RΛIN’s Research Atlas / Geometry / Interactive guide
Hypersphere Explorer: A 4D Sphere in Cross-Section
A hypersphere is a higher-dimensional counterpart of a spherical surface. In four-dimensional Euclidean space, that boundary is called a 3-sphere; the filled object it encloses is a 4-ball. Move the slider to explore their cross-sections and see why those names describe different things.
ATO Observatory · Mathematical demonstration · Published November 4, 2026
Move the slice through 4D space
The parent radius is 1. The slider changes w, a fourth spatial coordinate. The right-hand drawing represents the resulting 3D slice on a flat screen.
Drag or use the arrow keys. Try the center, an edge, and a position outside the object.
At w = 0.60, the solid 4-ball has a solid 3-ball cross-section with radius 0.80.
Keep a slice, compare the results
Share a link to your chosen slice or download a CSV table of 25 positions from w = −1.20 to +1.20. Empty sections have a blank radius, rather than a misleading zero. The CSV samples the radius-1 object in 0.10 steps; it is a calculation, not experimental data.
Download the 4D geometry field notes (PDF, 2 pages) for a printable comparison with the tesseract projection explorer.
Why is a “4D sphere” called a 3-sphere?
The names count different dimensions. Four coordinates locate the surrounding space. On the boundary, the fixed-radius equation removes one local degree of freedom. A point on that boundary therefore needs three local coordinates. The distinction is the same as a two-dimensional spherical surface sitting in ordinary three-dimensional space.
| Object | Defining condition, radius 1 | Intrinsic dimension |
|---|---|---|
| Circle, S¹ | x² + y² = 1 | 1 |
| Spherical surface, S² | x² + y² + z² = 1 | 2 |
| 3-sphere, S³ | x² + y² + z² + w² = 1 | 3 |
| Solid 4-ball, B⁴ | x² + y² + z² + w² ≤ 1 | 4 |
People often say “4D sphere” for a round object in four-dimensional space. The boundary and the filled solid are different objects, so this guide names each explicitly. A mathematical 4-sphere, S⁴, has four intrinsic dimensions and is usually defined in five-dimensional Euclidean space. Here, w is a spatial coordinate; it is not time.
How to calculate a hypersphere cross-section
Start with a solid 4-ball of radius R: x² + y² + z² + w² ≤ R². Fixing w selects a three-dimensional slicing space. Moving w² to the right leaves x² + y² + z² ≤ R² − w². For |w| < R, this is a solid 3-ball whose radius is r = √(R² − w²).
For the 3-sphere boundary, replace ≤ with = throughout. The ordinary slice is then a 2-sphere surface. At |w| = R either object meets the slicing space in one point. When |w| > R, the right side is negative and there is no intersection.
| Slice position w, with R = 1 | Radius r | Section of the solid 4-ball |
|---|---|---|
| −1.20 | Not defined | Empty |
| −1.00 | 0 | One point |
| −0.60 | 0.80 | Solid 3-ball |
| 0.00 | 1.00 | Largest solid 3-ball |
| 0.60 | 0.80 | Solid 3-ball |
| 1.00 | 0 | One point |
| 1.20 | Not defined | Empty |
For example, w = 0.60 gives r² = 1 − 0.36 = 0.64, so r = 0.80. Equal positive and negative offsets have equal radii because the formula uses w². These values are calculated from the model; they are not measurements of an extra physical dimension.
What does this picture actually show?
There are two reductions here: first a 4D object is intersected with a 3D slicing space; then that 3D section is drawn on a 2D screen. The circular outline and curved guides communicate an ordinary ball or spherical surface. The radius graph is a separate chart of the calculation.
A projection answers a different question. Dropping w from every point of the solid 4-ball produces a full 3-ball of radius R, rather than selecting one slice. To compare these operations on a shape with corners, open the tesseract projection and cross-section example.
Rotating a centered, unmarked 4-ball does not change its shape or the slice at a fixed offset: the defining sum of squares is unchanged. Rotation becomes visibly informative when a shape has edges, markings, or internal structure. Explore those cases in the 4D shape viewer.
How this connects to the yin–yang question
RΛIN’s investigation asks whether a specified four-dimensional construction can generate the yin–yang pattern, including its opposite-colored eyes and recursive structure. A round 4-ball gives us a precise starting vocabulary. Its spherical slices alone do not generate that pattern or establish the proposed connection between its two centers.
The next step is to specify the internal structure and the observation rule, then test what they predict. Read Is the yin–yang a 4D shape? for the candidate already tested, the result, and the open questions. Use the research claim template to separate an assumption from a prediction and a result.
Definitions and further reading
California State University, Northridge: Math 592D / Phys 640, Special Assignment 1 (PDF) gives the defining equation and angular coordinates of S³ in ℝ⁴. Marianne Freiberger, “The hypersphere in four dimensions,” Plus, May 6, 2025 develops the circle-to-hypersphere analogy. The slice equation and worked values above are derived explicitly on this page.
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