# Attractive Centers — geometry explorer **Live app:** https://ismaa3iil.fyi/attractive-centers/explorer/ Free software by Ismail Hammoudeh, MIT license. The license covers this directory, not the PDF papers in its parent directory. ## Use - **Triangle:** drag any vertex. Gold is the standard chest; green is the hull of the selected attractive family. Hover over green points for their names. - **Four planar points:** independently toggle parallelogram, bisector, network, and stability constructions. Stability is OFF initially: it needs absorption. - **Tetrahedron:** drag empty space to rotate; drag a vertex in the screen plane. Exact x, y, z coordinates are editable. Scroll to zoom. Arrow keys rotate; the `0` key fits the view. Planar coordinate scales are equal. Vertex positions do not jump when overlays are selected. The viewport remains fixed during dragging; “Fit view” recenters it. No account, backend, external library, or telemetry is used. Calculations run in a Web Worker. Brief theory and live percentages are displayed beside the plot. ## Mathematics and scope Every half-space is written `n ⋅ x ≥ b`. The construction requires symmetry, attractivity along straight vertex motions, and continuous extension to the degenerate configurations used as predecessors. It does **not** assume that arbitrary nonparallel displacements transport a common supporting inequality. ### Triangle: outer chest Intersect the convex hull with (1) the three strips between the perpendicular bisector of each side and its parallel altitude, and (2) the anticomplementary triangle of the **internal** Napoleon centroids. The internal equilateral apex lies on the same side of a side as the opposite vertex. For an equilateral triangle, this construction reduces to the centroid. The strip overlay is clipped to the original triangle; the Napoleon triangle is shown unclipped. ### Triangle: attractive inner family The green polygon is the convex hull of X(1), X(10), X(2), and 49 power centers M_p with equally spaced p in `[4−2√2, 4+2√2]`. M_p minimizes `sum(norm(x−p_i)^p)`; the computation uses its gradient, Hessian and a damped Newton method. Papers III and IV provide the attractivity results. “Best” means the extreme points of this sampled family for the current triangle, not a proof of extremality in the class of all attractive centers. The power interval is the sharp **universal** guarantee for the power family; this does not assert that every endpoint is a boundary point of each triangle's Heart. X(1)–X(10) supplies the sharp segment on its centroidal ray. Numerical power samples whose convergence check fails are omitted and counted in the UI. No center has been clipped or projected into the standard chest to hide a failure. The older 50-point ETC base was generated by scans with subsequently identified evaluation problems. It is not included as certified data. Noninteger power centers also need not be analytic at every configuration: this green region uses the regularity/extension conventions for radial centers in Paper III, and is not labeled as an exact maximal **analytic** Heart. ### Four points: four constructions 1. **Parallelogram.** For each pair partition `ij | kl`, put `m=(p_i+p_j)/2`, `q=(p_k+p_l)/2`, `d=m−q`. Retain the slab `min(d⋅m,d⋅q) ≤ d⋅x ≤ max(d⋅m,d⋅q)`. Translating the other pair to equal midpoints gives a centrally symmetric predecessor; parallel restoration proves the bound. Equal midpoints force the center itself, not merely a vacuous zero-normal inequality. 2. **Bisectors.** Set `u=p_j−p_i`, `m=(p_i+p_j)/2`. If both other points have the same sign σ of `u⋅(p−m)`, impose `σu⋅(x−m) ≥ 0`. Project the other points onto the bisector, then restore them in the same parallel direction. If both lie on the bisector, symmetry gives equality; the implementation imposes both half-spaces. 3. **Two-order network.** For every labeling and both equilateral apices C0 on AB, let D0 be the centroid of ABC0, `u=C−C0`, `v=D−D0`. The initial four-point center is D0. If C moves first, its intermediate center belongs to `R_C = parallelogram_chest(A,B,C,D0) ∩ {u⋅(y−D0)≥0}`. Restoring D therefore gives `v⋅x ≥ min_{y in R_C} v⋅y`. Reverse the order to obtain `u⋅x ≥ min_{y in R_D} u⋅y`. Intersect the resulting inequalities for all 48 scenarios. This is a finite network relaxation, not the optimal chest. The intermediate center is retained; it is not replaced by an invalid OR of endpoint inequalities. This method uses parallelogram chests at intermediate nodes, even when the separate final-configuration parallelogram option is off. 4. **Stability / absorption (conditional).** This assumes a family across cardinalities satisfying `X4(A,B,C,X3(A,B,C)) = X3(A,B,C)`. For an omitted point p and the remaining triangle chest T, some q in T must satisfy `(p−q)⋅(x−q) ≥ 0`. At fixed x this expression is convex in q, so its maximum over the polygon T is attained at a vertex. Thus use a finite OR over vertices. Intersect the four OR clauses and the hull, then take the convex hull to obtain a conditional chest. A clause containing q=p is identically true. We intersect this convexified chest with other checked chests; convexification can weaken the result, but does not invalidate it. The optional ABC triangle chest is only a visual reference. It is **not** automatically a four-point chest and is never imposed on the four-point center. ### Three dimensions Intersect the tetrahedron's four inward facet half-spaces with the three opposite-edge slabs and the useful bisector planes. Vertices are feasible triple-plane intersections; faces are ordered in their planes. Volume is the sum of tetrahedra from an interior point to triangularized faces. These are outer constructions, not a claim to the OSC. Planar network and absorption algorithms are not silently extended to tetrahedra. ### Percentages: a reproducible reference For `A=(0,0), B=(21,0), C=(5,12), D=(26/3,4)`, the hull area is 126. Each “alone” row includes the hull and **only that final-region construction**: | Construction | Hull area retained | Hull area excluded | |---|---:|---:| | Parallelogram | 58.410491% | 41.589509% | | Bisectors alone | 21.239749% | 78.760251% | | Network alone | 31.369615% | 68.630385% | | Stability alone (conditional) | 82.260924% | 17.739076% | | First three intersected | 13.412708% | 86.587292% | For ABC alone, the standard chest retains 5.930191% and the sampled inner family retains approximately 0.398093% of the triangle's area. An inner hull's complement is **not** ruled out for attractive centers; its percentage is a size comparison only. Areas/volumes are floating-point values, not interval certificates or percentages of centers. ## Run locally and reproduce Serve this directory over HTTP (module workers cannot run from `file://`): ```sh python -m http.server 8772 # Open http://localhost:8772/ node tests.mjs julia --startup-file=no tests.jl ``` The browser engine is plain JavaScript so it can run directly on GitHub Pages. The scientific companion is genuine Julia using only `LinearAlgebra` from the standard library. Example: ```julia include("AttractiveCenters.jl") using .AttractiveCenters p = [[0.,0.], [21.,0.], [5.,12.], [26/3,4.]] r = explore_four(p; stability=false) 100 * polygon_area(r.chests.selected) / polygon_area(r.chests.hull) t = explore_triangle(p[1:3]) # Undo normalization for the chest vertices: world = [t.center + t.scale*x for x in t.chest.polygon] v = explore_tetra([[0.,0.,0.],[2.,0.,0.],[.3,1.8,0.],[.5,.6,2.2]]) ``` High-level `explore_*` functions normalize coordinates and return the inverse map `(center, scale)`. Low-level functions assume moderate coordinate scales. Regression tests cover reference percentages, equality/symmetry, similarity and permutation invariance, and seeded random configurations. Collinear triangles/four-point inputs and flat tetrahedra get explicit messages in the browser. Near-degenerate inputs still have the limitations of floating-point geometry; this is research software, not a formal proof assistant. An optional WebMCP tool configures the same vertex state when the hosting browser supports it. It waits for computation, validates input, and never publishes or transmits geometry. Unsupported browsers simply use the normal UI. ## Papers The theory follows the four [Attractive Centers papers](../), especially Paper I (symmetry-displacement chests), Paper III (radial attractivity), and Paper IV (analytic inner families and nonlocal constraints).