Can a Cube Pass Through a Hole Smaller Than Its Own Face?
A cube can be cut with a hole large enough for an identical cube to pass through it — a classic geometry result called Prince Rupert's Cube. Model the cube's rotated cross-section to find the largest square hole it creates.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Prince Rupert's Cube is the geometric fact that a hole can be cut through a cube large enough for a second, equally-sized cube to pass through it. The key is projection: as the cube is tilted and rotated, its 2D outline (the shadow it casts, or the cross-section a flat template would need to match) changes shape and size continuously. At certain orientations, that outline is larger than the cube's own face. This exploration models the cube's 2D outline as a function of its rotation angle, to find the orientation that produces the largest possible square cross-section, and how large a hole it allows.
Guiding Questions
- As the cube rotates about a chosen axis, how does its 2D outline (its projection onto a flat plane) change shape?
- Express the outline's dimensions as a function of the rotation angle, using trigonometry.
- At what angle is the projected outline largest, and how does its size compare with the cube's own face?
- How does the answer change if you rotate about a different axis, or use a different solid, such as a regular tetrahedron?