OANA RINALDI · FIELD NOTES · 3 MIN READ
A shadow, a slice, a world
How to read a four-dimensional scene without mistaking its image for its shape.
The shape on screen is evidence
A stone block seems to narrow as you turn. A collectible grows out of empty space. Before deciding that the world is changing its rules, ask a smaller question: which part of the object is visible now?
Aetherfold stores puzzle positions using four coordinates. Its playable view is a three-dimensional section through that space, rendered by an ordinary perspective camera onto your screen. There are therefore two separate operations: selecting a section of the 4D world, and drawing that 3D section in a 2D image. Moving the camera changes the second operation. Turning the slice changes the first.
Projection and intersection answer different questions
A projection maps points into a space with fewer coordinates. Several distinct points may land in the same place. Imagine looking straight down at two balconies: their outlines can overlap in a photograph even though one is above the other. A 4D-to-3D projection has a similar ambiguity involving its discarded direction.
A section selects only points belonging to a chosen hyperplane. In a slice defined by w = 0, an object entirely at w = 3 contributes nothing. It has not been destroyed; your section simply misses it. This is why the familiar drawing of a cube inside another cube is not a literal picture of two nested rooms. It is one way of projecting a tesseract’s connected structure.
| Representation | What it helps reveal | What to watch for |
|---|---|---|
| Projection | Connections across an entire object | Overlap can hide separation |
| Section | The geometry present in one hyperplane | Missing parts still exist elsewhere |
| Unfolding | How boundary cells are attached | A net is neither the original object nor a playable slice |
Why a sphere changes size
Consider the solid 4D ball x² + y² + z² + w² ≤ R². Fixing w = d leaves x² + y² + z² ≤ R² − d². The section is a 3D ball with radius √(R² − d²). At the centre it is largest; at the boundary it reduces to a point; beyond the boundary there is no intersection.
The drawing above represents that spherical section on a flat screen. Move the slider slowly and predict when it will disappear. The same radius relationship appears in the game’s collectible rendering.
A cube does not always shrink
Different solids produce different sequences. For an axis-aligned tesseract defined by −1 ≤ x,y,z,w ≤ 1, every interior slice w = d with −1 < d < 1 is the same full cube. It does not gradually inflate like a sphere. Rotate the tesseract relative to the slice and the section can acquire different faces and change shape. Position and orientation both matter.
In play, follow a feature over several small turns. One frame can be ambiguous; a sequence tells you which surfaces persist, which recede and where a connection begins to appear.
Research connection: Oana’s geometry and 4DVisualizer reports examine complementary representations. For an independent visual treatment, see Thomas Banchoff’s Beyond the Third Dimension. The worked examples here are newly written for Aetherfold.