Drag the dots to change λ and λ̃ . Hold Shift to see the allowed region in each panel, or click Regions to keep both visible. Drag panel headers to rearrange them and their lower-right corners to resize.
Each dot chooses a 2-dimensional subspace, λ ⊂ C or λ̃ ⊂ C⊥ , while the cyclically symmetric C ∈ Gr>0 (3,6) and the weighted graph stay fixed. The matrices show bases of these planes. Red hatched triangles have reversed orientation.
Angle conditions: at every interior vertex, the angles of black triangles sum to 180° , and those of white triangles also sum to 180° . At every boundary vertex, each color’s angle sum is at most 180° .
The origami–amplituhedron correspondence says, for this example: the picture is embedded and satisfies these angle conditions ⇔ (λ, λ̃) belongs to the momentum amplituhedron slice shown here—the product of the two green hexagons . See Theorem 1.3 in arXiv:2410.09574
Close Two planes, one fixed C Drag either point, or click anywhere in its panel. Hold Shift while dragging to shade the admissible region of each individual plane: all six consecutive minors are positive and the winding is 2π for λ, or 4π for λ̃. Each region and point color depends only on that plane. Regions keeps both regions visible. Red triangles have reversed their orientation relative to the cyclic drawing.
Drag panel headers to arrange them; drag lower-right corners to resize. Glide controls how slowly the points follow your cursor. The outlined point is the target; the solid point is the value being drawn.Matrices (or M ): show/hide the matrix displays. R : cyclic reset. F : fullscreen. Arrow keys nudge the focused parameter point (Alt for a smaller step).Scroll over a parameter panel to change its range. Scroll over the drawing to zoom. Drag the drawing to pan. Fit resets its view. Coordinates (j = 0, …, 5) cj = cos(πj/3), sj = sin(πj/3), 1j = 1j = cos(2πj/3), ej = sin(2πj/3), aj = (−1)j ⊥ = span(d, e, a)
The matrices beneath the panels show the actual 2 × 6 bases used in the drawing, including the common SL₂ change of basis when enabled. Entries are rounded to two decimals (one decimal in narrow panels); computation uses full precision. Each matrix represents its row span. The origins are the cyclically invariant 2-planes. Each panel is an affine chart in Gr(2,3) ≅ ℝP²: λ’s chart omits the planes containing 1; λ̃’s chart omits those containing a. All 20 ordered 3 × 3 minors of the displayed basis of C are positive.
Graph and construction The fixed reduced top-cell graph has trip permutation i ↦ i + 3 modulo 6. Its dual has 10 vertices and 12 triangles, and the cyclic drawing has threefold symmetry. The graph and its linear extension maps are computed using Pavel Galashin’s SageMath implementation. No Sage installation or internet connection is needed to run this file.
With the rows of λ denoted (λ₁, λ₂), and those of λ̃ denoted (λ̃₁, λ̃₂), the boundary edge increments are (λ₁ⱼ + iλ₂ⱼ)(λ̃₁ⱼ − iλ̃₂ⱼ). Interior edge increments are products of two complex-valued functions on the graph, obtained by extending the boundary data. Summing the signed increments determines the vertices of the drawing. The chosen bases vary continuously; their signs are never reset after a minor vanishes.
Balance SL₂ A common matrix g ∈ SL₂(ℝ) acts on both 2 × 6 bases. Writing S = gᵀg, the total absolute triangle area has the form tr(MS) for a 2 × 2 positive semidefinite matrix M. When M is positive definite, its minimum is attained at S = √det(M) M⁻¹. On an embedded drawing this is the same area objective as the area minimization in the SageMath implementation. After triangles flip, the absolute areas are added separately to keep the drawing useful. Eigenvalues of S are capped at 20 and 1/20 near degeneracy; the drawing reports when this cap is used.
What the parameter colors mean The parameter panels use equation (1.8) of the paper. Green means ⟨i,i+1⟩ > 0 for λ (respectively [i,i+1] > 0 for λ̃), for all i modulo 6, and winding 2π (respectively 4π). Winding is the sum of the six counterclockwise angles, each in (0,π), from column i to column i+1. Amber points lie on the boundary within numerical tolerance. Red means the condition fails. Moving one point never changes the other plane’s region or point color. Changing the SL₂ basis also leaves these conditions unchanged.
For the present cyclic C and the displayed charts, both admissible regions are the open hexagon
|y| < √3/2, √3|x| + |y| < √3
Indeed, λ’s consecutive minors are √3/2 + x(sᵢ₊₁ − sᵢ) + y(cᵢ − cᵢ₊₁). For λ̃ replace (x,y) by (u,−v). Their positivity gives the displayed hexagon. Its strict interior is convex and contains the origin. Along a straight path to the origin all consecutive minors stay positive, so the winding cannot change; it is 2π for λ and 4π for λ̃ at the origin. The boundaries are drawn as straight segments directly from these inequalities, without sampling a grid. Boundary points are excluded.
Angle conditions and the origami–amplituhedron correspondence At an interior vertex, add the angles of all incident black triangles and separately the angles of all incident white triangles. Both sums must be 180°. At a boundary vertex, each sum must be at most 180°; for the nondegenerate t-embeddings represented by the open green hexagons, both sums lie strictly between 0° and 180°. Equality occurs on the boundary of the admissible region.
For the example displayed here, the origami–amplituhedron correspondence identifies the pictures that are embedded and satisfy these angle conditions with the pairs (λ, λ̃) in the indicated momentum amplituhedron slice. With C fixed, this slice is the product of the two green hexagons: each point must be in its own hexagon. A picture may remain embedded after leaving this slice, while violating a boundary-angle condition.
What the drawing checks The drawing separately checks triangle orientations, angle sums of each color, and nonincident edge intersections. Red hatched triangles have reversed orientation. A plane can leave its admissible region before any triangle flips: a boundary angle condition can fail first. Thus “no flips” alone does not assert a t-embedding. The status checks both the angle conditions and global injectivity of the displayed picture.
Definitions and construction: Galashin, Amplituhedra and origami , equation (1.8), Definitions 1.2–1.3 and §5.2.