Topologically Interlocking Blocks inside the Tetroctahedrille

TL;DR

This work develops a modular framework for topological interlocking by constructing non-convex blocks from tetrahedra and octahedra within the tetroctahedrille, a tetrahedral-octahedral honeycomb lattice. It introduces three concrete blocks—the Kitten, Cushion, and Shuriken—each with explicit tetrahedral-octahedral decompositions and symmetry properties, and demonstrates various TI assemblies built from copies of these blocks. The authors extend these blocks with generalized (m,n)-shurikens, grid-based cushion assemblies, and spiraling kitten configurations, and show how truncation and continuous deformation yield new TI blocks without destroying interlocking capabilities. A key contribution is the combination of lattice-based construction with assembly graphs to systematically describe block connectivity and motion constraints, supplemented by practical demonstrations of approximating objects and forming space-filling configurations. The approach provides a versatile toolkit for designing rigid TI structures with potential applications in architecture and geometric modeling, supported by deformations that can realize more complex vault-like or curved interlocking systems.

Abstract

A topological interlocking assembly consists of rigid blocks together with a fixed frame, such that any subset of blocks is kinematically constrained and therefore cannot be removed from the assembly. In this paper we pursue a modular approach to construct (non-convex) interlocking blocks by combining finitely many tetrahedra and octahedra. This gives rise to polyhedra whose vertices can be described by the tetrahedral-octahedral honeycomb, also known as tetroctahedrille. We show that the resulting interlocking blocks are very versatile and allow many possibilities to form topological interlocking assemblies consisting of copies of a single block. We formulate a generalised construction of some of the introduced blocks to construct families of topological interlocking blocks. Moreover, we demonstrate a geometric application by using the tetroctahedrille to approximate given geometric objects. Finally, we show that given topological interlocking assemblies can be deformed continuously in order to obtain new topological interlocking assemblies.

Figures (35)

• Figure 1: Attaching a tetrahedron to an octahedron so that two faces are identified
• Figure 2: Approximations of the unit sphere with scaled tetrahedra and octahedra
• Figure 3: (Left, middle) Folding plans for the tetrahedron and octahedron with $+,-$ signs according to the poling; (right) a block constructed by assembling tetrahedra and tetrahedra with inserted magnets
• Figure 4: An approximation of the Stanford Bunny http://graphics.stanford.edu/data/3Dscanrep/ (left) and an approximation generated using the tetroctahedrille (right)
• Figure 5: An STL-model of the cathedral in Aachen given by Philipp Hoffmann available at https://www.thingiverse.com/thing:4820139 (left) and an approximation generated using the tetroctahedrille (right)

Theorems & Definitions (9)

• Remark 2.1
• Remark 2.2
• Definition 4.1
• Definition 4.2: topological22
• Definition 5.1
• Remark 5.2
• Remark 5.3
• Remark 5.4
• Definition 5.5