Projection Methods: Visualizing the Fourth Dimension

How we render 4D objects in 3D space—stereographic, perspective, and cross-sectioning techniques explained.

Since 4D objects cannot be viewed directly, their structure must be inferred through projection—the same way we understand 3D objects through 2D images. This guide explains the techniques used to visualize polychora.

The Fundamental Challenge

Our eyes are 2D sensors that reconstruct 3D scenes. We cannot add another dimension to this process. Instead, we use mathematical projections to reduce 4D objects to 3D representations, which we can then view normally.

This is analogous to how 3D objects cast 2D shadows. A 4D object casts a 3D "shadow" that we can examine from all angles.

Perspective Projection

The Concept

Perspective projection mimics how our eyes see: objects farther away appear smaller. For 4D→3D projection:

  1. Place a 4D object in 4-space
  2. Choose a projection point (like your eye, but in 4D)
  3. Draw lines from each vertex to the projection point
  4. Where these lines cross a 3D hyperplane, plot the projected points

The Result

The projected 3D object shows:

  • Parts of the 4D object closer to the projection point appear larger
  • Parts farther away appear smaller
  • The 3D object seems to have one cell "containing" another

Example: Tesseract

A tesseract projected this way shows a small cube inside a large cube, with edges connecting corresponding vertices. This is misleading—both cubes are the same size in 4D! The "inner" cube is simply farther away in the w-direction.

Schlegel Diagrams

What They Are

A Schlegel diagram is a specific type of perspective projection used in polytope theory. The projection point is placed just outside one face (or cell), looking inward.

For Polychora (4D→3D)

A Schlegel diagram of a tesseract:

  • One cubic cell becomes the outer boundary (a cube)
  • The opposite cubic cell appears as a smaller cube inside
  • The remaining 6 cubic cells appear as truncated pyramids connecting them

Why Schlegel Diagrams Are Useful

  • All cells are visible (none hidden behind others)
  • The combinatorial structure (what connects to what) is preserved
  • Easy to understand face/cell adjacency

The Distortion

Schlegel diagrams severely distort size and shape. In the tesseract Schlegel diagram, all 8 cubic cells are congruent in 4D, but they appear as wildly different sizes in the diagram. Angles and proportions are not preserved.

Stereographic Projection

The Concept

Stereographic projection maps a sphere onto a plane. For 4D visualization, we use the 4D analogue:

  1. Place the polytope on the surface of a 4D hypersphere
  2. Choose a projection point (typically the "north pole")
  3. Project each point onto a 3D hyperplane

Key Properties

  • Conformality: Angles are preserved locally
  • Circles become circles: Great circles on the hypersphere project to circles in 3D
  • Smooth curves: Edges that are straight in 4D become graceful curves in 3D

Why This Viewer Uses Stereographic Projection

This 4D polytope viewer primarily uses stereographic projection because:

  1. Visual elegance: The curved edges create beautiful, organic forms
  2. Equal treatment: No single cell dominates visually
  3. Rotation clarity: As the polytope rotates in 4D, the flowing curves make the motion intuitive
  4. Mathematical significance: Preserves the conformal structure

The Characteristic Look

In stereographic projection:

  • Edges curve gracefully rather than going straight
  • The overall shape appears spherical
  • 4D rotation creates mesmerizing transformations

Orthographic Projection

The Concept

Orthographic (or parallel) projection uses parallel projection lines rather than converging ones. Choose a viewing direction in 4D, then project every point perpendicularly onto a 3D hyperplane.

This is like viewing from infinitely far away—no perspective distortion.

Properties

  • No size distortion by depth: Objects don't shrink with distance
  • Parallel lines stay parallel: Good for technical analysis
  • Overlapping elements: Parts of the polytope may overlap confusingly

Cross-Sectioning (Slicing)

The Concept

Instead of projecting the entire 4D object, we can slice it with a 3D hyperplane—like passing an object through a plane and observing the intersection.

Dynamic Cross-Sections

As a 4D object passes through 3D space, the cross-section changes:

Tesseract passing vertex-first:

  1. Point (first vertex touches)
  2. Growing tetrahedron
  3. Octahedron (at the middle)
  4. Shrinking tetrahedron
  5. Point (last vertex exits)

Tesseract passing face-first:

  1. Square (first face touches)
  2. Growing cube
  3. Cube (entire face inside)
  4. Shrinking cube
  5. Square (last face exits)

Hinton's Colored Cubes

Charles Howard Hinton developed a system of colored cubes to help visualize cross-sections. By memorizing the sequence of colors as different parts of a tesseract pass through 3D space, practitioners could develop 4D intuition.

Rotation vs. Projection

4D Rotation

True 4D rotation happens before projection. A 4D object can rotate in 6 independent planes:

  • XY, XZ, XW (rotations involving X)
  • YZ, YW (rotations involving Y)
  • ZW (rotation purely in the "extra" dimensions)

What You See

When a 4D object rotates in a plane involving W (like XW, YW, or ZW), the 3D projection transforms dramatically:

  • Cells appear to turn inside-out
  • Parts that were "inside" move "outside"
  • The object seems to flow through itself

This is not the projection changing—it's genuine 4D rotation revealing different aspects of the object.

3D Rotation (of the Projection)

You can also rotate the 3D projection itself, like turning a sculpture. This helps you understand the 3D shape of the projection but doesn't reveal new 4D information.

Choosing the Right Method

Method Best For Limitation
PerspectiveUnderstanding depthSevere size distortion
SchlegelSeeing all cellsShape/angle distortion
StereographicAesthetic visualizationCurved edges can confuse
OrthographicTechnical analysisOverlapping elements
Cross-sectionStep-by-step understandingOnly partial view at once

This Viewer's Approach

This interactive viewer combines several techniques:

  1. Stereographic projection for the primary visualization
  2. Real-time 4D rotation to reveal the object's structure
  3. 3D rotation (via mouse drag) to examine the projection
  4. Adjustable projection parameters to customize the view

By rotating a polytope through all 6 4D rotation planes, you can observe how its 3D projection transforms—the closest we can come to "seeing" a 4D object.

Further Reading