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FLAT SHEET | BIRD'S-EYE VIEW
Math & Dimensions
Manifold Playground
Explore curved surfaces, 2D–10D cubes, and a knot that unties through a fourth dimension.
One extra direction. Twice the vertices.
Start with a square. Make two copies and connect matching corners to build the next dimension. Select a diagram to explore it below.
All diagrams show the full edge graph in two dimensions, normalized to fit each card. Apparent crossings are not extra vertices; projected lengths and angles are not the original ones.
Explore surfaces and higher dimensions
Geometry explorerNegative K0Positive K
Fixed color scale: −4 to +4 model units⁻². Colors saturate outside that range.
Switch between surfaces and cubes. Projection and slice reveal different aspects of the same shape.
What you are seeing
Explore surfaces and higher dimensions
Each extra dimension doubles the vertices. Projection shows the whole cube; a slice can change shape or disappear as hidden coordinates move. The gold vertex keeps its identity through rotations.
Orthogonal rotations of a hypercube with coordinates from −1 to +1, sequential perspective into 3D, and solid cross-sections at fixed hidden coordinates. A ghost vertex can lie outside the slice. These are geometric models; screen crossings need not be intersections in the original space.
A knot takes an extra direction
3D → 4D demonstrationPlay the deformation, then peek toward W. Crossings in this picture need not be intersections in four dimensions.
Current state
Hidden directions
What you are seeing
A loop, with more room to move
The trefoil begins entirely in XYZ. During the deformation it gains a W coordinate, then returns to XYZ as a circle. Its centerline stays distinct in 4D even where the 3D picture crosses itself.
The optional explorer adds smooth bends in V, U, T, and S and lets you rotate any pair of directions. Projection shows the whole loop; Slice keeps only portions within the selected hidden-coordinate bands. This is a loop embedded in a larger space, not a knotted surface.
Equations & model notes
A trefoil-to-circle deformation in four dimensions
K(t) = (sin t + 2 sin 2t, cos t − 2 cos 2t, −sin 3t)
C(t) = (2.4 sin t, 2.4 cos t, 0)
(X,Y,Z) = (1 − s)K(t) + sC(t)
W = 2 sin(πs) sin t
pⱼ = 0.6 sin(πs) sin((j − 1)t), j = 4,…,n − 1
Symbols and units
t ∈ [0, 2π) labels a position around the closed loop; s ∈ [0,1] is deformation progress. Coordinates are dimensionless. n is the ambient dimension, from 4 to 8. The added coordinates p₄,… are V, U, T, S; they vanish at both endpoints. W is p₃. The loop itself remains one-dimensional.
Why the centerline does not cross itself
For 0 < s < 1, equality of W forces equal sin t. The only different candidate parameters have opposite cos t. Their Y values differ by 2(1 + 1.4s) cos t, which is nonzero unless the parameters already coincide. Thus Y and W together distinguish every loop point in the interior of the deformation. Their derivatives cannot both vanish. The endpoints are the original trefoil and a circle. Adding coordinates and applying orthogonal rotations preserve this property. The visible tube has illustrative thickness; this argument concerns its centerline.
Rotations and projection
p′ₐ = pₐ cos θ − pᵦ sin θ; p′ᵦ = pₐ sin θ + pᵦ cos θ
p′ᵢ = d pᵢ / (d − p_last); d = 1.8B
Plane rotations compose in selector order. The demonstration drops W after an X–W rotation; the resulting XYZ geometry is then viewed by the ordinary 3D camera. The explorer applies perspective from the last coordinate down to W. A radius bound starts at B = 4 and grows by 1.8/√(1.8² − 1) at each projection step, keeping each eye outside the bounded curve. Orbiting the 3D camera is a separate operation.
Slice and numerical limits
|p′ⱼ − hⱼ| ≤ ε, for every j = 3,…,n − 1
Slice clips each of 960 sampled centerline segments against all hidden-coordinate bands simultaneously, then displays the surviving XYZ segments and points. ε is the displayed half-thickness. Positive ε gives a finite-thickness neighborhood of a true 3D slice. At ε = 0, intersections are computed for the sampled polyline with floating-point tolerance; tangencies and very small features of the smooth curve may be missed. Empty slices are expected, especially in higher dimensions. The optional gray guide is not a slice intersection. Unlike the cube explorer, this is not a solid-volume section.
There is no rope tension, collision response, or conserved strand length. The deformation is prescribed, and display thickness or projected overlaps are not evidence of centerline contact. More ambient dimensions do not form a hierarchy of new ordinary knot types.
Trefoil background · Wolfram MathWorld. The particular deformation and its injectivity argument above describe this implementation.