Design Your Own 3D Optical Illusions

The illusion in this lesson is based on the paper Anomalous Mirror Symmetry Generated by Optical Illusion by Dr. Kokichi Sugihara. You can read the paper for the complete details, but we will summarize it here.

The paper presents objects that appear flipped left to right when placed in front of a mirror. One such object is the "impossible arrow" (Figure 1). The shape only looks like an arrow when viewed from the correct angle. When viewed from the top or the side (Figure 2), it does not look like an arrow at all.

The object looks like an arrow pointing to the right, but its reflection seems to show an arrow pointing to the left.

Figure 1. The "impossible arrow" illusion. The arrow appears to point to the right, but its reflection in the mirror points to the left.

Figure 2. Top and side views of the "impossible arrow" reveal that it does not look like an arrow at all when viewed from these angles.

This illusion works because your brain assumes the top of the object is flat. However, as shown in the side view in Figure 2, the top and bottom surfaces of the arrow are not flat. The shape of these surfaces is carefully defined so it looks like the outline of a two-dimensional arrow only when viewed from a certain angle.

Dr. Sugihara's paper describes how you can start with the outline of a 2D shape in the x-y plane, defined by top and bottom parametric curves a₁ and a₂ (Figure 3). The curves are defined over the interval [-1,1] as functions of the parameter t. They can be defined by piecewise functions, but they must be continuous and meet at their endpoints on the x-axis, forming a closed curve. They are not required to be symmetrical about the x axis, although the arrow in Figure 3 is.

Figure 3. A two-dimensional drawing of an arrow in the x-y plane, with top curve a₁ and bottom curve a₂.

The paper presents equations that convert these 2D curves into 3D curves that look like the 2D shape when viewed from an angle θ (Figure 4). The view angle is defined downward from a horizontal line that is parallel to the y-axis, so looking at the shape directly from the side would be a view angle of 0°, and looking at the shape from directly above would be a view angle of 90°.

A dashed line parallel to the y axis is offset in the positive z direction, with an angle theta defined downward from that line.

Figure 4. Definition of the view angle θ.

The equations to convert the the 2D curves a₁ and a₂ into 3D parametric curves c₁ and c₂ are:

Equation 1: $\; \mathbf{c_1(t)} = \left( t, \frac{a_1(t)-a_2(-t)}{2}, \alpha \frac{a_1(t)+a_2(-t)}{2} \right)$

Equation 2: $\; \mathbf{c_2(t)} = \left( t, \frac{a_2(t)-a_1(-t)}{2}, \alpha \frac{a_2(t)+a_1(-t)}{2} \right)$

where

Equation 3: $\;\alpha = \tan \theta$

Note that since x = t for both c₁ and c₂, you can substitute x for t in the above equations if your students have not yet learned about parametric curves.

The resulting 3D curves c₁ and c₂ only look like the original 2D shape when viewed along the y-axis at a downward angle θ (Figure 5). When the shape is rotated 180°, its original orientation appears preserved (e.g., an arrow pointing to the right), resulting in the "anomalous mirror symmetry" illusion (Figure 6).

All three views are aligned along the y axis but at three different downward view angles: 90 degrees, 45 degrees, and 0 degrees. The 2D arrow shape is only visible at the 45-degree view angle.

Figure 5. Views of the 3D curves c₁ and c₂ while changing the view angle θ.

Rotations about the z axis of 0 degrees, 45 degrees, 135 degrees, and 180 degrees. The 2D arrow is only visible in the 0 degree and 180 degree views. In both cases it looks like the arrow is pointing to the right.

Figure 6. Views of the 3D curves c₁ and c₂ with a constant view angle θ=45° while rotating the shape about the z-axis.

While this is sufficient to generate the illusion in a 3D graph, if you want to 3D print the shape, you need to generate an STL file (originally developed for a type of printing called stereolithography). An STL represents the surface of a solid object using triangles. So, for example, an STL file of a cube will consist of twelve triangles, two on each of the six faces of the cube (Figure 7).

Figure 7. Left: A cube plotted in 3D space. Right: An STL representation of the same cube, with each face broken up into two triangles.

A triangularized representation of the 3D shape based on the curves c₁ and c₂ can be created first by offsetting the curves in the z direction (Figure 8), then filling in the sides, top, and bottom of the shape with triangles (Figure 9). This results in a solid 3D version of the illusion (as opposed to a wireframe version) that can be viewed in a 3D graph or exported as an STL for 3D printing or further editing in a CAD program.

Figure 8. 3D curves offset in the z direction to create the top and bottom edges of the 3D shape.

Figure 9. 3D shape with the surface filled in with triangles.

In this lesson, your students will define their own 2D shapes using piecewise functions (linear, quadratic, polynomial, trigonometric, etc.—you can decide what is appropriate for your students). They will then enter their shapes into a working MATLAB or Python program that automatically generates the triangularized 3D version of the shape as shown in Figure 9, as well as the intermediate steps (as in Figures 5, 6, and 8). You can choose whether your students just view their shapes in the output MATLAB or Python plots (which can be rotated on the computer screen) or also 3D print them.