Zindler Carousels in Floating-Body Geometry

• Zindler carousels are geometric–dynamical structures on convex plane domains where sliding inscribed equilateral polygons represent equilibrium conditions.
• They reduce the continuum floating-body problem to finite polygonal configurations, highlighting key roles of rational perimetral densities such as 1/2 and 1/6.
• Experimental and affine-differential studies reveal rigidity phenomena and links to bicycle curves and elastica, deepening insights into shape equilibrium.

Searching arXiv for recent and foundational papers on Zindler carousels and related floating-body geometry. {"query":"all:(\"Zindler carousel\" OR \"Zindler carousels\" OR \"floating body\" planar Ulam perimetral density)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} I’m going to retrieve arXiv records relevant to Zindler carousels, Zindler curves, and the planar floating body problem. Zindler carousels are geometric–dynamical structures attached to strictly convex plane domains in the planar floating-body problem at rational perimetral density. If $K$ has perimeter $P$ and arc-length parametrization $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$, then for $\sigma=1/N$ one sets $\mu=P/N$ and defines

$v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$

These points form a Zindler carousel when the inscribed polygon $V(t)$ with vertices $v_1(t),\dots,v_N(t)$ is equilateral with side length $\ell$ independent of $t$. As $P$0 varies, the equilateral $P$1-gon slides along $P$2, and this sliding polygon is equivalent to the condition that $P$3 float in equilibrium in every orientation with perimetral density $P$4 (Asipchuk et al., 11 Apr 2026).

1. Geometric definition and floating-body equivalence

In the perimetral model, mass is distributed uniformly along the boundary $P$5. The equilibrium axiom states that for every direction of the waterline, the line splits the boundary into two arcs whose lengths are in a fixed ratio $P$6. For $P$7, the carousel construction replaces this global equilibrium condition by a moving inscribed equilateral $P$8-gon with vertices equally spaced in arc length along $P$9 (Asipchuk et al., 11 Apr 2026).

The precise equivalence is that $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$0 floats in equilibrium in every orientation with perimetral density $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$1 if and only if there exists $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$2 such that for every $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$3, the points $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$4 form an inscribed equilateral $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$5-gon with side length $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$6. The side midpoints

$\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$7

then satisfy the midpoint-parallel property

$\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$8

which encodes torque balance along the waterline. In this formulation, force balance is matched by the fixed arc-length ratio, while torque balance is expressed by the kinematics of the moving side midpoints (Asipchuk et al., 11 Apr 2026).

This equivalence makes the carousel a reduction device: a continuum hydrostatic condition becomes a finite-dimensional geometric constraint. The approach is especially effective for rational densities, where the waterline geometry can be encoded by polygonal configurations rather than by arbitrary chord families.

2. The classical case $\gamma:\mathbb{R}/P\mathbb{Z}\to\mathbb{R}^2$9: Zindler curves, equal chords, and related geometries

Zindler curves are the $\sigma=1/N$0 case. For a strictly convex curve $\sigma=1/N$1 of perimeter $\sigma=1/N$2, the defining property is that every chord whose endpoints are separated by half the perimeter has the same length: $\sigma=1/N$3 is independent of $\sigma=1/N$4. This equal-chord property is equivalent to floating in equilibrium at perimetral density $\sigma=1/N$5, and it implies that the midpoint motion is parallel to the chord, which is the classical Zindler characterization (Asipchuk et al., 11 Apr 2026).

A complementary formulation uses the parametrization

$\sigma=1/N$6

with

$\sigma=1/N$7

Here the boundary points at parameters $\sigma=1/N$8 and $\sigma=1/N$9 are joined by a chord of constant length $\mu=P/N$0. This diameter bisects the perimeter and, under sufficient convexity, bisects the enclosed area as well. The two halves satisfy

$\mu=P/N$1

with $\mu=P/N$2 independent of $\mu=P/N$3, and their centroids obey

$\mu=P/N$4

Thus the line joining the two centers of gravity is always normal to the chord and has constant length (Wegner, 2019).

The same paper places Zindler geometry within a broader family of curve theories. It emphasizes that Bor–Levi–Perline–Tabachnikov identified the coincidence between the floating-body equation in two dimensions and the elastica-under-pressure equation, so many buckled rings furnish floating-body boundaries. It also notes that Zindler curves are bicycle curves: the one-parameter family of fixed-length chords sliding along the boundary corresponds to the rear-wheel tangent construction, with the chord midpoints tracing the envelope (Wegner, 2019). This shows that the carousel mechanism is not merely hydrostatic; it is also a chordal kinematics shared by floating bodies, elastica under pressure, and the bicycle problem.

3. Rational densities and the hexagonal rigidity mechanism at $\mu=P/N$5

For general $\mu=P/N$6, after normalizing the side length to $\mu=P/N$7, the interior angles $\mu=P/N$8 of the inscribed equilateral $\mu=P/N$9-gon satisfy

$v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$0

with

$v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$1

This already converts the floating-body problem into an ODE on polygonal angle data (Asipchuk et al., 11 Apr 2026).

The case $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$2 is special because the carousel is an inscribed equilateral hexagon. The paper shows that the hexagon is centrally symmetric, and consequently $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$3 itself is centrally symmetric with fixed center $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$4 independent of $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$5. This collapses the $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$6 system to a $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$7 system and then to a $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$8 system for $v_i(t):=\gamma(t+(i-1)\mu),\qquad i=1,\dots,N.$9 and $V(t)$0: $V(t)$1 The reduced system is Hamiltonian with

$V(t)$2

and the area $V(t)$3 of the inscribed hexagon satisfies

$V(t)$4

Hence area is conserved; this is identified as Auerbach’s invariant (Asipchuk et al., 11 Apr 2026).

Convexity restricts the dynamics to

$V(t)$5

On $V(t)$6, $V(t)$7 is strictly concave with a unique maximum at $V(t)$8, corresponding to the circle. Passing to

$V(t)$9

the dynamics reduce further to a one-dimensional equation

$v_1(t),\dots,v_N(t)$0

with period

$v_1(t),\dots,v_N(t)$1

The paper proves the sharp bounds

$v_1(t),\dots,v_N(t)$2

A direct geometric computation also yields

$v_1(t),\dots,v_N(t)$3

where $v_1(t),\dots,v_N(t)$4 and $v_1(t),\dots,v_N(t)$5 is the perimeter of $v_1(t),\dots,v_N(t)$6 (Asipchuk et al., 11 Apr 2026).

The rigidity step combines these period bounds with rotational symmetry quantization. If $v_1(t),\dots,v_N(t)$7, then the minimal period satisfies

$v_1(t),\dots,v_N(t)$8

The resulting inequalities force $v_1(t),\dots,v_N(t)$9, hence $\ell$0, but the analytic upper bound gives $\ell$1, a contradiction. Therefore, for $\ell$2, no noncircular convex body can float in equilibrium in every orientation; the only solution is the disk (Asipchuk et al., 11 Apr 2026).

Placed in context, this extends earlier rigidity results for $\ell$3 and $\ell$4, while the same paper recalls numerical evidence of nonexistence for $\ell$5 and $\ell$6.

4. Affine Zindler carousels and homothety rigidity

A recent affine-differential reformulation replaces Euclidean arc length and Euclidean chord invariants by equi-affine arc length and signed affine distance between linear elements. For a non-degenerate curve $\ell$7, the affine arc-length parameter is

$\ell$8

and for a Euclidean arc-length parametrization $\ell$9,

$t$0

In parallel, for a convex body $t$1, the body of flotation $t$2, the body of buoyancy $t$3, and the body of illumination $t$4 are defined from water lines, buoyancy centroids, and silhouette cones, respectively (Zawalski, 16 Jul 2025).

In this setting, the key homothety theorem states that

$t$5

if and only if

$t$6

where $t$7 is a chord of flotation and $t$8 is the signed affine distance between the endpoint linear elements. A second theorem shows that this homothety is equivalent to constancy of the cut-off affine arc length: $t$9 or, equivalently,

$P$00

These are the affine counterparts of Auerbach’s classical Euclidean equal-length and equal-cutoff statements (Zawalski, 16 Jul 2025).

Affine Zindler carousels are then formulated for rational perimetral density $P$01 using a $P$02-tuple $P$03. The side conditions are

$P$04

together with the equal affine arc-length condition given by vanishing of

$P$05

For the affine density $P$06, the paper proves that if $P$07 is homothetic to $P$08 and every chord of flotation cuts off exactly one third of the total affine arc length, then the centroid $P$09 of the associated “3-chair” triangle is fixed and $P$10 is homothetic to $P$11 with center $P$12 and ratio $P$13. By the affine-sphere argument and the Blaschke–Deicke theorem, this forces $P$14 to be an ellipse (Zawalski, 16 Jul 2025).

5. Experimental realization and the visible carousel of waterlines

An experimental realization has made the classical $P$15 theory directly observable. In that setting, a Zindler curve is a closed planar curve for which every chord dividing the enclosed area into two equal parts has the same length. As the body rotates through an angle $P$16, the waterline sweeps a one-parameter family of equal-area chords. Following Bracho–Montejano–Oliveros, this family is called a carousel. When those chords all have identical length $P$17, the body exhibits orientation-independent equilibrium at effective density $P$18 (Pontiggia et al., 2 Apr 2026).

The hydrostatic formulation is explicit. If $P$19 is the total area and $P$20 is the submerged area, then

$P$21

so at neutral density,

$P$22

The potential energy is

$P$23

and neutral rotational equilibrium requires $P$24 for all $P$25. The paper states that the Zindler condition guarantees that the submerged centroid $P$26 remains on the vertical through the body centroid $P$27, so no restoring torque arises. For an infinitesimal rotation,

$P$28

which makes the constancy of $P$29 the mechanism behind torque neutrality (Pontiggia et al., 2 Apr 2026).

The realized object is a heart-shaped Zindler curve inspired by Auerbach’s 1938 construction. It consists of a thin 3D-printed black profile of the heart’s contour, less than $P$30 mm thick, sandwiched between two transparent PMMA plates. Liquid density was tuned with water–ethanol mixtures; near-neutral behavior was observed at $P$31, consistent with a small capillary downward force. When oriented arbitrarily with thin rods and released, the floater did not drift to a preferred angle. Superposition of many orientations showed bright waterline segments forming the carousel, a triangular caustic envelope, and reconstructed submerged centroids $P$32 lying on a circle centered at $P$33. Quantitatively, the waterline length was nearly constant, with mean $P$34 mm and $P$35 mm variation, close to the $P$36 cm theoretical value, and at $P$37, $P$38 lay on the waterline within $P$39 mm (Pontiggia et al., 2 Apr 2026).

Away from $P$40, the neutral carousel gives way to an energy landscape. For $P$41 and $P$42, the experiment found three stable orientations approximately $P$43 apart; the stable angles for $P$44 were shifted by about $P$45 relative to those for $P$46. Release from a non-equilibrium angle produced damped oscillations toward the nearest minimum, with fitted angular frequencies $P$47 Hz and $P$48 Hz, and one higher $P$49 Hz oscillation consistent with a stiffer vertical mode. The same study emphasizes that density inhomogeneity and capillarity are not negligible perturbations near neutrality (Pontiggia et al., 2 Apr 2026).

6. Conceptual scope, distinctions, and open problems

Zindler carousels are not the same as constant-width geometry. A shape of constant width is defined by constant separation of parallel support lines, whereas Zindler curves require that all area-bisecting chords have the same length. The circle satisfies both properties, but the families are distinct; the heart-shaped Auerbach example is explicitly cited as non-circular and not of constant width (Pontiggia et al., 2 Apr 2026).

The equilateral constraint in the carousel definition is essential. For $P$50, the constant side length $P$51, independent of $P$52, is what makes the hexagonal reduction possible; inscription on $P$53 forces the chord endpoints to follow the boundary, and the midpoint-parallel property guarantees torque balance. The paper on $P$54 identifies two obstacles for larger $P$55: the angles $P$56 generally cannot be expressed uniquely in terms of a small set of interior angles $P$57, and the resulting ODE system typically has higher dimension, making global analysis and period bounds harder (Asipchuk et al., 11 Apr 2026).

Several problems remain open. Numerical evidence suggests monotonicity of the period function $P$58 along Hamiltonian level sets, but a rigorous proof is still open. For general rational densities $P$59 with $P$60, the carousel must incorporate alternating arc constraints rather than equal spacing, which complicates both the geometry and the dynamics. In the affine theory, the density $P$61 admits a complete rigidity theorem, but analogous statements for other densities such as $P$62 are described as more involved, and an explicit synthetic proof is not obtained there (Zawalski, 16 Jul 2025).

This suggests a layered picture of the subject. At $P$63, non-circular Zindler curves exist; at $P$64, convex rigidity yields the disk; in the affine $P$65 setting, the homothety hypotheses force the ellipse; and experimentally, the $P$66 carousel can be visualized directly as a family of equal-area, equal-length waterlines. Across these settings, the unifying content of the term “Zindler carousel” is a sliding family of chords or polygon sides whose constrained motion encodes equilibrium, invariance, and, in several regimes, rigidity (Asipchuk et al., 11 Apr 2026)