Project a tesseract exactly

Four dimensions are not implicitly three

CoordinateProjection names the information being discarded. The Cayley rotation and affine projection remain exact; only the final camera snapshot is numeric.

Runnable RiXRun cell

.Plugin.Load("nd"); .Plugin.Load("scene3d"); tesseract := .nd.Hypercube(4, 2); rotation := .nd.CayleyRotation(4, 1, 4, 1/3); xyz := .nd.CoordinateProjection(4, [1,2,3]); projected := .nd.Project(tesseract, .nd.Compose(xyz, rotation)); camera := .scene3d.OrthographicCamera([4,4,3], [0,0,0]); scene := .nd.ToScene3D(projected, {= camera=camera, style={= color="#7c3aed", width=2 } }); .scene3d.Snapshot(scene, {= size=[560,400] })["value"];

Calling ToScene3D on the original 4D value fails and asks for an explicit projection. Slicing and projection remain separate concepts.

The projective endpoint of a Cayley parameter is also ordinary RiX code:

Runnable RiXRun cell

halfTurn := .nd.CayleyRotation(4, 1, 4, .Complex[:infinity]); halfTurn["matrix"][1];

Parameterize an affine slice and take an exact section

Runnable RiXRun cell

.Plugin.Load("nd"); plane := .nd.AffineSlice([0,0,1],[[1,0,0],[0,1,0]]); square2d := .nd.Polyline([[-1,-1],[1,-1],[1,1],[-1,1]],{= closed=1 }); embedded := plane.Parameterize(square2d); section := .nd.Section(.nd.Hypercube(3,2),.nd.Hyperplane([0,0,1],0)); {: plane.Parameterize([2,3]),embedded[:dimension],section[:points] };

Section uses only the supplied vertices and edges. The four returned points are exact; no unrecorded face connectivity is inferred.

Evaluate fields and inspect fibers

Runnable RiXRun cell

.Plugin.Load("nd"); sum := .nd.Field(2,1,(p)->[p[1]+p[2]],{= name="sum" }); domain := .nd.Polyline([[-1,1],[0,0],[1,-1],[1,1]]); fiber := sum.Fiber([0],domain); samples := sum.Sample([[-1,-1],[-1,1],[1,-1],[1,1]]); scene := .nd.ToScene3D(samples); {: sum.Evaluate([2,3]),fiber[:points],scene };

Omit the finite domain to get an explicit implicit-preimage record rather than an unbounded hidden search.

Explore a projection family

Runnable RiXRun cell

.Plugin.Load("nd"); family := .nd.ProjectionFamily(3,2, (t)->.nd.Compose( .nd.CoordinateProjection(3,[1,2]), .nd.CayleyRotation(3,1,2,t) ), (-1):1 ); {: family.At(0)[:matrix],family.At(1/3)[:matrix],family[:parameterDomain] };

Every selected projection is dimension-checked and retains the family and parameter in its provenance.

Link projections of a bounded four-dimensional cover

Runnable RiXRun cell

.Plugin.Load("nd"); .Plugin.Load("calculus"); x := .calculus.Variable(:x); y := .calculus.Variable(:y); z := .calculus.Variable(:z); w := .calculus.Variable(:w); region := .nd.ImplicitRegion(x^2+y^2+z^2+w^2-1, {= x=(-1):1,y=(-1):1,z=(-1):1,w=(-1):1 },{= maxCells=7 }); .nd.LinkedRegions(region,[.nd.CoordinateProjection(4,[1,2,3]),.nd.CoordinateProjection(4,[1,3,4])], {= maxVisibleCells=3,size=[280,210],legendOffset=[0,80] });

Selecting a visible cell links its retained source ID in both views. Affine projection preserves enclosure evidence while allowing overlap and dependence loss. Unresolved cells and the hull of omitted cells remain visible.