GEOMETRY IN MOTION
One shape. Two kinds of center.
Move the vertices and follow the centers as the power changes.
Power center Mp
Hull-power center Up
Distance between centers
COMPARE WITH KNOWN TRIANGLE CENTERS
Three nearby ETC centers
Ranked by behavior across 100 triangle shapes, checked on 40 independent shapes.
Loading the ETC comparison data…
Power center Mp
| ETC | Grid % | Check % | Here % |
|---|
Hull-power center Up
| ETC | Grid % | Check % | Here % |
|---|
At p = 2 both centers are exactly the centroid X(2). The other two entries are nearby alternatives.
Attractivity belongs to each ETC rule and is independent of the comparison power. Of 82 distinct neighbors, 33 have exact finite failure certificates, three are proved attractive, and 46 have no failure in a 983-triangle screen. The positive proofs cover X(2), X(18236), and X(56203); the latter two cover finite motions with proper triangle endpoints. A numerical pass is not an all-shape proof. Nearby ETC centers can fail even when the hull-power center is proved attractive.
Revised article · PDF · LaTeX source · Neighbor research notes
Every percentage uses the longest side as 100%. Grid % determines the rank; Check % tests it independently; Here % describes your current triangle. These are the best three among 70,853 complete finite records in PITC’s 72,807-entry snapshot. Rankings at other powers are not interpolated. Proximity does not establish identity or transfer attractivity.
Download all 33 comparisons · CSVComparison method, attractivity tests, and attribution
No explicit barycentric formula for a power center is needed: its numerically solved Cartesian position is compared directly with each ETC formula on the same triangles. The score is √mean[(distance / longest side)²]. The 100-shape equal-area angle grid and 40-shape holdout are the same as PITC’s positional comparison. The independent holdout does not determine the ranking. Unsupported or singular formulas are excluded rather than assigned a centroid.
The independent attractivity screen differentiates each ETC formula with respect to all three vertices and checks the symmetric response matrices on 983 proper triangles, including thin and obtuse families. Negative responses are converted to finite motions and verified with exact rational interval arithmetic. Square roots are enclosed using integer square roots; the final dot-product upper bounds are strictly negative. X(7934) also has a short algebraic counterexample on an isosceles triangle. X(18236) and X(56203) have exact positive-coefficient certificates after substituting the triangle inequalities. Download response results · CSV. Download exact negative certificates · JSON. Download positive certificates · JSON.
Research and rankings: Ismail Hammoudeh, with extensive use of ChatGPT. ETC definitions and original formulas: Clark Kimberling and the Encyclopedia’s contributors. Only the selected formulas are included; the full ETC database is not distributed. External formula notices.
The journey of the centers
Coordinate values against power. Click the chart to select p.
LAST VERTEX MOTION
Attractivity probe
After a drag, compare the center displacement with the direction you moved the vertex.
The quantity is h · (center after − center before), evaluated at the current power. A negative value is a numerical indication of failure for that motion.
Mathematics, proof status, and how this explorer works +
The two definitions
The power center minimizes the sum of distances to the vertices raised to p. The hull-power center minimizes the average of distance to every point in the solid, raised to p.
Mp = arg minx ∑i ‖x − vi‖p
Up = arg minx E ‖x − ∑i λivi‖p
The barycentric weights λ are uniform on the reference simplex. This same measure is retained at flat shapes and coincident vertices. At p = 2 both centers are the vertex centroid.
What has been proved
The universal atomic interval is exactly [4 − 2√2, 4 + 2√2]. For triangle hull-power centers, every power in [4 − 2√2, 16], and the separate powers 20 and 21, has an all-shape proof. The interval between 16 and 21 is still open.
The common hull interval for all simplex dimensions reaches 4 + 2√2 + 1/2,000,000. Observed upper failure branches occur near 21.63336 for triangles and 19.95468 for a tetrahedral family. The triangle explorer extends to 24.99 to compare centers beyond the observed failure branch; its numerical range is not an attractivity claim.
The displayed centers are floating-point approximations. The selected hull center is checked with a second quadrature order. That comparison estimates numerical stability; it is not a rigorous error bound.
Research and authorship: Ismail Hammoudeh. Developed with extensive use of ChatGPT. The explorer and its numerical tools are free, open-source software under the MIT license. External ETC definitions retain Kimberling’s attribution. No account, external service, or paid mathematics package is needed.