Equimodular Elliptic Type Polyhedra

• Equimodular elliptic type is a flexible Kokotsakis polyhedra class defined by elliptic function relations and characterized by equal moduli and amplitude matching.
• The methodology employs closed-form equations, strict period conditions, and QS-net constructions for explicit numerical and algebraic validations.
• These structures enable the computational synthesis of mechanisms with precise angular and geometric controls, supporting both self-intersecting and non-self-intersecting designs.

The equimodular elliptic type is a canonical class of flexible Kokotsakis polyhedra with a quadrangular base, distinguished by rich geometric and algebraic properties. Within Izmestiev’s eight principal types of flexible Kokotsakis polyhedra, the equimodular class encompasses both “elliptic” and “conic” subtypes, with the elliptic subtype notable for being governed by elliptic rather than purely trigonometric relations. Recently, explicit algebraic constructions and characterizations of the equimodular elliptic type have been developed, including constructible classes such as quasi-symmetric nets (QS-nets), which facilitate a synthetic and computational approach to this class of flexible mechanisms (Nurmatov et al., 24 Nov 2025).

1. Definition and Characterization

A Kokotsakis polyhedron of equimodular elliptic type is a 3×3 net of points $V_{ij}\in\mathbb{R}^3$, with convex quadrilateral faces, central face $A_1A_2A_3A_4$, and associated flat angles $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$ and dihedral angles $\theta_i\in(0,\pi)$, $i=1,\dots,4$.

An elliptic-type net satisfies the non-degeneracy condition: $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$ for all sign combinations and all $i$. This prevents trigonometric singularities.

The equimodular elliptic type imposes further conditions ((Nurmatov et al., 24 Nov 2025), Definitions 2.3 and 2.4):

• (a) Equal moduli: For $i=1,2,3,4$,

$M_i:=a_ib_ic_id_i=M$

where

$$a_i=\frac{\sin\alpha_i}{\sin(\sigma_i-\alpha_i)},\quad b_i=\frac{\sin\beta_i}{\sin(\sigma_i-\beta_i)},\quad c_i=\frac{\sin\gamma_i}{\sin(\sigma_i-\gamma_i)},\quad d_i=\frac{\sin\delta_i}{\sin(\sigma_i-\delta_i)},$$

and $A_1A_2A_3A_4$0.

• (b) Amplitude matching:

$A_1A_2A_3A_4$1

where $A_1A_2A_3A_4$2, $A_1A_2A_3A_4$3.

• (c) Period condition: Certain amplitudes $A_1A_2A_3A_4$4 satisfy

$A_1A_2A_3A_4$5

for sign choices $A_1A_2A_3A_4$6, $A_1A_2A_3A_4$7 the elliptic period lattice.

This structure distinguishes the equimodular elliptic type from all other classes in the established flexible polyhedra classification.

2. Distinction from Other Flexible Types

Izmestiev’s classification enumerates eight types of flexible Kokotsakis polyhedra. The equimodular class, split into elliptic and conic subtypes, is unique for its elliptic function parameterization of all flexions, unlike types governed by merely trigonometric/rational relations. Only the equimodular elliptic type admits a nontrivial four-parameter family of continuous flexions in closed form ((Nurmatov et al., 24 Nov 2025), Section 2).

This type encompasses the richest geometric structures, including families not realized by constructions of other types. The closed-form description enables explicit synthesis, computational analysis, and the construction of mechanisms with prescribed flat or dihedral angular data, subject to the algebraic existence criterion (see Section 3).

3. Algebraic Existence Criterion

A 3×3 net with prescribed flat angles $A_1A_2A_3A_4$8 and dihedral angles $A_1A_2A_3A_4$9 is of equimodular elliptic type if and only if there exist real parameters

$\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$0

with $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$1 such that nine explicit algebraic equations (indices mod 4) are satisfied ((Nurmatov et al., 24 Nov 2025), Proposition 2.8): $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$2

$\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$3

$\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$4

$\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$5

$\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$6

plus the period condition involving Jacobi-delta amplitudes and the elliptic lattice. The variables $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$7, $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$8, and the definitions of $\alpha_i,\,\beta_i,\,\gamma_i,\,\delta_i\in(0,\pi)$9 are as given in the original analysis.

Geometrically, these equations encode face compatibility (planar angle sum), matched moduli and amplitudes, and Bricard's equations linking flat and dihedral angles.

4. Quasi-Symmetric Nets (QS-nets) as Constructive Subclass

A quasi-symmetric net (QS-net) is a 3×3 net in which the flat angles are arranged with specific pairwise equality or complementarity patterns ((Nurmatov et al., 24 Nov 2025), Definition 2.2): $\theta_i\in(0,\pi)$0 QS-nets admit an explicit closed-form parameterization of their flexion (Theorem 2.9), and every elliptic QS-net is of equimodular elliptic type, except at discriminant vanishing points where the realization degenerates.

Under the QS ansatz, the flexion can be written in terms of a single real parameter $\theta_i\in(0,\pi)$1: $\theta_i\in(0,\pi)$2 with $\theta_i\in(0,\pi)$3 as defined in the original source. For real $\theta_i\in(0,\pi)$4 with $\theta_i\in(0,\pi)$5 and non-vanishing denominators, the one-parameter family defines a flexible net in $\theta_i\in(0,\pi)$6.

5. Exceptional Cases and Loss of Flexibility

Flexibility of the equimodular elliptic type is generically preserved, but degenerates precisely at values $\theta_i\in(0,\pi)$7 where the discriminant

$\theta_i\in(0,\pi)$8

or denominators in the closed-form expressions vanish. Geometrically, this occurs when certain strips become planar or folded so that one dihedral angle reaches $\theta_i\in(0,\pi)$9 or $i=1,\dots,4$0, degenerating the net. These exceptional configurations delimit the interval of real $i=1,\dots,4$1 for isometric flexion.

6. Computational Synthesis and Verification

An explicit numerical pipeline for discovering and verifying equimodular elliptic nets is as follows ((Nurmatov et al., 24 Nov 2025), Section 3):

1. Prescribe $i=1,\dots,4$2, $i=1,\dots,4$3 with $i=1,\dots,4$4.
2. Solve the algebraic system for unknowns $i=1,\dots,4$5, respecting imposed linear symmetries.
3. Use random initial guesses in admissible ranges (e.g., $i=1,\dots,4$6; $i=1,\dots,4$7).
4. Employ a robust root solver (e.g., Levenberg–Marquardt) on the squared polynomial system.
5. For each solution:
   • Discard if out of range.
   • Recover angles and check them for geometric validity.
   • Test ellipticity and period conditions to high numerical precision ($i=1,\dots,4$8 to $i=1,\dots,4$9).
   • Eliminate duplicates under sign symmetries.
6. Output all valid parameter sets and reconstruct the geometry.

Calculated moduli $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$0, period, and angle matches reach numerical tolerances of up to $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$1–$$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$2, ensuring reliably validated constructions.

7. Explicit and Numerical Examples

The explicit and numerical construction of equimodular elliptic type polyhedra is illustrated by several examples ((Nurmatov et al., 24 Nov 2025), Section 4):

• Closed-form QS-net: For

$$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$3

with the specified symmetry relations, formula (12b) provides a family of non-self-intersecting QS-nets, flexible for all $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$4.

• Numerical ($$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$5): With $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$6, $$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$7, the pipeline yields

$$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$8

corresponding to fully validated net data.

• Numerical ($$\alpha_i \pm \beta_i \pm \gamma_i \pm \delta_i \neq 0 \pmod{2\pi}$$9), dual case: For $i$0, $i$1, a solution with $i$2 is handled via the “dual” elliptic period lattice, yielding valid flexions within a restricted parameter range.

These constructions can yield both self-intersecting and non-self-intersecting mechanisms depending on choice of signs in the closed-form dihedral families.

Table 1: Core Conditions for Equimodular Elliptic Type Nets

Condition Type	Equation/Relation	Description
Ellipticity	$i$3 (all $i$4)	Prevents singular trigonometric cases
Moduli Equality	$i$5	Fourfold product criterion
Amplitude Matching	$i$6	Partial product matching
Period Condition	$i$7	Elliptic function closure

The equimodular elliptic type thus occupies a pivotal position in the geometric theory of flexible Kokotsakis polyhedra, characterized by deep symmetry, closed-form isometric flexion, and explicit algebraic realization, with constructions and verification now accessible via computational techniques (Nurmatov et al., 24 Nov 2025).