2 Meso-structure of the novel TLM deduced by space group *P*6*mm*

The meso-structure of TLM were composed of the intersected struts, and these struts could be obtained by symmetric and translational operations, as shown in Fig. 1. It was found that TLM’s meso-structures had similar symmetries such as rotation and mirror reflection with some crystal structures. The symmetric group was a mathematical tool to describe the symmetry and periodicity of crystal structures. Therefore, using symmetric groups for the meso-structure design of TLMs could be considered as a feasible method in this study.

The space group *P*6*mm* con\vatained the space point group 6*mm* and space lattice *hP*, and the corresponding meso-structure of TLM could be deduced based on the symmetric and translational operations that described by space group *P*6*mm*. The unit cell of TLM’s meso-structure could be deduced by the space point group 6*mm*, and the distribution of struts in 3D space could be expressed by the group elements:\(E,{c_6},c_{6}^{2},c_{6}^{3},c_{6}^{4},c_{6}^{5},{\sigma _v},{c_6}{\sigma _v},c_{6}^{2}{\sigma _v},c_{6}^{3}{\sigma _v},c_{6}^{4}{\sigma _v},c_{6}^{5}{\sigma _v}\). The element *E* represented the position of initial strut. The related geometric parameters of initial strut were defined as follows: *l* was the effective length of strut; *d* represented the diameter of strut;\(\alpha\)was the inclination angle of strut. When the strut was rotated\(\pi /3\),\(2\pi /3\),\(\pi\),\(4\pi /3\)and\(5\pi /3\)successively around the rotation axis\({C_6}\), the positions of the obtained struts were expressed as the elements\(c_{6}^{n}\). The positions of the struts that were obtained through mirror reflections could be described by the elements\({\sigma _v}\)and\(c_{6}^{n}{\sigma _v}\). The symmetric operations of space group *P*6*mm* and the unit cell of corresponding TLM’s meso-structures were shown in Fig. 2(a). Set the rotation axis\({C_6}\)as *z*-axis in 3D coordinates *o-xyz*, and the reflection plane\(\sigma\)was the *yoz* plane. The symmetric operations could be expressed as follows:

\(E=(x,y,z)\) , \({c_6}(x,y,z)=(\frac{1}{2}x - \frac{{\sqrt 3 }}{2}y,\frac{{\sqrt 3 }}{2}x+\frac{1}{2}y,z)\), \(c_{6}^{2}(x,y,z)=( - \frac{1}{2}x - \frac{{\sqrt 3 }}{2}y,\frac{{\sqrt 3 }}{2}x - \frac{1}{2}y,z)\), \(c_{6}^{3}(x,y,z)=( - x, - y,z)\), \(c_{6}^{4}(x,y,z)=( - \frac{1}{2}x+\frac{{\sqrt 3 }}{2}y, - \frac{{\sqrt 3 }}{2}x - \frac{1}{2}y,z)\), \(c_{6}^{5}(x,y,z)=(\frac{1}{2}x+\frac{{\sqrt 3 }}{2}y, - \frac{{\sqrt 3 }}{2}x+\frac{1}{2}y,z)\), \({\sigma _v}(x,y,z)=( - x,y,z)\), \({c_6}{\sigma _v}(x,y,z)=( - \frac{1}{2}x+\frac{{\sqrt 3 }}{2}y,\frac{{\sqrt 3 }}{2}x+\frac{1}{2}y,z)\), \(c_{6}^{2}{\sigma _v}(x,y,z)=(\frac{1}{2}x+\frac{{\sqrt 3 }}{2}y,\frac{{\sqrt 3 }}{2}x - \frac{1}{2}y,z)\), \(c_{6}^{3}{\sigma _v}(x,y,z)=(x, - y,z)\), \(c_{6}^{4}{\sigma _v}(x,y,z)=(\frac{1}{2}x - \frac{{\sqrt 3 }}{2}y, - \frac{{\sqrt 3 }}{2}x - \frac{1}{2}y,z)\), \(c_{6}^{5}{\sigma _v}(x,y,z)=( - \frac{1}{2}x - \frac{{\sqrt 3 }}{2}y, - \frac{{\sqrt 3 }}{2}x+\frac{1}{2}y,z)\).

The translational operations of space lattice *hP* was described as\(\left\{ {T|{{{t}}_n}={n_1}{{a}}+{n_2}{{b}}+{n_3}{{c}}} \right\}\)(\({n_1}\),\({n_2}\)and\({n_3}\)were integers,\({{a}}\),\({{b}}\)and\({{c}}\)were translational vectors). The single-layer structure could be formed by translating the unit cell in the following directions:\((2{{a}},0,0)\),\(( - 2{{a}},0,0)\),\((0,2{{b}},0)\),\((0, - 2{{b}},0)\),\((2{{a}},2{{b}},0)\),\(( - 2{{a}}, - 2{{b}},0)\), as shown in Fig. 2(b). The meso-structure of novel TLM was obtained by translating the single-layer structure along the vector\({{c}}\), as shown in Fig. 2(c).

Figure 3(a) showed the schematic diagram of the structure’s unit cell, and all the struts were mirror symmetry based on the plane\(\sigma\). *L* and *H* represented the edge length of bottom plane and the height of unit cell. The volume of unit cell and it’s all struts was showed as follows:

$$\left\{ {\begin{array}{*{20}{c}} {L=\frac{{2\sqrt 3 }}{3}l\cos \alpha ,H=l\sin \alpha } \\ {{V_c}=2\sqrt 3 {{(l\cos \alpha )}^2}l\sin \alpha =2\sqrt 3 {{\cos }^2}\alpha \sin \alpha {l^3}} \\ {V=12 \times \pi l{{(\frac{d}{2})}^2}=3\pi {d^2}l} \end{array}} \right.$$

1

The relative density of novel TLM’s meso-structure could be obtained by the following equation:

$${\rho ^*}=\frac{V}{{{V_c}}}=\frac{{\sqrt 3 \pi }}{{2{{\cos }^2}\alpha \sin \alpha }}{(\frac{d}{l})^2}$$

2

The relative density of novel TLM’s meso-structure could be influenced by the geometric parameters of strut, and the relationship was shown in Fig. 3(b). It was found that the relative density of novel TLM’s meso-structure could be increased with the diameter-to-length ratio (*d/l*) and inclination angle (\(\alpha\)) of strut.