IVAN BESEVIC / RESEARCH

Mathematics in motion.

Interactive animations: a geometric cube-root construction, actual Pandrosion 449 runs and stereographic projection onto a sphere.

Reading a formula, watching a construction and following numerical evaluations offer complementary ways to understand a result. These scenes connect the papers, published code and limitations of the methods.

01 / CONSTRUCT

Two square roots.
One cubic correction.

Construct an approximation to ∛2 with two mean proportionals, then read the result in the Pandrosion rectangle.

FIRST MEAN PROPORTIONAL
ℓ = √(109/13)
SECOND MEAN PROPORTIONAL
q = √(2 + ℓ/108)
Lengths determine the points; no known cube root is used to construct the figure.

Construction from Geometric Root Calculators, Section 4.2, equation (12). Rational preparations are grouped together; the six sequences are not the manuscript’s drawing-operation count. Values are displayed in finite precision.

View the construction on Zenodo ↗
02 / SOLVE

Follow an actual
449 run.

The landscape represents |z³ − 1|. Its three minima are the known roots. Replay the best candidates observed during calls to the original solver.

z³ − 1 = 0 · one complex variable

A recorded trajectory, a landscape to explore.

3D loads when you request it. Animations respect your reduced-motion preference.

03 / EXPLORE

Infinity comes closer.
The plane remains the plane.

A line from the North Pole N connects a planar point P to its image on the sphere. As P moves farther away, its image approaches N without reaching it at finite coordinates.

N

An unbounded plane, a sphere without its North Pole.

Move the sliders or open the scene to explore the projection.

Compactifying the representation helps organize exploration. It does not make a finite-budget search exhaustive, remove numerical difficulties near the pole or prove convergence for every system.

Return to the solver and its limitations →

Diagrams and explanations are accessible without 3D. Three.js scenes are hosted on this website, without a visualization CDN. Three.js license.