Geometry / Interactive explanation
Tesseract Explorer: See a Four-Dimensional Cube
Sixteen vertices. Thirty-two edges. One extra coordinate. Rotate a mathematical tesseract and watch how its projection changes.
Turn through a fourth coordinate.
Change the XW and YW angles. Violet lines connect vertices that differ in W. Animation starts only when you choose it.
Take the geometry with you
Download the free 4D geometry field notes (PDF, 2 pages) for prediction exercises, coordinate counts and a hypersphere slicing worksheet. The SVG button saves your current diagram; the CSV contains the 16 original vertices before rotation.
To compare a projection with a slice, open the hypersphere cross-section explorer. Record your observation in the research claim template.
What is a tesseract?
A tesseract is the four-dimensional analogue of a cube. Its vertices are every choice of (x, y, z, w) where each coordinate is either −1 or +1. That gives 2⁴ = 16 vertices. Join vertices that differ in exactly one coordinate: each vertex has four neighbors, so the edge count is 16 × 4 ÷ 2 = 32.
The boundary contains 24 square faces and eight cubic cells. “Four-dimensional” here means four mathematical spatial coordinates. The display is a representation on a two-dimensional screen; it is not evidence of physical travel through another dimension.
Why does it seem to fold through itself?
Rotation changes the coordinates before the image is projected. Distinct points in four dimensions can overlap in the displayed view. Lines crossing on the screen do not necessarily intersect in the original object. An inner cube appearing to pass through an outer cube is a property of this representation.
Projection and cross-section answer different questions
A projection maps every point into fewer coordinates. A cross-section keeps only points on a chosen slice, such as w = 0. This lightweight explorer displays projections. The full Observatory also offers a cross-section mode and additional forms.
The calculation
x′ = x cos θ − w sin θ w′ = x sin θ + w cos θ Orthographic: (x′, y′, z′, w′) → (x′, y′, z′) Perspective: (x′, y′, z′) × 3 / (3 − w′)
The second rotation applies the same rule in the YW plane. A fixed three-dimensional camera then maps the result to the screen. There are six coordinate planes in four dimensions; these two controls expose a manageable subset.
Try three observations
- Set both angles to zero and compare perspective with orthographic projection. Some distinct vertices can overlap.
- Rotate XW by 90°. The roles of the X and W coordinates exchange, with a sign change.
- Share a view, reopen the link and compare it with the original. The angles and projection mode are encoded in the URL.
Source and attribution
The coordinate construction and rotation approach follow standard linear algebra. See Oliver Knill’s Harvard Math 21b tesseract demonstration. This ATO implementation is a new educational interface around established mathematics.
RΛIN / ATO · Public research and interactive design · Updated 5 November 2026. Research direction and original concepts by RΛIN; editorial and software development use AI assistance. Sources and limits are stated alongside each example.
4D projection vs. cross-section: a worked example
A projection records where points land after a coordinate is removed. A cross-section keeps only points satisfying a chosen slice condition. They can produce similar pictures while answering different questions.
Consider the solid tesseract with each coordinate x, y, z and w between −1 and +1. Orthographic projection drops w: every (x, y, z, w) becomes (x, y, z). The result is a solid three-dimensional cube. The distinct points (0, 0, 0, −1) and (0, 0, 0, +1) both become its center.
The slice w = 0 is also a solid three-dimensional cube, but it contains only points whose fourth coordinate is zero. Neither of those two example points belongs to the slice. For this axis-aligned solid, slices with −1 ≤ w ≤ +1 have the same cubic extent; outside that interval the slice is empty. Rotating the solid relative to the slicing plane can change the section’s shape.
The explorer above draws the tesseract’s vertices and edges as a wireframe projection. Its orthographic and perspective settings change the projection rule; they do not perform a slice. For the slicing interface, open the 4D shape viewer in the Observatory and choose Cross-section through W.
This distinction matters for the yin–yang’s proposed four-dimensional origin: one two-dimensional image does not uniquely identify the object or the observation that generated it. Compare that with the pairwise-projection experiment, where different relations produce the same summaries.
Coordinate construction: Oliver Knill’s Harvard tesseract demonstration. The solid-cube slice comparison above follows directly from the stated coordinate bounds.
Keep exploring
Compare a projection with a hypersphere cross-section, or choose another question in the Research Atlas.
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