A theorized new class of polyhedral hydrocarbons of molecular formula CnHn and their bottom-up scaffold expansions into hyperstructures

We address the use of Euler's theorem and topological algorithms to design 18 polyhedral hydrocarbons of general formula CnHn that exist up to 28 vertexes containing four- and six-membered rings only; compounds we call “nuggets”. Subsequently, we evaluated their energies to verify the likelihood of their chemical existence. Among these compounds, 13 are novel systems, of which 3 exhibit chirality. Further, the ability of all nuggets to perform fusion reactions either through their square faces, or through their hexagonal faces was evaluated. Indeed, they are potentially able to form bottom-up derived molecular hyperstructures with great potential for several applications. By considering these fusion abilities, the growth of the nuggets into 1D, 2D, and 3D-scaffolds was studied. The results indicate that nugget24a (C24H24) is predicted to be capable of carrying out fusion reactions. From nugget24a, we then designed 1D, 2D, and 3D-scaffolds that are predicted to be formed by favorable fusion reactions. Finally, a 3D-scaffold generated from nugget24a exhibited potential to be employed as a voxel with a chemical structure remarkably similar to that of MOF ZIF-8. And, such a voxel, could in principle be employed to generate any 3D sculpture with nugget24a as its level of finest granularity.

www.nature.com/scientificreports/ formed cages with ions, in which these systems have the potential to be applied in magnetic resonance, acting as contrast agents, with semiconductive and ferromagnetic properties 13 . Cubane derivatives can also be employed as additives, for example, in fuel, due to their tensioned structures 14 . In addition, 4-methyl-cuban-1-amine and 4-methyl-cuban-1-methylamine compounds exhibited antiviral biological activity 15 . Finally, if synthesized in larger amounts, heptanitrocubane would perhaps be one of the most effective non-nuclear explosives possible 16 . Poater et al. studied several structural and energy aspects of a class of packed carbon nanoneedles, that were conceptualized by stacking up units of 4, 6, and 8 carbons with potential applications to nanomedicine by acting as drug carriers through nonpolar biologic media 17 . The ability of the polyhedral hydrocarbons to be structurally fused was further examined by Katin et al. 18 The authors studied a material based on polyprismanes and concluded that these systems are similar to the carbon nanotube 18 . In addition, the interactions of the orbitals between the parallel rings of these materials seem to be the main component associated with the stability of the systems 19 .
Karpushenkava et al. 20 , studied both structural and vibrational properties of a set of polyhedral hydrocarbons of the C n H n cage class in gas phase. The authors concluded that when the energy associated with the cage tension is either negative or slightly positive, the corresponding compounds could be synthesized. An unique exception was verified for a triprism compound with a cage energy of + 55.2 kJ mol −1 (ref 20 ).
Wang et al., reported three stable isomers of the type C 24 H 24 . In their article, G3(MP2) calculations revealed that the optimized geometries of these systems have a positive value for Δ f H 21 . These geometries are unstable when compared to their fullerene isomers. In addition, one of the structures formed with Si has the potential to be a semiconductor material and, by replacing the CH groups with nitrogen atoms, high-energy density materials can be prepared 21 .
On the other hand, DFT methods were also employed by Shamov et al. 22 to predict both structural and energy properties of a set of C n H n compounds, with n being 12, 16, 20, and 24. Both C 12 H 12 and C 20 H 20 compounds were synthesized, and the energetic properties indicated that C 16 H 16 and C 24 H 24 could be prepared. In this sense, Ohno et al., investigated both dimers and trimers of the regular prisms, with 6, 10, 12, 14, 16, 18 and 20 faces, connected by cubane-shaped bridges 23 . Their results also revealed that these compounds are able to be formed in organic reactions at low temperatures. Moreover, due to the metastable nature of the regular prismatic compounds, they could be potentially employed, for example, in energy storage 23 .
In this article, we employ Euler's theorem to deduce polyhedra containing four-and six-membered rings that exist up to 28 vertexes, that we call "nuggets". We then evaluate their energetics in order to conjecture the likelihood of their existence. Finally, because all nuggets can be fused together in several manners, either through their square faces, or through their hexagonal faces, we investigated the fusion abilities of this set of nuggets to investigate the perspectives for their growth into 1D, 2D, and 3D-scaffolds.

Results and discussion
The nuggets structural possibilities from Euler's theorem. Our intention was to design hydrocarbon polyhedra that could be potentially stable. Although there are polyhedral hydrocarbons of the type C n H n with triangular faces, such as the tetrahedron 24 and the triprism 25,26 , as well as ones with pentagonal faces, such as the dodecahedron and the pentaprism 9,26 , we decided to restrict our work to polyhedra whose faces are polygons with an even number of vertices. Such systems can have alternating double bonds, thus potentially displaying energy stabilization due to electronic delocalization.
Let us first consider polygonal hydrocarbons of formula C n H n . The smallest polygon with this formula is triangular C 3 H 3 . However, C 3 H 3 is a radical system. The same happens with C 5 H 5 , as shown in Fig. 1. Actually, all neutral polygonal C n H n hydrocarbons with n being an odd number must be radical systems.
On the other hand, when n is an even number with n ≥ 4, the C n H n polygonal hydrocarbons are neutral systems, with cyclobutadiene, C 4 H 4 , and benzene, C 6 H 6 , displaying planar structures and thus being the most important members of this class. But, when n is equal to or larger than 8, the compounds become non-planar 27 . Figure 2 shows images of these polygonal compounds up to n = 10.
Because we intend to grow the polyhedra into 1D, 2D, and 3D-scaffolds by fusing together their polygonal faces, we will restrict the polyhedra in this work to those with square and hexagonal faces only, since it would be very difficult, if not impossible, to fuse together two significantly non-planar and twisted faces. In these polyhedral compounds, each carbon atom must be bound to a single hydrogen atom as well as to three other carbon atoms as well.
Euler's theorem 28 defines a relation between the numbers of faces, edges and vertices for any simple polyhedron: the polyhedra of our interest. Simple polyhedra are topologically equivalent to a sphere, that is, these systems are polyhedra that have no central cavities as "donuts". Therefore, if inflated, in the limit, these systems would become spheres. There are two possibilities for a hydrogen atom bonded to a carbon atom in a carbon polyhedron: either it is located inside or outside the polyhedron. If the hydrogen atoms appear in the interior of the polyhedron, steric effects would be very significant due to the congestion between other hydrogen or carbon atoms, especially for the smaller polyhedra. Moreover, if all hydrogen atoms always point inwards, at least one hydrogen atom would have an HCC angle less than 90°, which is not reasonable from the point of view of chemical bonds. Therefore, to be chemically realistic in applying Euler's theorem, we will focus on carbon polyhedra with the hydrogen atoms of the CH bonds always pointing outwards.
Euler's theorem for simple polyhedra relates the number of faces (F), edges (E), and vertices (V) by the formula: where V is the number of vertices, E is the number of edges and F is the number of faces. www.nature.com/scientificreports/ When the polyhedron has only square and hexagonal faces, such as the nuggets, then where F 4 is the number of square faces, and F 6 is the number of hexagonal faces. Of course, each square face of the polyhedron delimits four edges, and each hexagonal face delimits six. However, if the edges are counted from each polyhedral face, they would be counted twice, since each and every edge of the polyhedron is shared by exactly two faces. Accordingly, the relation between the number of edges, E, and the number of square and hexagonal faces of such a polyhedron is given by the following equation: For our polyhedra, the number of vertices is represented by the union of three edges. That is, each carbon atom is chemically bonded to exactly three other carbon atoms, i.e. V = V 3 ; the fourth bond being to a hydrogen atom. And each edge is bounded by two distinct end points: the vertices. Therefore, the relation between the number of edges and the number of vertices is given by:   www.nature.com/scientificreports/ By simplifying the term 6F 6 on both sides of Eq. (7), we finally obtain that F 4 = 6. This reveals that any simple polyhedron that has only square and hexagonal faces must always have 6 square faces for an arbitrary number of hexagonal faces, except one. This exception is because Euler's formula is a necessary, but not sufficient condition for a polyhedron to exist. As can be intuited from Fig. 3 a configuration of one hexagon and six squares cannot be possibly closed into a polyhedron without forming at least a second hexagonal face. Consequently, the number of hexagonal faces must be either 0 (for the cube), or equal or greater than 2 for a constant number of six square faces.
Design and computational details. The software Blink 29 developed by the research group of one of us (SL) was used to generate a set of unique nuggets from the cube up to the three different solids with 28 vertexes, all with 6 square faces and up to 10 hexagonal faces. From graph encoded manifold and UNIVs data 30 the Blink software is capable of generating several representations of graphs, only formed by faces with even numbers of vertices-squares, hexagons, octagons, etc. In this article, the Blink software was employed to map the topologically different and possible shapes of up to n = 28 vertices. Among all possibilities generated, we selected, according to chemical criteria, a subclass that we call nuggets that is composed of those that have structural forms containing six squares and an arbitrary number of hexagons, either equal to zero, or greater than or equal to two, generating a set of three-dimensional representations of the nuggets. From this class, we selected the first 18 that led to chemically different structures 29,30 .
Hence, we generated a set of all different such polyhedra, starting with the cube, C 8 H 8 , up to those containing 28 vertices, of empirical formula C 28 H 28 , a number which we found to be reasonable to explore from a chemical point of view. Figure 4 shows the chemical structures of all 18 nuggets obtained, identified by the number of vertices, that is of carbon atoms, which is identical to the number of hydrogen atoms, and an additional letter in case there are more than one such nuggets for a given number of vertices.
Being fully aware that predicting the properties of unusual molecules is risky, in order to calculate structural, vibrational and energy properties of the set of 18 nuggets, we needed to choose a quantum chemical model chemistry that would be at the same time both accurate enough and workable, given the size of the systems that we want to study, to be able to make educated inferences on the prospects of their chemical realities. We thus chose the ωB97XD functional by Chai and Head-Gordon because of its inclusion of a version of empirical Grimme's D2 dispersion as well as long-range correction with superior results 31 , together with the 6-31G* basis set of Petersson et al. 32 , for ease of computation of the larger hyperstructures formed by the molecular building blocks. Accordingly, all geometries of the designed nuggets, as well as the more complex 1D, 2D, and 3D systems were fully optimized by ωB97XD/6-31G* calculations via both Spartan'14 33 and Gaussian09 34 softwares. All structures have been characterized to be minima with frequency calculations.
Nuggets exhibiting polyhedral chirality. From Fig. 4, nugget 24b , nugget 26b , and nugget 28b exhibit polyhedral chiral properties, as can be seen, in an illustrative manner, in Fig. 5, below, where we represent their respective pair of enantiomers. . Planar configurations of one hexagon and six squares, from which one can intuit that it would be impossible for any of them, or any other one for that matter to be closed as a 3D polyhedron without creating at least a second hexagonal face. www.nature.com/scientificreports/ Nuggets as voxels. Voxels are the three-dimensional (3D) equivalents of pixels. Analogously to pixels, which can be used to generate any 2D images by juxtaposition, voxels can be likewise used to generate any 3D sculptures. Voxels can be virtual as in computer 3D graphics or real as in 3D printers. For a carbon polyhedron to be able to efficiently function as a voxel, it should possess the important property of 3D space-filling. That property being satisfied, they could in principle perhaps function as solid controllable building blocks that could be used to assemble any arbitrary 3D structures by juxtaposition.
Of all nuggets that we studied, in only three of them, the carbon atoms define space-filling polyhedra that could function as chemical voxels: nugget 8 (cubane), nugget 12 (hexaprismane or [6]-prismane) and nugget 24a (a truncated octahedron hydrocarbon).
Let us first consider nugget 8 (cubane), of point group O h . Cubane's chemical stability with respect to selfdecomposition in the absence of any other reagents is something that can be inferred from its corresponding calculated energy change of reaction. Accordingly, let us consider the possibility of a nugget 8 , cubane, molecule dissociating into either 2 molecules of cyclobutadiene (C 8 H 8 → 2C 4 H 4 ), or into 4 molecules of ethyne Fig. 6.
The ΔE ωB97XD/6-31G* values for these reactions are equal to + 368.8 kJ mol −1 and 551.2 kJ mol −1 ; large values that prevent such dissociation from occurring despite cubane's highly tensioned cubic structure. These ΔE ωB97XD/6-31G* values indicate that these entropy-favored self-decompositions, are unlikely to occur spontaneously. These  www.nature.com/scientificreports/ findings are consistent with the fact that, as previously mentioned, cubane (nugget 8 ) has already been prepared 6 . Further, cubane growth in three dimensions is predicted to be a stable allotrope of carbon. Actually, a carbon allotrope with this 3D-structure could be very well used as an energy storage compound and would probably exhibit a larger mass density when compared with all other allotropes of carbon, including diamond.
Let us now examine the case of nugget 12 , the hexaprismane, which has the structure of a prism with two parallel hexagonal faces linked through six square faces (Fig. 4). Hexaprismane can be thought of as a face-to-face dimer of benzene. The calculated energy of dissociation of nugget 12 into two benzene molecules (C 12 H 12 → 2C 6 H 6 ) Fig. 7a, yields a ΔE ωB97XD/6-31G* = − 389.8 kJ mol −1 , indicating that, in this case, the spontaneous chemical selfdecomposition of hexaprismane is predicted to be highly likely to occur. As a reinforcement to this affirmation, the thermal cycloaddition of two benzene molecules [6 + 6] is symmetry forbidden 35 . Indeed, so far and despite many attempts, nugget 12 , C 12 H 12 , the hexaprismane, has never been synthesized. These facts point further in the direction that the growth of nugget 12 to three dimensions would quickly spontaneously transform such a hypothetical solid into superimposed layers of graphene, such as graphite. Recently, a vertical stacking of graphene has been evolved into materials with highly tunable electronic properties and unique functionalities: the van der Waals heterostructures (vdWHs) 36 . So, for all practical purposes, it is very unlikely that the hexaprismane hydrocarbon nugget 12 could ever be of practical use as a chemical voxel. Nevertheless, the geometric concept of an hexaprismane polyhedron as a chemical voxel has recently been realized by the synthesis of isoreticular pillar layered metal organic frameworks exhibiting properties such as catalytic activity 37 . Two other self-dissociation reactions that could be thought of for the hexaprismane nugget 12 would be: (i) self-dissociation   www.nature.com/scientificreports/ into 3 cyclobutadiene molecules, that is: C 12 H 12 → 3C 4 H 4 with a ΔE ωB97XD/6-31G* value of + 843.2 kJ, and (ii) selfdissociation into 6 ethyne molecules, C 12 H 12 → 6C 2 H 2 , with a ΔE ωB97XD/6-31G* value of + 1116.8 kJ, as can be seen in Fig. 7b,c, respectively. These two large positive calculated values reveal, as expected, that the self-decomposition of hexaprismane nugget 12 into two benzene molecules is the one most likely to occur spontaneously. The third and last carbon voxel is nugget 24a , which has the geometric form of a truncated octahedron: a space-filling Archimedean solid displaying many geometric properties, nugget 24a is a hydrocarbon, not the C24 fullerene which presents the same carbon structure 38 , which is geometrically equivalent to both the B 12 N 12 Fullerene reported by Matxain et al. 39 as well as to ZIF-8, a very stable and largely researched metal-organic framework, MOF 40 .
Due to its high symmetry, and much less strained chemical bonds than either cubane or hexaprismane, nugget 24a is a possibility to be considered as a carbon voxel. Let us now proceed by first examining its three possible forms of self-decomposition of nugget 24a : (a) into 4 benzene molecules, with a ΔE ωB97XD/6-31G* value of − 154.3 kJ; (b) into 6 cyclobutadiene molecules, with a ΔE ωB97XD/6-31G* value of + 2311.6 kJ; and (c) into 12 acetylene molecules, with a ΔE ωB97XD/6-31G* value of + 2858.9 kJ, Fig. 8a-c, respectively.
These results indicate that nugget 24a , although possibly unstable with respect to a self-decomposition into 4 benzene molecules, can be expanded as voxel into a 3D solid that would constitute an allotrope form of carbon. By being constituted by carbon atoms only, and noncoplanar vicinal six-membered rings, it cannot be split into benzene molecules or into graphene layers that would benefit from electron delocalization for stabilization. The geometric arrangement of the carbon-only hexagons in a such a perfectly packed 3D solid, placing each and every carbon atom in a condition of equilibrium of forces, would most certainly prevent its dismantling. Its infinite 3D expansion leads to a carbon-only solid compound which would constitute an allotrope of carbon. So much so that a sample has been found and properly characterized as a natural, super-hard, and transparent crystalline polymorph of carbon from the Popigai impact crater in Russia, formed because of a natural shockwave event 41 , and established to be consistent with such structure 42 .
Stability of the nuggets. Now, we turn our attention to the structural stabilities of the non-voxel nuggets.
Due to their molecular formula, their self-dissociation into ring compounds is a bit more complex, necessarily being at least into a mixture of benzene and cyclobutadiene, according to where n = 6p + 4q, with n, p, and q being integers. Further, there can be multiple combinations of p and q integer numbers that solve this expression for a given integer value of n. However, due to their geometric shapes, it is not always possible for these nuggets to be disassembled into combinations of benzene and cyclobutadiene molecules according to any stoichiometrically possible pair of values of p and q. Indeed, some of these disconnections could C n H n → pC 6 H 6 + qC 4 H 4 www.nature.com/scientificreports/ be shape forbidden. Finally, self-dissociations could also happen into ethyne molecules according to C n H n → (n/2) C 2 H 2 , a reaction that would always be possible since n is necessarily an even number and there are no geometric restrictions for any edges to be detached from the polyhedra. Table 1 shows ωB97XD/6-31G* calculated energies of reaction for all possible shape-allowed self-dissociations of all studied nuggets. From Table 1, complete dissociations into ethyne molecules are unlikely to happen for all nuggets, the same happening for self-dissociations producing any number of cyclobutadiene molecules. Thus, we can divide the nuggets into two groups, according to their energies of self-dissociation reaction ΔE ωB97XD/6-31G* .
The first group of nuggets is comprised by the ones with at least one of the calculated ΔE values being negative: nugget 12 (hexaprismane), nugget 18 , and all nuggets 24 (including the truncated octahedron, nugget 24a ). These are the nuggets that may perhaps be less stable.
The second group of potentially more stable nuggets comprises nuggets 8, 14, 16, 20 (a,b,c), 22, 26 (a,b,c) and 28 (a,b,c). This group includes nugget 28b which exhibits polyhedral chirality. As far as we know, so far, none of them have been reported in the literature, not even as a theoretical possibility. These results reveal that most of the designed nuggets are seemingly energetically stable and, probably, not easily capable of self-dissociation into simpler organic compounds.
On the other hand, the nuggets of formula C 20 H 20 , C 24 H 24 , C 26 H 26 , and C 28 H 28 possess structural isomers. Table 2 shows the energy of isomerization for all energetically favorable possibilities between these isomers. From Table 2, the most stable isomers for each of the molecular formulas are nugget 20c , nugget 24b , nugget 26a , and nugget 28a . However, transformation of one of the isomers into the other, involves fracturing a relatively rigid polyhedron through rearrangements of chemical bonds, thus rendering this type of transformation not likely.

Vibrational frequencies.
We now turn to examine the rigidity of the carbon scaffolds of the nuggets, that is, how they would vary from being hard and inflexible to soft and malleable as the number of vertices (carbon atoms) increases. We regard rigidity as a desirable property in a constrained geometry polyhedral compound, contributing to its structural stability and to other properties such as less susceptibility to thermal relaxation of excited states. Accordingly, in this work, we use the lowest calculated vibrational frequency of each nugget as a measure of its rigidity, the larger this frequency, the more rigid the compound. Indeed, the lowest frequency vibration, generally corresponds to a collective movement of all atoms of the molecule, fluttering in a synchronized manner along the corresponding normal coordinate. Table 3 shows frequency values for the lowest vibrational modes for each of the 18 nuggets, after geometry optimization, from ωB97XD/6-31G* density functional theory, DFT, calculations.
For comparison purposes, Table 3 also shows the lowest vibrational frequency of other compounds, where one can see that, as expected, cyclic compounds are generally more rigid than linear ones. Further, the presence of double bonds certainly increases rigidity in otherwise similar compounds.
Let us first consider the case of nugget 8 (cubane, C 8 H 8 ), which can be regarded as having been formed by two piled up cyclobutadienes. Cubane (ν ωB97XD/6-31G* = 628 cm −1 ) is more rigid than a cyclobutadiene (ν ωB97XD/6-31G* = 547 cm −1 ), indicating a sturdier structure. On the contrary, nugget 12 (ν ωB97XD/6-31G* = 394 cm −1 ), the [6]-prismane, which can be regarded as having been formed by two piled up benzene molecules, is actually more flexible than benzene, which has a ν ωB97XD/6-31G* value of 414 cm −1 . In general, it can be argued that the sturdier the structure, the more difficult it is for it to get disassembled. Accordingly, as previously discussed, nugget 12 would probably easily self-dismantle into two benzene molecules.
If we consider all other nuggets, from nugget 14 to nugget 28c , one of them, nugget 24a stands out as being the most rigid, having a very large lowest ν ωB97XD/6-31G* of 372 cm −1 . Nugget 24a is certainly special, displaying a very symmetric structure. This points to a molecular structure with much more balanced forces in each atom than those of the other nuggets. This reinforces the possibility of its 3D expansion, as discussed above, as likely being a very stable carbon allotrope that will probably be found to exhibit unique physical properties.
All other nuggets display rigidities that are seemingly large enough to guarantee their structural stabilities. As one would expect, the more prolate ones (the "c" ones) are less rigid than the more spherical ones (the "a" ones).
Naturally, as the number of carbon atoms in their structures increases, the nuggets tend to become less and less rigid. Nevertheless, their rigidities are, of course, still larger by a large difference than those displayed by the n-alkanes, and even by the cyclic alkanes with the same number of carbon atoms. All of this points to the direction that they could all be synthesized, as the synthetically challenging cubane indeed has been 6 .
As rigid as they are, the nuggets can then be fused together to form even larger structures, generating an assortment of shapes and forms that can bring about regular and irregular solids, porous structures, etc., with many potential applications to materials science. To examine such possibilities, let us now turn to their energetic properties of fusing.

Energetics of nugget-nugget face-fusion reactions.
To be able to design novel 1D, 2D, and 3D-scaffolds from the set of nuggets considered in this article, let us now study the ability of these systems to perform face-fusion reactions. Because the nuggets present both square and hexagonal faces, their growths must occur via the fusion reactions of either two square or two hexagonal faces. However, not all these face-fusions may take place because some of the faces of these nuggets, mostly the hexagonal faces, are not exactly flat surfaces, but slightly skew polygons, whose vertices are not all coplanar. In such cases, for a fusion to occur, a requirement of spatial complementarity may not always be possible because the hexagonal faces tend to be all concave. On the other hand, square faces in these polyhedra are almost all invariably planar. Therefore, face-fusion reactions are generally predicted to occur more frequently through square faces, rather than via the usually more skewed hexagonal faces. www.nature.com/scientificreports/ Let us first consider the most probable face-fusion reactions between two identical nuggets only. Of course, between two square faces, the fusions may occur in up to 4 different relative orientations of one face with respect to the other. Likewise, with hexagonal faces, the fusions may occur in up to 6 such different relative orientations, Table 1. Energy values for the shape allowed dissociation reactions of the studied nuggets into either C 6 H 6 and/or C 4 H 4 compounds, or into C 2 H 2 . All values were calculated by the model chemistry ωB97XD/6-31G*.
The energy values are given in kJ mol −1 units because they refer to chemical reactions involving one mole of reactant only.

Compound
Reaction ΔEω B97XD/6-31G * (kJ mol −1 ) www.nature.com/scientificreports/ all leading to a huge number of possibilities. Table 4 shows the energies of reactions, one for each type of fusion (whenever possible) that displayed the least ωB97XD/6-31G* energy values of reaction for each pair of identical nuggets. Results on Table 4 indicate that while there are 18 square face fusions, the number of hexagonal face fusions possible is only 5. The values of energy of hexagonal face-fusion reactions range from − 185.5 kJ for nugget 24a to 638.8 kJ to nugget 12 , with the same numbers for square face fusion reactions ranging from − 80.2 kJ, for nugget 26b , to + 427.4 kJ for nugget 8 , cubane. Although the larger the nugget, the more likely it is to display negative face-fusion energies of reaction, we notice an exception to this rule: among the 18 nuggets designed in this article, two identical molecules of the carbon voxel nugget 24a are predicted to perform hexagonal face-fusion reactions with the largest negative value of ΔE ωB97XD/6-31G* = − 185.0 kJ. Therefore, of all nuggets studied, nugget 24a www.nature.com/scientificreports/ is predicted to exhibit the largest aptitude to be applied to growth as 1D, 2D, and 3D-scaffolds, especially when one considers its voxel characteristics.

Growth of nuggets into patterns.
Upon face-fusion reactions, nuggets can grow into either regular or irregular structures. Let us first consider possible fused compounds displaying structures with regular patterns. The simplest of these patterns are tessellations: covering of the space with nuggets, without overlaps or gaps. Tessellations can occur in one, two or three dimensions, and are the result of face-fusion reactions of a nugget, or of a combination of nuggets, made up by their translations, rotations or reflections. The carbon voxels, nugget 8 , nugget 12 and nugget 24a would be natural candidates. However, as explained above, only nugget 24a would make such a chemically feasible tile for this purpose. Let us therefore turn to consider the growth of nugget 24a in 1 dimension. The idealized self-fusion reaction of two of them via one of its all-equivalent hexagonal faces, 2C 24 H 24 → C 42 H 36 + C 6 H 12 , ΔE ωB97XD/6-31G* is − 185.0 kJ, where C 6 H 12 refers to cyclohexane leads to a generator of the simplest 1D scaffold extension. Figure 9 shows its optimized geometry together with the released cyclohexane for easier visualization.
Next, to evaluate the ability of nugget 24a in generating 2D-scaffolds, the following idealized fusion reaction was now considered: C 24 Fig. 10 (right) and whose predicted energy of reaction is − 85.7 kJ. Due to its 2D-structural arrangement its stability is substantially more accentuated when compared with the formation of the essentially linear C 60 H 48 1D compound obtained by fusing together the 1d-generator compound in Fig. 9 with another nugget 24a . This is because now a larger quantity of viable fusion reactions was carried out.
The infinite 3D expansion of this polyhedron will lead to a carbon-only compound that would constitute an allotrope of carbon 42 . A solid model image of a piece of this allotrope can be seen in Fig. 12 below. It is noteworthy that, by acting as a space filling carbon voxel in this manner, at least in principle, nugget 24a could be employed to generate any 3D sculpture with itself as its finest granularity level.
Another seemingly rigid allotrope of carbon can also be made from nugget 24a in the form of a regular skew apeirohedron. Similarly, but not exactly like the one advanced by Zhou et al. 43 , this will be formed by joining the carbon voxels nugget 24a through hexagonal pyramidal bridges linking hexagonal faces of one to square faces of others, in a manner so that each external square face of the hexagonal prismatic bridge shares an edge with a square face of one of the polyhedra while its opposite edge is shared with a hexagonal face of the other. Figure 13 exemplifies such a hexagonal prismatic bridge between two nuggets 24a . In this case, the idealized chemical Table 4. Energy values of the most stable fusion reactions between two identical nuggets, either via square faces releasing C 4 H 8 , or, whenever possible, via planar hexagonal faces releasing C 6 H 12 . All values were calculated by employing the level of calculation ωB97XD/6-31G*.  (Table 4), these bridged connections of hexagonal faces are more energetically favorable than connections via square faces. Therefore, the regular skew apeirohedron can then be formed by linking together, in this manner, each nugget 24a by 4 of its 8 hexagonal faces according to Fig. 14 below 44 . This putative allotrope of carbon, adding to previous exotic carbon allotropes 45 , would be very stable and rigid. Its density, however, would be evidently smaller than that of the space filling allotrope shown in Fig. 12. The presence of zeolite-like nanoporous cavities inside its structure could be a singular feature, that could perhaps prove to be the origin of many emerging and interesting properties.
Other types of polyhedral solids, with larger cavities, can also be conceptualized, such as the one made, this time by nugget 16 , via square face-fusions, and whose projection in one plane reveals a semiregular or Archimedean tessellation, that can be grown indefinitely Fig. 15. Such a compound, if ever obtained, would also likely behave as a load resisting skeleton due to its symmetric nature. Furthermore, this structure could also be grown   www.nature.com/scientificreports/ in 3D leading to lengthy tubular cavities that could prove eventually useful. Structures such as these, with large cavities in the middle, suggest applications to materials science as catalysts, porous powders, etc. Many more combinations can be conceptualized by connecting the nuggets. Figure 16 shows a helix compound made by fusion of nugget 28b via two of its quasi-planar hexagonal faces. Such a compound, whose form resembles a twisted rope, would exhibit helicity, a form of chirality.
Besides, these regular and aesthetically appealing structures, several other large structures can be conceived by binding together several of the nuggets, leading to a myriad of hydrocarbon structures that would extend far beyond what is being here presented. The geometric possibilities of molecular structures that could in principle be formed based on these nuggets are truly vast: "symmetries, spirals, trees, waves, foams, tessellations, meanders, cracks, and stripes with fractal dimensions" 46 .

Conclusions
Euler's theorem and topological strategies were employed in order to theoretically design a set of 18 hydrocarbon nuggets of general formula C n H n containing four-and six-membered rings, that exist up to 28 vertexes. From Euler's theorem we demonstrated that all such polyhedra must contain exactly six four-membered rings, for an arbitrary number of six-membered rings equal or greater than two. Among these 18 nuggets, 13 are novel systems, with 3 of them exhibiting polyhedral chirality.
We also showed that, with the exception of hexaprismane, which is predicted to easily self-dissociate into two benzene molecules, and therefore unlikely to be synthesizable; and also with the exception of nugget 18 , which is presumably expected to dissociate into three benzene molecules, all other nuggets are likely to be relatively stable and not self-dissociate or degrade.
Subsequently, vibrational properties revealed that the designed nuggets are sufficiently rigid. In this sense, the nuggets with 28 carbons are predicted to exhibit a structural rigidity, in average about 100 times greater than that of the linear alkane n-octacosane C 28 H 58 .
We also explored the expansions of these nuggets into larger structures by face-fusion reactions involving mainly hexagonal and sometimes square faces.
Nugget 24a , the carbon voxel, resembles the most a fullerene (6 and 5-membered rings, however) in terms of the spherical shape, and possesses a chemical structure similar to the MOF ZIF-8. Due to its energetically favorable face-fusion reactions, Nugget 24a is deemed to be the most suitable one to have a large potential to be applied to growth as 1D, 2D, and 3D-scaffolds. Accordingly, any 3D sculpture could be generated with nugget 24a  www.nature.com/scientificreports/ at its finest granularity level if sufficient synthetic control is one day discovered; or perhaps by carving from the innovative carbon allotrope presented in Fig. 11.
In conclusion, as mentioned in the previous section, the nuggets could be in principle expanded into all sorts of forms: "symmetries, spirals, trees, waves, foams, tessellations, meanders, cracks, and stripes of fractal dimensions" 46 . Their scaffolds may be decorated with strategically placed substituents as quantized perturbations, to promote attractive forces between them for a potential use in molecular tectonics. Perhaps they can form designer hyperstructures made layer by layer in a precisely chosen sequence where electronic or even exotic phenomena, typically requiring exceptionally low temperatures, can be explored. In summary, these are structures that should   www.nature.com/scientificreports/ be considered as possibilities and of interest to researchers from all areas of carbonaceous nanomaterials (e.g., fullerene, nanotube, graphene, etc.). Finally, we also present the perspective of novel carbon allotropes, both space filled, as well as with cavities, hinting at interesting properties if synthesized or found as it appears to be the case with the natural, super-hard, and transparent crystalline polymorph of carbon from the Popigai impact crater in Russia, formed because of a natural shockwave event 41,42 .