SHADOWS → CONSTRAINTS → FORM

One more dimension.
One more degree of freedom.

Specify projections. Recover the largest 4D solid consistent with them. Inspect what the shadows leave undetermined.

A 3D slice at W = 0Drag to orbit · cyan → violet encodes W · dots are grid cells

−1W coverage · drag the bars or slider+1

W coordinate 0.00

FORWARD CHECK

Do the reconstructed shadows agree?

● matched● missing target● extra, inactive view

Project overview

Higher-dimensional inversion

Live · ready

Extra dimensions add freedom, but shared projections still impose shared constraints.

Target dimensionalityExperiment

Reconstruction grid

Read the hidden coordinate

A slice fixes spatial W. “All W projected” collapses W onto XYZ and overlays the slices; it is not a time average or a physical time evolution.

Color coding

Checking constraints

The construction · no black box

H = Ω ∩ ⋂ Pᵢ⁻¹(Sᵢ)

Sᵢ is a requested silhouette or volume; Pᵢ is a linear orthographic projection. Every forbidden target cell removes its entire inverse image from the 4D domain Ω = [−1,1]⁴.

Forward-project the surviving hull H. If an enabled target cell is still missing, no subset of this hull can restore it. The targets are incompatible within this finite grid and domain. Exact continuous geometry can differ at cell boundaries.

When all active targets match, H is the largest feasible solution. It usually contains more than the hidden source. Extra views reduce ambiguity; the shape is not uniquely recovered just because the projections match.

From shadows to animated fields

In 3D, several prescribed 2D shadows can carve away material needed by another view. Three orthogonal directions have no universal compatibility guarantee; additional directions impose further constraints.

Moving to 4D allows 3D projections with an extra null-space direction. This provides more freedom, but arbitrary target volumes need not be compatible. The two targets in the conflict preset deliberately demand incompatible data in the same view.

W is a spatial coordinate here. Calling it time does not ensure useful animation: some slices may be empty. The coverage strip makes this visible. The linked 5D Explorer and Compounded 5D Explorer experiments explore procedural fields and periodic generators, aiming for structure across coordinate directions. Looping and evenly populated slices are additional design objectives, not consequences of shadow inversion. Coordinate symmetry is different from strict rotational isotropy; this is not a physical spacetime model.