bottom and discard the upper three-sided corner that includes the top (click to enlarge the diagram).
- Make two more partial cubes, or hollow corners, the same way.
- Cut the poster board into nine squares, each measuring 4 × 4 inches (10 × 10 cm).
- Use three of the squares to construct a partial cube or corner in the following fashion (click to enlarge diagram below): Tape two squares together at one edge. Open the two squares into a right angle, and then tape the third square on top of the first two squares. You can use the edge or corner of a table to get a more precise angle.
- Make two more partial cubes, or corners, the same way.
- Cut a milk carton as shown in the diagram below. You should keep the lower three-sided corner that includes the bottom and discard the upper three-sided corner that includes the top (click to enlarge the diagram).
- Arrange the three partial cubes, or corners, as shown in the photos below (click to enlarge) and tape them together. The photo on the left shows the "outside" of the three-cube milk carton assembly, and the photo on the right shows the "inside."
- Hold the three-cube assembly as shown in the photo at the top of this Snack. The hollow open sides of each corner should be facing out and down, and the entire assembly should be tilted slightly backward.
- Position the light so it shines up into the hollow assembly.
Stand back about 5 feet (1.5 m) and close one eye. With a little mental effort, you can see the corners that you have constructed as solid, outward-protruding cubes, rather than as the hollow corners that they are.
Walk back and forth in a direction parallel to the box, or just turn your head back and forth. Notice that the cube on top, and possibly all three cubes, seem to be following you as you move.
The first step to successfully seeing the partial cubes turn with you lies in your ability to perceive them as solid, six-sided objects. This perception has a lot to do with being raised in a society that recognizes cubes as a common shape. Your brain is used to seeing cubes, so even though this partial cube only has three sides, it assumes the cube is solid.
As you move past the assembly, your view of the corners changes in a way that would not make any sense if the corners were stationary cubes. Your eye-brain system is accustomed to seeing things that are near you move faster than things that are farther away. When you’re riding in a car, for example, nearby objects seem to whiz by, whereas distant objects seem to follow you at a slower pace. Because you perceive this inside corner to be the outside of a solid cube, your brain “sees” the corner farthest from you as being the closest. To maintain this misconception, your brain perceives a rapid rotation of the cube as your angle to the corner changes (click to enlarge the diagram below).
The diagram above shows how this illusion works. In the real situation, as your eye moves to the right, it sees more of side A. In order to see more of side A of the imagined corner, the perceived cube must be seen to rotate as you move.
This Snack is an example of an ambiguous figure—a figure that is subject to more than one interpretation. The Necker Cube (pictured below) is a classic ambiguous figure, and there are many, many others. The Necker Cube can be seen with either the lower left face in front or the upper right face in front.
