A Kokotsakis polyhedron is a polyhedral mesh in three-dimensional Euclidean space formed by a central n-gonal face (the base), n quadrilateral faces each sharing one edge with the base, and n triangular faces inserted between every two adjacent quadrilaterals; it is called flexible if it admits a continuous deformation that preserves the rigidity of every face. This work investigates flexible Kokotsakis polyhedra with a quadrangular base (n = 4) of equimodular elliptic type, filling a significant gap in the literature by providing the first explicit constructions of this type together with an explicit algebraic characterization in terms of flat and dihedral angles. A straightforwardly constructible class of polyhedra - called quasi-symmetric nets (QS-nets) - is introduced, characterized by a symmetry relation among flat angles. It is shown that every elliptic QS-net has equimodular elliptic type and is flexible in real three-dimensional Euclidean space (rather than only in complex configuration spaces), except for a few exceptional choices of dihedral angles, and that its flexion admits a closed-form parameterization. Examples are constructed that are non-self-intersecting and belong exclusively to the equimodular elliptic type. To support applications in computational geometry, a numerical pipeline is developed that searches for candidate solutions, verifies them using the explicit algebraic characterization, and constructs and visualizes the resulting polyhedra; numerical validations achieve high precision. Taken together, these results provide constructive criteria, algorithms, and validated examples for the equimodular elliptic type, enabling the design of a broad range of flexible Kokotsakis mechanisms.

翻译：Kokotsakis多面体是三维欧氏空间中的一种多面体网格，由一个中心n边形面（基底）、n个各与基底共享一条边的四边形面、以及n个插入每两个相邻四边形之间的三角形面组成；若该多面体存在连续变形且保持每个面的刚性，则称其为可弯折的。本文研究具有四边形基底（n=4）的等模椭圆型可弯折Kokotsakis多面体，通过提供该类型首批显式构造，并结合平面角与二面角给出显式代数刻画，填补了文献中的重大空白。本文引入一类可直接构造的多面体——称为拟对称网格，其由平面角之间的对称关系刻画。研究表明，除少数特殊的二面角选择外，每个椭圆型QS网均具有等模椭圆型，且在实三维欧氏空间（而非仅复构造空间）中可弯折，且其弯折运动具有闭式参数化。本文构造了属于且仅属于等模椭圆型的非自交实例。为支持计算几何应用，本文开发了一套数值流程：搜索候选解、利用显式代数刻画进行验证、并构造与可视化所得多面体；数值验证达到高精度。综合而言，这些结果为等模椭圆型提供了构造性准则、算法及验证实例，从而能够设计大范围的柔性Kokotsakis机构。