Regular Dodecahedron-Based Network Structures

Abstract

The packing and assembly of Platonic solids have fascinated mathematicians for ages. Recently, this fundamental geometrical problem has also attracted the attention of physicists, chemists, and engineers. This growing interest is due to the rapid advancements in various related fields, ranging from the formation of colloidal crystals and the design of metal–organic frameworks to the development of ultra-lightweight metamaterials, which are closely tied to the fast-evolving 3D printing technology. Numerous reports have focused on the assembly of Platonic polyhedra, particularly tetrahedra, for which an optimal packing strategy remains unidentified to this day. However, less attention has been given to the dodecahedron and its networks. This work introduces a new type of framework, designed from regular dodecahedra combined with icosahedron-based binders. The relatively simple design protocol employed here results in a remarkable variety of intriguing networks, which could be potentially useful in fields such as architecture, regenerative medicine, or aeronautics. Additionally, the dodecahedral networks presented in this study led to the discovery of intriguing structures resembling distorted graphene sheets. These structures exhibit features characteristic of both graphene and diamond.

1. Introduction

The packing of Platonic and Archimedean solids [1,2,3] remains a subject of continuous interest to mathematicians, physicists, chemists, and engineers. The problem of packing and tiling space with congruent polyhedra was included in the famous list of 23 mathematical problems formulated by David Hilbert in 1900, who predicted its importance in number theory, physics, and chemistry. Since then, many scientists have explored this field, and a multitude of new concepts have emerged. Computational and experimental techniques have been developed to examine topics such as three-dimensional tilings and the assembly of polyhedra [4,5], spontaneous self-assembly observed in colloidal crystals [6,7,8,9,10,11,12], the description and classification of nets (emerging from complex molecular clusters) [13,14], or topological interlocking assemblies [15,16,17,18,19]. Additionally, topological assemblies and particle packing have proven essential in designing high-performance, ultra-lightweight materials [20,21,22,23], a dynamically developing field in material science that is closely tied to the rapid advancements in 3D printing technology. These low-density, porous structures can exhibit remarkable mechanical properties despite their ultra-light weight. Notably, their cellular architecture determines their mechanical performance and directional-dependent stiffness. These artificially engineered metamaterials should be considered at two distinct length scales: the lower scale, which corresponds directly to the intrinsic properties of the bulk material used, and the micro/macro scale, at which the geometrical features manifest and may play a dominant role [24]. Thus, contrary to conventional composites, the effective properties of metamaterials can be altered without the addition of dopants and can be modulated simply by adjusting the microarchitecture. A variety of scaffold geometries have already been explored [25,26,27] and engineers are continuously designing and developing new architectures with unique features. Although some studies have been conducted on dodecahedron packing [1,5,28,29] and clustering [30], reports on dodecahedron-based lattices and scaffolds are scarce. It is the icosahedral symmetry of the dodecahedron that prevents the formation of periodic structures, which is why these specific building units are rarely observed in bulk materials. However, icosahedral ordering is abundant in spherically confined geometries, such as small atomic or particle clusters, including Mackay-type clusters [31], since it minimizes the energy and maximizes the packing density. These unique structures were experimentally grown by Wang et al. [9], who revealed the spontaneous assembly of colloidal particles into well-defined shell architectures under emulsion droplet confinement. An interesting contribution concerning dodecahedron self-assembly was discussed by Marson et al. [28], who computationally examined colloidal crystals grown from Platonic polyhedral sphere clusters (PSCs). They showed that dodecahedral PSCs can adopt an FCC configuration, as well as other competing phases at higher volume fractions, including a γ-brass phase, cI52-Cu₅Zn_8, and the β-manganese structure. In this work, we introduce several new network structures based on regular dodecahedra. Additionally, we present a simple protocol in which combining two different types of Platonic solids allows the design of a variety of intriguing structures. Furthermore, the dodecahedral networks introduced here were used to design a lattice that displays structural features characteristic of both graphite and diamond.

After the discovery of quasicrystals [32], crystallographers realized that matter can organize into highly ordered, condensed lattices lacking translational symmetry. Engel et al. [33] have shown, by employing molecular dynamics simulations, that different variants of icosahedral quasicrystals can be self-assembled from a one-component system of particles. For the first time, a regular dodecahedral quasicrystal was grown from the icosahedral phases of Zn-Mg-RE (rare-earth element) alloys, as reported by Tsai et al. [34]. These exotic structures attracted the attention of engineers, particularly those working in photonics, who investigated the optical bandgap properties and wave propagation within quasicrystalline 3D models [35,36]. It is worth mentioning that the models introduced here can also exhibit features characteristic of quasicrystals and can become candidates for photonic bandgap materials. In 1988, Paquette et al. [37] successfully synthesized dodecahedrane (C20H20), the most highly symmetric hydrocarbon. This organic compound crystallizes into a face-centered cubic structure and, similar to buckminsterfullerene (C60), exhibits an extremely high melting point. This exceptional thermal stability is largely due to the high icosahedral (I_h) symmetry of both molecules. The regular dodecahedron features ten three-fold symmetry axes passing through pairs of opposite vertices and six five-fold symmetry axes passing through the centers of opposite pentagonal faces. Since all 12 faces of dodecahedrane are regular pentagons, the carbon–carbon bond angles are all 108°. This spatial bond arrangement is close to the mechanically robust tetrahedral geometry (109.47°) observed in diamond. Therefore, lattices based on the regular dodecahedron may have unique potential in the field of metamaterials, which are designed to withstand substantial loads. However, there is a challenge in designing such frameworks due to the dihedral angle of the dodecahedron (φ ~ 116.565°), which prevents the formation of closed structures when three individual dodecahedra are clustered together (3φ < 2π). To overcome this geometric obstacle, an additional solid can be introduced to bind adjacent dodecahedra. We have demonstrated that a regular icosahedron-based polyhedron is an ideal candidate for bridging an assembly of dodecahedra.

2. Materials and Methods

Structural modeling was performed using Wolfram Mathematica 7 Software. Regular dodecahedrons with the following initial Cartesian coordinates were used: (±1, ±1, ±1), (0, ±ϕ, ±1/ϕ), (±1/ϕ, 0, ±ϕ), (±ϕ, ±1/ϕ, 0), where ϕ is the golden ratio. Accordingly, the regular icosahedron was resized precisely to match the regular pentagonal faces of the dodecahedron, i.e., the icosahedral binder and dodecahedron always share one pentagonal face. The following Cartesian coordinates were used to define the icosahedron/binder geometry: (0, ±1/ϕ, ±1), (±1/ϕ, ±1, 0), (±1, 0, ±1/ϕ). Objects are shown in Figure S1 in the Supplementary Information. The presented frameworks were constructed upon translation and rotation transformations of individual structural elements. It has been shown that the introduced networks consist of repetitive structural motifs. It is worth emphasizing that all presented structures can be arbitrarily scaled. Structural motifs were designed upon translation of icosahedron-based binders with respect to the central dodecahedron with translation vectors $\overrightarrow{T_i}$ with the following Cartesian coordinates (0, ±1, ±ϕ), (±1, ±ϕ, 0), (±ϕ, 0, ±1). All these vectors are presented in Figure S2 in the SI. Subsequently, the frameworks introduced here were constructed upon structural motif transformations and assembling. Required translation and rotation transformations are illustrated in Figure S4 in Supplementary Materials.

3. Results and Discussion

Among the Platonic solids, the cube is the only one that can tile 3D Euclidean space. Other polyhedra from this group cannot fill space entirely. However, significant attention has been focused on tetrahedron packing, as its optimal spatial arrangement remains undiscovered [38]. The highest packing density reported so far, approximately 0.8563, was achieved by Chen et al. [39] using a dimer arrangement discovered by Kallus [40]. An alternative approach to filling space with Platonic solids is the well-known tetrahedral–octahedral honeycomb, which forms one of the most unique structures: the diamond lattice. This tiling strategy was later extended by Conway et al. [41], who introduced a new family of tilings. In this work, we adopt a similar general strategy, combining two Platonic solids—namely, the regular dodecahedron and the regular icosahedron—to create an entirely new type of framework. While it is not possible to fill space completely with these two polyhedra, we demonstrate that this approach enables the design of a variety of different architectures, depending on the structural motifs used. So far, only gas hydrates [42] and silica clathrates [43] have been identified among lattices containing structural units in the form of regular dodecahedra. Zhu et al. [44] postulated new carbon allotropes based on hydrate and clathrate architectures. Their ab initio calculations confirmed that all of these allotropes would be thermodynamically stable. Furthermore, they highlighted remarkable mechanical characteristics, suggesting that these allotropes could potentially serve as lightweight 3D structures.

Figure 1a illustrates the strategy, which employs icosahedra as binders to connect adjacent dodecahedra. The inset figure highlights the geometry of the binder. It has the shape of a discus, bound by 10 equilateral triangular faces on the sides and 2 regular pentagonal faces on the top and bottom. As a result, it possesses a five-fold symmetry axis passing through the centers of the opposite pentagonal faces. These pentagonal faces will be shared with adjacent dodecahedra in the formed networks. In principle, each dodecahedron can be decorated with 12 identical binders of this kind (see Figure S2 in the Supplementary Materials). This simple protocol offers a multitude of possibilities for constructing interesting frameworks. As shown in Figure 1, the architecture of the final network depends significantly on the structural motif used. For instance, the structure presented in Figure 1c corresponds to motif 1 while Figure 1d to motif 2. Interestingly, the first structure possesses a three-fold symmetry axis (see Figure S5 in the Supplementary Materials). It appears that using a structural motif consisting of a single dodecahedron, decorated with six binders, leads to the formation of tightly packed networks. In contrast, motif 3, which has only four binders, can be assembled into the perforated networks shown in Figure 1e. This structure exhibits aligned, hexagonally organized elliptical hollow spaces, which could potentially be used as transport channels in membrane technologies. Obtained systems structurally resemble zeolitic molecular sieves [45]. It is worth emphasizing that it is possible to design frameworks in which all the above-mentioned motifs coexist, forming interconnected sub-frames, each potentially exhibiting different mechanical or transport properties (see Figure S6 in the Supplementary Materials). Similar to quasicrystals, they exhibit local order without translational symmetry and are constructed by combining only two polyhedra: a regular dodecahedron and an icosahedron-based binder. This allows for the arbitrary design of trusses with tailored properties, such as structures that are partially permeable yet mechanically stable. Figure 2 illustrates another family of frameworks, which relies on the 3-fold symmetry of the dodecahedron. These networks resemble structures observed in organic chemistry and can be arranged into two variants: (a) graphene-like connected dodecahedra (Figure 2a,b) and (b) a sheet of densely packed dodecahedra (Figure 2c). Two kinds of structural motifs can be distinguished here (see Figure S7 in the SM), both resembling the architecture of a flower with a central dodecahedron decorated with three or six petals. Moreover, these quasi-two-dimensional sheets of interconnected dodecahedra can be assembled into stacks resembling graphite-like forms (Figure 2d), with additional intercalated dodecahedra (shown in green) binding the layers together. Accordingly, this binding strategy leads to the formation of dodecahedral pillars, all aligned parallel, as clearly visible in the YX projection. However, this stacking protocol represents a very specific variant, because, in principle, each subsequent layer can be deposited in three different ways; that is, the outermost layer always offers three equivalent pentagonal faces for anchoring the binding dodecahedra. Therefore, it is also possible to design randomly bound stacks. Interestingly, the same sheets can also be assembled into the intertwined form illustrated in Figure 2f. As shown in Figure 2e, in order to design this architecture, two independent, perpendicular stacks (here colored in red and blue) have to be arranged first and subsequently merged together, leading to interlocking topology. It is worth mentioning that adjacent layers within the stacks are translated with respect to one another along the z-axis with the vector $\left|\overrightarrow{T}\right|={\displaystyle \frac{L_0}{2}}$.

Figure 3a illustrates another type of framework, designed solely from regular dodecahedra. These dodecahedra are assembled into hexagonal rings, forming graphene-like sheets. The structure presented on the left-hand side consists of tightly bound dodecahedral pairs, where each pair shares one pentagonal face. In contrast, the structure depicted in the middle also involves pairs that are not fully bound together, meaning there is a small wedge-like gap between adjacent dodecahedra. This gap arises from the dihedral angle of the dodecahedron (φ ~ 116.565°). When three dodecahedra are clustered together, a small radial element is missing to form a closed structure: 2π − 3φ ≈ 10.3°. Figure S8 in the Supplementary Information shows how to construct an additional variant of densely arranged dodecahedra, as presented in the middle of Figure 3a. The mesh (colored in red) represents a network formed by connecting the central points of the dodecahedra. Interestingly, the resulting pattern resembles a distorted graphene sheet. Figure 3b compares two different carbon arrangements: the graphene arrangement (left-hand side), where all bond angles are 120°, and the diamond arrangement (right-hand side), where the bond angles equal the tetrahedral angle of approximately 109.47°. The pattern obtained here displays a hybrid arrangement, appearing as a transitional form between the two. By using this hybrid structure, it is possible to design a highly interesting scaffold that combines characteristics of both diamond and graphene. This structure is presented in Figure 3c and consists of mutually translated layers connected at sites marked by red dots. In the language of organic chemistry, each carbon atom occupying these specific sites would potentially be sp3 hybridized, forming bonds between adjacent sheets. The binder (connecting two dots) length, which determines the interlayer distance, has been adjusted to match the average distances between adjacent dodecahedra within the lattice shown in the middle of Figure 3a. This novel structure might be of interest in the context of phase transitions between carbon allotropes, specifically the graphite-to-diamond transition, which occurs under high-temperature and pressure conditions [46,47]. There has been a long-standing debate regarding the atomistic mechanism behind this structural transformation, which may result in two coexisting phases: cubic diamond (CD) and hexagonal diamond (HD, also known as lonsdaleite) [48]. It is tempting to investigate (for instance, using density functional theory) whether the model presented here could predict a phase transition, potentially leading to either CD or HD.

4. Conclusions

This work introduces several intriguing frameworks based on regular dodecahedra. It is important to emphasize that so far, only gas hydrates and silica clathrates have been identified among lattices containing structural units in the form of regular dodecahedra. Other types of lattices based on the regular dodecahedron have not yet been presented in the literature. These represent a new class of frameworks that include structural elements with icosahedral symmetry. This high-symmetry component could be a key factor leading to nearly orientationally uniform properties, which may be essential in the design of photonic crystals or mechanically robust matrices. To create these scaffolds, we combined two Platonic solids: the dodecahedron and the icosahedron. The geometry of the icosahedron was used to define a binder, which is employed here to create lattices of spatially assembled dodecahedra. This straightforward strategy enabled the design of a wide variety of interesting structures, potentially useful in fields such as architecture, regenerative medicine, aeronautics, and photonics. We would like to stress that all the lattices presented here are fully scalable and can be easily manufactured using rapidly advancing 3D printing technology. We have shown that the final framework architecture depends entirely on the structural motif used. Five different motifs were defined and examined in terms of their structural assembly. Furthermore, we have demonstrated that it is possible to design superstructures comprising grains of sub-frameworks, each displaying a different architecture. In effect, it is feasible to design scaffolds exhibiting satisfactory mechanical properties combined with the desired permeability, which could be exploited in sieve technology or regenerative medicine. These locally ordered structures reveal features characteristic of quasicrystals, which lack translational symmetry. Additionally, we have introduced a unique framework consisting of intertwined, honeycomb-like dodecahedral scaffolds. This architecture is particularly interesting because each individual sheet-like component involved possesses a certain degree of orientational freedom and can be translated with respect to the whole lattice until the steric barrier of an adjacent component is encountered. Therefore, quite unique elastic properties can be expected in this intertwined system: specifically, the plastic regime can precede the elastic regime, which is in contrast to the behavior of classical bulk materials. Accordingly, this architecture could potentially be useful in constructing impact energy absorbers. Finally, we introduced a highly unique framework constructed from graphene-like dodecahedral networks. We demonstrated that its architecture integrates structural motifs characteristic of both graphene and diamond. The structure resembles periodically distorted graphene sheets, which can subsequently be organized into stacks to form an interconnected, sandwich-like structure. Interestingly, this architecture could potentially serve as a hypothetical model for the transition phase between graphite and diamond, occurring under high-pressure and elevated temperature conditions.