A fit that hides a change

Rigid-body collisions: the opening demo, unpacked

The three balls from Part I provide our first mechanical blowup. In an elastic collision between two balls of masses $m_1, m_2$, conservation of momentum and kinetic energy uniquely determines the outgoing velocities:

For an approaching pair, momentum and energy conservation together with the outgoing-contact condition select the usual elastic impact. With three simultaneous contacts, conservation alone leaves multiple outgoing possibilities. A contact model or a limiting procedure must provide additional information.

Configuration space in a wedge: Write the gaps between neighbors as $g_1 = x_2 - x_1 - 2r \ge 0$ and $g_2 = x_3 - x_2 - 2r \ge 0$. In mass-weighted coordinates $q_i = \sqrt{m_i} x_i$, kinetic energy is standard Euclidean $\frac{1}{2}|\dot{q}|^2$. Removing uniform translation reduces the three-ball system to a single point bouncing inside a planar wedge between the constraint walls $g_1 = 0$ and $g_2 = 0$, with wedge angle:

A sequence of pairwise collisions is a billiard path bouncing between the walls. Simultaneous impact is the corner $g_1=g_2=0$; the equation $g_1g_2=0$ describes the union of both walls.

What the bar keeps. Fix the incoming velocities and vary the ratio of two small positive gaps. The pairwise impact model gives intervals with common limiting outcomes. Equal masses give the same outgoing velocity pattern for this experiment; a light middle ball can give different patterns. With a uniform prior over the displayed gap ratio, each interval’s width is its outcome’s probability share. A different prior gives different shares.

Figure 18. Left: three balls colliding simultaneously, with the gap-ratio bar colored by outcome. Right: the same motion mapped to a single point in a wedge. Drag the marker or click an outcome below to test incoming directions.

A second difficulty: inelastic collapse. For equal masses and constant restitution $e$, the three-ball model admits trajectories with infinitely many impacts in finite time below $7-4\sqrt3\approx0.072$. The figure follows one such setup with a finite event cutoff. All’s Well That Ends Well^[19] gives a separate terminating impact algorithm. Its guarantee concerns that collision model; it is not an application of Hironaka’s theorem.

Figure 19. Inelastic collapse. Below $e \approx 0.072$, the model can accumulate infinitely many impacts in finite time. The computed trace stops at its event cutoff; separation is not asserted in the collapsing case.

A fit that needs an unreachable limit

Here is another way two parts can cancel. Change the data: now three cells should be 1 and the rest 0. This target, W, needs three simple patterns for an exact match. Insist on using only two, and an optimizer can get arbitrarily close without ever reaching it.

The trick is to make the two parts large and almost opposite. One supplies large positive numbers; the other supplies large negative numbers. Most of that cancels, leaving a small, increasingly accurate answer. Watch the sum improve while the individual pieces grow.

This is called border rank 2: the target is a limit of two-part predictions, although its rank is 3. Every finite two-part fit still has some error.

Figure 20. Press Play. This recorded fitting run tries to match W using only two simple parts. Look at each part, then their sum, then the target. The error plot falls while the size plot rises. The construction above explains why such fits can keep improving without reaching a finite exact answer.

Describe the limit directly. Keep a common pattern, the direction in which the two parts differ, and a small separation h. After expanding their difference, the formula has a finite value even at h = 0. This new description includes the limiting tensor that the original two-part model could only approach.

Figure 21. Orange uses the original two-part parameters. Violet uses the description that includes the limiting pattern at h = 0. Compare the falling loss with the size of the parameters. The second description gives this optimizer access to a finite exact endpoint.

Symmetric tensors as binary cubics: two direction records

A symmetric $2 \times 2 \times 2$ tensor corresponds to a homogeneous binary cubic with four coefficients. On the chart where the leading coefficient is nonzero, normalize it to one and write $t^3 + a t^2 + b t + c$. For distinct roots, its real symmetric rank is determined by the cubic discriminant $\Delta$:

• $\Delta > 0$: three distinct real roots, real symmetric rank 3;
• $\Delta < 0$: one real root, real symmetric rank 2;
• $\Delta = 0$: repeated root (tangent developable boundary);
• Triple root $(t - r)^3$: rank one (the twisted cubic curve $(a, b, c) = (-3r, 3r^2, -r^3)$).

The boundary $\Delta = 0$ is swept out by tangent lines to the twisted cubic—a tangent developable creased along the rank-one curve into a sharp cuspidal edge $(e^2, 2e^3)$. Transverse to the rank-one locus, the boundary has a cusp. The point-blowup calculation from Part I applies along that locus: the strict transform becomes smooth first, and the exceptional boundary requires further steps. Recording the repeated root $r$ gives the smooth Nash lift for this cuspidal boundary. This is a special example, not a termination argument for general tensors.

Figure 22. Monic cubics $(a, b, c)$. The creased boundary $\Delta = 0$ unfolds into a smooth sheet under Nash's lift by $r$. The slice panel on the right shows Hironaka resolving the cuspidal crease step-by-step.

Cross fields: topology constrains what can be repaired

A quad mesh needs a cross field: four directions spaced by quarter turns at each regular point. Tangent boundary alignment on a disk forces total cross-field index $+1$. The representation $u=e^{4i\theta}$ has boundary degree 4. Four simple index-$+1/4$ defects realize that total; they can also merge into a higher-index defect. Topology fixes the total index, not a universal lower bound of four distinct points.

Figure 23. Drag singular points on a disk. The bars show a point-vortex energy approximation with calibrated core costs. Compare a merged degree-4 defect with four degree-1 defects. The leading core cost is quadratic in degree, while pair interactions depend logarithmically on separation.

Core structure and energy: Standard numerical optimizers minimize the Ginzburg–Landau energy:

Here $u$ is a complex representation of the cross, not the cross angle itself. Away from the cores, $|u|\approx1$; at a zero, its argument—and hence the cross direction—is undefined. For an isolated axisymmetric degree-$d$ vortex, the leading energy is $\pi d^2\log(1/\varepsilon)$. This favors splitting a degree-4 core into four degree-1 cores when the boundary and interactions permit it.

Figure 24. A Ginzburg–Landau vortex. Left: cross field over $|u|$. Middle: energy per logarithmic radial interval, which approaches a plateau outside the core. Right: four simple cores cost substantially less than one 4-fold core, driving core splitting.

Three strategies for frame fields:

• Ginzburg–Landau Penalty: Allows the field amplitude $|u|$ to soften to zero at the core. The penalty parameter $\varepsilon$ controls the core size.
• Drill a hole (Blowup): Changes the domain to an annulus: the winding can live on the inner boundary. Shrinking a circular boundary while retaining its angle is related to a real oriented blowup. A finite drilled hole is not an algebraic resolution.
• Lift to a sphere (Burer–Monteiro): Changes the target to unit vectors $v\in S^2$. The field can leave the equator; at a pole its planar projection is zero, so projecting back still leaves an undefined cross direction. This relaxation can improve optimization without removing the original topological obstruction.

Figure 25. Three strategies side by side: Ginzburg–Landau splitting (left), drilled hole blowup (middle), and Burer–Monteiro sphere lift (right).