Generalized Matryoshka: Computational Design of Nesting Objects

Abstract

This paper generalizes the self‐similar nesting of Matryoshka dolls (“Russian nesting dolls”) to arbitrary solid objects. We introduce the problem of finding the largest scale replica of an object that nests inside itself. Not only should the nesting object fit inside the larger copy without interpenetration, but also it should be possible to cut the larger copy in two and remove the smaller object without collisions. We present a GPU‐accelerated evaluation of nesting feasibility. This test can be conducted at interactive rates, providing feedback during manual design. Further, we may optimize for some or all of the nesting degrees of freedom (e.g., rigid motion of smaller object, cut orientation) to maximize the smaller object's scale while maintaining a feasible nesting. Our formulation and tools robustly handle imperfect geometric representations and generalize to the nesting of dissimilar objects in one another. We explore a variety of applications to aesthetic and functional shape design.

Scrappy: Using Scrap Material as Infill to Make Fabrication More Sustainable

“…Our system also involves integrating foreign objects, but for a diferent goal. To investigate the design of nesting objects, Jacobson created a generalized way of approximating the maximum scale at which one model fts within another [13]. Our inclusion engine is based on their algorithm, but solves a diferent problem.…”

“…Although quite distinct in motivation, this geometric problem bares many resemblances to the "Generalized Matryoshka" algorithm by Jacobson [13]. In that work, the goal is to create Russian nesting dolls for arbitrary input shapes: the outer object is split in half and the inner object is inserted within it.…”

“…We propose several modifcations to Jacobson's algorithm so that we can leverage their overall modelling of the problem and optimization technique. For readability, we give a brief overview of Jacobson's method, but refer the reader to that paper [13] for details.…”

“…The implemented system shown consists of a Fusion 360 plugin to monitor the modelling process, search for ftting scrap objects in the background, and compute a hollowed 3D model and printing instruction fles to fabricate the model with the selected scrap inside (Figure 1). Our insertion engine that handles the search for possible scrap inserts is based on an existing 3D nesting algorithm [13]. A web-based scrap library keeps track of scrap objects, helps users fnd them in the workshop, and allows them to add new scrap.…”

Packable Springs

“…In [LHAZ15], space‐saving furniture is designed by applying a minimum amount of modification to an object such that it can be folded. Jacobson [Jac17] generalizes the self‐similar nesting of Matryoshka dolls to arbitrary solid objects and finds the largest scale replica of an object that nests inside itself. Yu et al [YCC17] approximate shapes by telescopic structures for applications where mechanisms must be compact in size.…”