Biological membranes, typically phospholipid bilayers, can fold into highly ordered cubic membranes, as observed in mitochondria. Pappas et al.^[51] and Borysko et al.^[52] were among the first to describe zigzag patterns in certain mitochondria of the giant amoeba Chaos carolinensis. Subsequently, Daniels et al.^[53] systematically demonstrated that starvation induces the formation of these zigzag structures in amoeba mitochondria. Using electron tomography and computer simulations, Deng et al.^[54,55] first confirmed that the inner mitochondrial membranes of starved Chaos carolinensis can form cubic membrane structures of the TPMSs, and correlated these 3D configurations with dietary status and cellular energy adaptation. Mitochondrial membranes in tree shrew photoreceptor cells can fold into multilamellar G-type cubic structures of 8-12 layers^[56]. Bykowski et al. related the nanomorphology of prolamellar bodies (PLBs) to pigment composition and growth conditions in Arabidopsis thaliana^[57] and demonstrated that PLB cubic membranes in etiolated angiosperm seedlings (including Arabidopsis thaliana, cucumber and maize) are predominantly of D-type structure rather than the hexagonal structure as previously thought^[58]. These studies reveal that cubic membranes are widespread in protists, plants and animals, providing a foundation for understanding membrane structure-function relationships.

A striking manifestation of TPHSs is their vivid structural colors, most prominent in butterfly wing scales. Saranathan et al. used small-angle X-ray scattering (SAXS) to identify the SG structure in five butterfly species from the families Papilionidae and Lycaenidae^[40]. Drawing an analogy to the self-assembly of ABC triblock copolymers, they proposed that a core-shell DG forms in which one network is the extracellular space surrounded by the plasma membrane; chitin is selectively deposited there, and after apoptosis, only the SG-structured chitin remains^[40]. Early electron microscopy investigations have been instrumental in revealing the intricate photonic architectures in butterfly scales, laying the groundwork for later 3D structural elucidation^[59-62]. Because two-dimensional (2D) projected images cannot prove the existence of both enantiomers, Schröder-Turk et al. performed 3D electron tomography on Callophrys rubi, confirming a structure approximating a parallel surface of SG with space group I4₁32^[63]; both enantiomers were confirmed, with the LH structure dominant, indicating chiral preference during scale formation. In addition, Winter et al. found that these micron-scale domains exhibit preferred orientation^[64]. Inoue et al. statistically analyzed the chirality of SG structures in Teinopalpus imperialis and Parides sesostris using Callophrys rubi as a control: the percentage of LH crystallites is 89% in Parides sesostris, higher than the 68% in Callophrys rubi, while Teinopalpus imperialis displays purely LH SG structures^[65]. These findings reveal species-dependent chiral selection, but the underlying mechanisms still require further exploration.

Wilts et al.^[66] observed that in the butterfly Erora opisena, SG crystallite sizes followed a gradient along the long axis and orientations were random relative to the scale normal. This distribution, together with the chiral imbalance, challenged the pre-folding model proposed by Saranathan et al.^[40]: a single-crystal membrane cannot explain the random orientations, while a polycrystalline membrane cannot account for defect-free single crystals, and the LH and RH SG enantiomers should occur with roughly equal probability. They therefore proposed a co-folding model in which membrane folding and extracellular chitin deposition occur simultaneously; the molecular chirality of chitin may cause the chiral bias, while the tip-to-base crystallite-size gradient suggests a time-dependent assembly constituting a “time-frozen” sequence of developmental stages^[66]. By controlling the developmental time points of the butterfly Parides sesostris [Figure 3A-F] for observation, Jessop et al.^[67] fixed samples on day 13 (approximately 70% development) [Figure 3G] and day 15 (approximately 80% development) [Figure 3H and I], observing that the developing SG structure is not a homogeneous extrusion process, but a fibrous network of interwoven nanofibers ~ 25 nm in diameter; the number of fibers gradually increased and the fibers became more entangled near the lower lamina [Figure 3J-M]. This led to a two-stage model: scale cells first secrete and weave nanofibers into a preliminary network, which is then “smoothed” in later stages into the final solid SG; the specific smoothing mechanism remains unclear^[67].

The SG geometry endows butterfly wings with vivid structural colors. Although SG is a potential 3D photonic crystal with a complete bandgap (CBG), the low refractive index of chitin prevents a CBG. SG in Callophrys rubi exhibits orientation-dependent scattering, and the suppressed iridescence arises from spatial mixing of colors from crystals of multiple orientations^[68]. In Erora opisena, SG domains correlate moderately with reflectance^[69]. The scales of Morpho theseus contain a quasi-ordered tubular structure between a purely tubular state and the SG domain, possibly an early state of the G; color mixing makes the scales appear uniformly white^[22]. Thayer et al. analyzed 312 scale structures from 286 butterfly species, finding that blue structural colors dominate while red is extremely rare; and that 3D photonic crystals (including SGs and inverse opals) show the most restricted color range of 440 to 560 nm^[15]. These studies advance our understanding of structure-optical property relationships in butterfly scales, informing optical optimization and biomimetic design.

Coleoptera such as beetles and weevils are another major group that evolved TPHS structures. Bartl et al. discovered SD microdomains in the green scales of the weevil Lamprocyphus augustus; their multiple orientations enable effective reflection of light from different incident directions^[70], highlighting the different optical structural preference between Lepidoptera and Coleoptera. Wilts et al. found that the elytron scales of Entimus imperialis are also composed of SD microdomains of different orientations^[71]. Although individual microdomains reflect light highly directionally, the scales are densely arranged in concave pits; when light from the pit walls and different microdomains spatially overlaps, the reflectance peaks at ~ 540 nm. At close range, weevils distinguish shiny spot patterns for recognition and mating, while from a distance they blend into a green background, evading predators^[71]. Wu et al. found that the colored and colorless domains of Entimus imperialis share the same SD structure. In the colorless domains, the air is replaced by SiO₂, reducing the refractive index contrast with chitin and producing narrow bandgaps^[72]. Saranathan et al. surveyed ~ 127 arthropod species and found TPHS structures other than SD in beetles, such as SG in Platyaplastes venustus and SP in the longhorn beetle Sternotomis pulchra; lineage-specific structure preferences may stem from differences in curvature regulation caused by variations in membrane-binding protein expression^[73]. In 72 Pachyrhynchus weevil species, photonic structures range from long-range order to complete disorder^[74] [Figure 4A]. Amorphous D structures can also open photonic bandgaps and produce structural colors^[75]. The direct comparison of SD structure and its amorphous counterpart in the red and blue scales of Pachyrhynchus congestus mirabilis showed that blue scales lack long-range order yet still open a partial bandgap, demonstrating that long-range order is not necessary for producing structural color^[76].

Besides the common G, D and P bio-optical structures, the I-WP surface was also discovered in Sternotomis callais^[23]. This finding expanded the types of photonic crystals in biology while raising new questions about how cells selectively fill subspaces. For G, D and P structures, the two subspaces can be superimposed via inversion operations or translation, meaning that either can become the extracellular space for chitin filling and the cell does not distinguish them^[23]. In contrast, the two subspaces of the I-WP structure have different volume fractions, and the cells always recognize the I-subspace as the extracellular space^[23]. Currently, no specific mechanism has been proposed to explain this phenomenon. Bauernfeind et al. studied two longhorn beetle species from the tribe Sternotomini, elucidating how polycrystalline to quasi-disordered amorphous I-WP structures produce non-iridescent colors^[77,78]. In the green and blue scales of Sternotomis virescens [Figure 4B-O], both amorphous I-WP structures have the same volume fraction, but the green-scale struts are longer and thicker (larger periodic size), shifting the reflected peak wavelength; the lack of long-range order causes isotropic scattering, meaning that short-range order provides the color while long-range disorder eliminates iridescence^[78]. In Sternotomis amabilis sylvia, the green and orange scales correspond to polycrystalline and quasi-disordered I-WP structures, respectively; the flattened scale bases bear transparent distal fins with micrometer-scale ridges that suppress the iridescence of the polycrystalline structure, whereas the quasi-disordered network scatters intrinsically isotropically^[77]. These structures have implications for photonic materials design.

SG structures formed by β-keratin were also discovered in the feathers of Chloropsis cochinchinensis sensu lato, showing weak iridescence and saturated blue-green structural colors, representing the first example of directly self-assembled SG crystals known to science^[79]. Donnay et al. proposed that the interfaces between the organic matrix and inorganic crystals in echinoderm skeletons are TPMSs, which were the first examples reported in nature of periodic minimal surfaces^[80]. Subsequently, TPHSs were confirmed in starfish (D)^[21,81], sea urchins (SD and SP)^[20,82,83] and crinoids (D)^[84]; the D structure in the extinct crinoid Haplocrinites boitardi pushes the fossil record of TPHSs in biomineralized skeletons back to the Devonian. Unlike the optical roles in butterflies, beetles and birds, the TPHS structure in the knobby starfish Protoreaster nodosus primarily exhibits mechanical properties - high stiffness, high strength and high specific energy absorption^[21,81,85]. Its skeleton contains a single-crystalline structure with precise crystallographic registration between the atomic-scale calcite and the micrometer-scale D microlattice. The {111} planes of the D microlattice are considered to undergo preferred failure under compression, yet lattice-level dislocations (density 100-1,200 cm^-2) generate a “pinning” effect during damage-band propagation, deflecting and constraining the bands. Spatial variations in structural parameters (such as branch length/thickness) and the non-cleavage “conchoidal” fracture of biogenic calcite collectively enhance damage tolerance. The ossicles exhibit non-catastrophic stress plateaus and a specific energy absorption of up to 9.76 kJ·kg^-1, outperforming many synthetic ceramic and even metallic foams^[21]. These studies demonstrate that nature selects TPHSs for their multifunctional properties.

The formation of biological TPHSs reveals four regulatory principles that guide artificial self-assembly. First, chiral selection: butterfly species differ in chiral preference of scale crystallites^[64,65]. In artificial systems, molecular chirality enables the chiral blocks to self-assemble into LH and RH SG networks^[86,87]. Second, curvature regulation: biological membranes select G, D, P^[54,55,73] or I-WP^[23] surfaces through a balance among spontaneous curvature, lipid arrangement and the curvature regulation of membrane-binding proteins. Artificial systems tune spontaneous curvature and packing frustration via molecular architecture design^[88-93], blending^[94-99] and end-group chemistry^[27,100]. Third, selective subspace filling: cells recognize and fill the I-subspace in Sternotomis callais^[23], and chitin is selectively deposited in the extracellular space of butterfly wing scales^[40]. Artificial systems achieve this by controlling nucleation-site density to fill one labyrinth of the D surface formed by block copolymers, yielding SD structure^[26]. Fourth, dynamic remodeling: Parides sesostris forms a woven fibrillar network that later smooths into SG^[67]; the random orientations and chiral bias in Erora opisena are explained by a co-folding model^[66]. Artificial systems regulate this via kinetic pathways such as solvent evaporation rate^[101-105], annealing history^[106,107], precursor hydrolysis/condensation kinetics^[26] and pathway-dependent phase transitions^[107,108]. This mapping bridges biological structures and artificial self-assembly. The corresponding synthesis strategies are discussed below.

Liquid crystals combine the fluid properties of conventional liquids with the ordered characteristics of crystals and are widely regarded as the fourth state of matter^[126-129]. They are classified into thermotropic and lyotropic types. Thermotropic liquid crystals undergo a temperature-driven transition from a disordered isotropic phase to an ordered anisotropic phase, as temperature changes alter intermolecular interactions and thus the degree of molecular order and alignment. Lyotropic liquid crystals form ordered mesophases when amphiphilic molecules are dissolved in a solvent at specific concentrations and self-assemble into aggregates or micelles, which further organize into ordered phases^[130,131]. Through these two approaches, a rich variety of liquid crystalline phases, including TPHS structures, can be obtained^[132-138].

Levelut et al. used time-resolved diffraction to confirm the DG structure in thermotropic liquid crystals^[139], and Zeng et al. reported a triple-network tricontinuous phase based on the P surface^[140]. Zeng et al. reported the first confirmed DD structure^[141], the first SD structure^[142], and the second single-network (SP) structure^[143] in thermotropic liquid crystal systems. The I-WP^[25] [Figure 5A-F] and a tetragonal gyroid with one network twisting synchronously while the other twists randomly^[144] were also reported, enriching the library of soft matter self-assembled structures. Chen et al.^[25], Cao et al.^[138,144], Cai et al.^[145] and Cao et al.^[146] also systematically studied TPHS self-assembly in liquid crystal systems. Using resonant soft X-ray scattering, Cao et al. verified two distinct molecular packing modes in DG structures self-assembled from rodlike and taper-shaped molecules: orientation is smoothly transferred through junctions in anisotropic channels filled with aromatic cores, whereas it is mismatched and discontinuous at junctions in isotropic channels filled with alkyl chains^[146]. Two intermediate phases (orthorhombic Fmmm and hexagonal P6₃/m) in the DD-DG transition of bolapolyphiles revealed a stepwise coordination-number change from 4 to 3^[145]. An SP → DG → SD alternating single/double network sequence demonstrated that such phases can alternate with continuous changes in molecular structure (e.g., side-chain length), providing ideas for as-yet-unrealized SG-structured liquid crystals^[147]. Tang et al. reported a four-network gyroid (4NG) with two networks per subspace, discovering a “supertwist” in which the double-helix twist direction opposes the subspace chirality [Figure 5G-O] - challenging the assumption that molecular orientational twist matches network chirality and providing new insight into chiral transfer in liquid crystals^[24].

Luzzati et al. first resolved a bicontinuous lyotropic G phase in divalent-cation soaps^[148,149]. Templer et al. systematically investigated the effects of chain length, water content and temperature on the G, D, P and hexagonal phases in a homologous series of 2:1 (mol:mol) saturated fatty acid/phosphatidylcholine mixtures, determining the location and area of the molecular pivotal surface and providing geometric constraints for interfacial-bending energetics^[150]. Tang et al. revealed the effects of temperature and pressure on the pivotal surface position in the G phase^[151]. Vacklin et al.^[152], Tang et al.^[153], Shearman et al.^[154], Tyler et al.^[155], Barriga et al.^[156] and Kulkarni et al.^[157] explored the phase stability and lattice parameter modulation through packing frustration energy, lipid chain geometry, thermodynamic conditions and chemical/physical synergistic regulation^[152-157]. Pressure-jump time-resolved SAXS and cryo-electron microscopy identified interlamellar attachments (ILAs) as a key intermediate in the lamellar to cubic transition^[158,159]. Van ’t Hag et al. reviewed the influence of over 130 additives on bicontinuous cubic phases. These strategies overcome the compositional limitations of binary systems and facilitate membrane protein crystallization and drug delivery^[160].

Lyotropic liquid crystal structures can be replicated by inorganic precursors, which may overcome soft-matter limitations and realize TPHS structures that are difficult to achieve in pure lyotropic liquid crystal systems. Gao et al. synthesized DD mesoporous silica AMS-10 using an anionic surfactant, an organosilane co-structure-directing agent and a silica precursor; adjusting the ionization degree of surfactant headgroup drove a structural evolution from tetragonal P4₂/mnm (cage type, AMS-9), cubic Fd$$ \overline{3} $$m (cage type, AMS-8), 2D hexagonal p6mm (cylindrical, AMS-3) to cubic Pn$$ \overline{3} $$m (DD, AMS-10)^[161]. Han et al. utilized cationic surfactants with bulky polar headgroups and hydrophobic side chains attached near the headgroup to build a mesoporous material whose silica pore wall follows a hexagonal minimal surface; varying ammonia and surfactant concentrations produced a structural evolution from a bicontinuous cubic phase (G) to a tricontinuous structure (the branched hexagonal minimal surface) and then to a 2D hexagonal phase within one templating system^[48], validating previous theoretical predictions^[49]. The tricontinuous branched hexagonal minimal surface was later also discovered in lyotropic liquid crystals^[162].

Thanks to their highly flexible structural design, block copolymers enable ordered mesostructures through microphase separation driven by entropy and enthalpy. Block incompatibility is described by the Flory-Huggins interaction parameter χ, while χN (where N is the degree of polymerization) represents the driving force for microphase separation. The standard mean-field theory model of ideal symmetric diblock copolymer melts (f = 0.5) assumes incompressibility, Gaussian chain statistics, monodispersity, conformational symmetry, negligible fluctuations and a single composition-independent Flory-Huggins interaction parameter. When χN exceeds the critical value of 10.495, the system enters an ordered state^[163-166]. However, in real experiments, fluctuations near the order-disorder transition shift the actual transition point from this mean-field prediction^[167]. The phases can be controlled by composition, volume fraction and χN^{[121,168-171]}. According to self-consistent field theory (SCFT) and experiments, the G phase in diblock copolymer melts is stable only within a narrow window between the lamellar and hexagonally packed cylindrical phases^[172-174]. This window narrows with increasing segregation strength (χN) but can be significantly broadened by conformational asymmetry. G formation is kinetically sensitive to thermal history and annealing conditions - solvent casting or short-time high-temperature annealing tends to produce kinetically trapped metastable states rather than the equilibrium G phase^[172-175].

In linear neat diblock copolymer systems, DG is consistently the thermodynamically favored network structure over DD and DP. This has been attributed to the higher node connectivity of DD and DP, which increases packing frustration and chain stretching. Using medial strong segregation (mSST) theory and SCFT, Dimitriyev et al. further attributed DG stability to its terminal-boundary geometry: the tubular medial surface of DG is a single twisted sheet without interior corners, allowing chain ends to spread over web-like surfaces and reducing the entropic penalty^[124], whereas that of DD and DP possesses multiple leaves intersecting at abrupt angles, tilting and extending the chains and sharply varying the interfacial curvature^[124]. They also pointed out that in conformationally symmetric linear diblock copolymers, cubic DD is not metastable but resides at a saddle point, enabling continuous transitions to DG or the T¹¹¹ phase, whereas mSST and SCFT show that homopolymer doping or conformational asymmetry can render DD metastable - fundamentally revising the understanding of DD stability^[125]. Lai et al., using polymeric SCFT, predicted that binary blends of homopolymer-like linear diblock copolymers (f_A = 0.95) with gyroid-forming linear diblock copolymers stabilize metastable DD and P phases over a broader composition window through segregation toward node centers and local segregation^[176]. Hou et al., using SCFT, first obtained DG, DD and DP simultaneously in pure AB-type block copolymers with the A′(A″B)₅ miktoarm star architecture, which combines enhanced spontaneous curvature, combinatorial entropy and local segregation to satisfy the geometric requirements of these networks^[88]. Single-network structures (SG, SD and SP) are harder to stabilize: in linear AB diblock copolymers, they are all metastable, tending to reduce domain sizes to maintain a CMC interface and thereby increasing interfacial area per unit volume and the enthalpic penalty^[123]. Park et al. predicted that SG can be stabilized in H-shaped (BA)₂A(AB)₂ block copolymers when the A backbone and A arm block sizes are moderately asymmetric^[93]. Xie et al.^[89] and Cheng et al.^[90] designed a B₁AB₂AB₃ linear pentablock copolymer in which the middle B₂ block pushes the longer B₃ segment away from the interface via a loop configuration (rather than the previously believed bridging configuration), together with the shorter terminal B₁ segment, relieving packing frustration in the matrix and achieving stable SG. Xie et al., using an inverse design algorithm based on the adjoint solution of sequence-SCFT with automatic differentiation and negative selection, calculated that forming the SG with the space group I2₁3 (a lower symmetry variant of I4₁32) and the P-like cubic network with the space group P$$ \overline{4} $$3n requires at least three and ten linear blocks, respectively^[177].

Various processing strategies and molecular designs have stabilized different TPHS structures in pure block copolymer systems. Alward et al.^[178], Thomas et al.^[179] and Kinning et al.^[180] first reported an ordered bicontinuous double-diamond (OBDD) phase in poly(styrene-isoprene) (PS-b-PI) star block copolymers. Hajduk et al. discovered another equilibrium phase in linear PS-b-PI, in which the minority component (PS) forms two three-fold-jointed networks, and the majority component (PI) serves as the G matrix^[175]. Later calculations revealed that OBDD is metastable under the standard model of linear diblock copolymers^[181,182], and it was subsequently revised as a G structure^[183]. Susca et al. prepared large-size DG single-crystal thick films via evaporation-induced self-assembly (EISA), co-assembling poly(isoprene)-b-poly(styrene)-b-poly(dimethylaminoethylmethacrylate) (PI-b-PS-b-PDMAEMA) with poly(methyl vinyl silazane) (PMVS) to obtain single-crystal regions up to 14 mm² and over 100 μm thick, pyrolyzable into mesoporous silicon (oxy)nitride monoliths^[184]. Lin et al.^[185] and Chu et al.^[186-188] stabilized the DD phase using the helical conformations of stereoregular syndiotactic polypropylene (sPP) and isotactic polypropylene (iPP) chains, confirming that configurational regularity is the key to stabilization. Fourier transform infrared (FTIR) spectra revealed that the stabilization originates from stretching-induced helical segments at the nodes rather than ordered packing, with polydispersity acting synergistically^[185,186]: a broad chain length distribution alleviates the entropic penalty of packing frustration, while helical conformations and the enthalpy of their associations further compensate it^[185]. Adding homopolymer h-sPP broadens the stable temperature window of DD, but upon heating it transforms into a hexagonally packed cylindrical phase rather than DG, via the reorganization of four-arm nodes and end-fusion^[95]. In non-stereoregular flexible systems, blending with homopolymers or polymers of different chain lengths is also an effective route to the DD phase^[96-99,189].

All of the above-mentioned structures are balanced. To synthesize unbalanced single-network phases, especially analogs of the CMC surfaces found in nature, one strategy is to employ triblock copolymers to form an alternating G (G^A) structure. Dolan et al. prepared G^A thin films from polyisoprene-b-polystyrene-b-poly(ethylene oxide) (PI-b-PS-b-PEO) via solvent vapor annealing (SVA); in situ grazing-incidence SAXS (GISAXS) showed that SVA proceeds through eight distinct stages and that the ternary triblock copolymer exhibits strong robustness to SVA parameters, which is promising for large-area ordered structures^[103]. Wang et al. employed two chiral triblock terpolymers, polyisoprene-b-polystyrene-b-poly(L-lactide) (PI-b-PS-b-PLLA) and polyisoprene-b-polystyrene-b-poly(D-lactide) (PI-b-PS-b-PDLA), achieving controlled network chirality in the G^A phase: PLLA and PDLA form LH and RH SG networks, respectively, transferring chirality from the monomeric molecular configuration to the multichain domain morphology. The achiral control sample polyisoprene-b-polystyrene-b-poly(D,L-lactide) (PI-b-PS-b-PLA) formed an alternating diamond (D^A) structure^[86] [Figure 6A-N]. Another approach to achieving lower symmetry is to shift the networks. Hsueh et al. used the DG-forming PS-b-PLLA as a template^[190]: after removing PLLA, silica precursors were infiltrated to obtain PS/SiO₂ nanohybrids; thermal treatment above the PS glass-transition temperature rearranged the two SiO₂ networks, and subsequent calcination released constraint, allowing more extensive random shifting that broke the inversion symmetry of DG and produced two incoherent SG networks^[190].

Compared with G, synthesizing D and P surfaces is considerably more challenging. La et al.^[91], Ha et al.^[92] and An et al.^[191] demonstrated that tuning the molecular design of branched block copolymers and the hydrophilic/hydrophobic molecular-weight ratio can drive the packing parameter above unity, enabling direct self-assembly of G, D and P cubosomes in dilute solution: the bulky branched hydrophilic block limits hydrophobic-chain stretching, increasing the packing parameter. Building on this, cosolvent regulation and blending enable precise, continuous tuning of the P → D → hexagonal phase transitions^{[91,92,94,191-193]}. Lin et al. also achieved D and P cubosomes using a linear polystyrene-b-poly(ethylene oxide) (PS-b-PEO, SEO) diblock copolymer in a dioxane/dimethylformamide mixed solvent at a high initial concentration, where water addition drives self-assembly through hydrophobic interactions of PS chains, and the high initial concentration and high hydrophobic volume fraction synergistically elevate the packing parameter^[102]. Lee et al. reported the first thermodynamically stable P structure in pure block copolymers: introducing diphosphonic acid linked via an ether bond at the PEO terminus of SEO regulates end-group hydrogen bonding [Figure 6O], inducing “medial packing” in which PEO chain ends aggregate at the microdomain center rather than undergoing skeletal aggregation, alleviating the higher packing frustration of P relative to DD. During cooling after annealing, PEO crystallization merges two three-arm nodes into one six-arm node. If the interaction is too strong (e.g., electrostatic interactions from amine-linked diphosphonic acid), the PEO ends cannot form medial packing or adjust to curvature requirements, preserving the initial curvature^[27] [Figure 6P].

Introducing inorganic precursors into the solution self-assembly of block copolymers significantly enhances the structural diversity and properties of TPHS materials. Wiesner has made pioneering contributions to this field. Finnefrock et al.^[194,195] and Toombes et al.^[196] utilized poly(isoprene-b-ethylene oxide) (PI-b-PEO) with (3-glycidyloxypropyl)-trimethoxysilane (GLYMO) and aluminum sec-butoxide to prepare mesoporous aluminosilicates via solvent casting-induced self-assembly. After symmetry breaking was taken into account, the true structure was a DG severely distorted by uniaxial contraction. Inorganic precursors can also yield SG materials: they co-assemble with the triblock terpolymer into a G^A structure, are selectively enriched within one block, and after calcination leave one set of inorganic networks^[197,198]. Han et al. first directly prepared SDD-structured macroporous silica using a triblock copolymer template [poly(tert-butyl acrylate)-b-polystyrene-b-poly(ethylene oxide) (PtBA-b-PS-b-PEO)] and tetraethyl orthosilicate (TEOS), which exhibited structural color but no CBG, owing to the low refractive index of silica^[43]. Replacing the inorganic source with titanium diisopropoxide bis(acetylacetonate) (TIA) and using a dinonylphenyl end-capped polyethylene glycol-b-polystyrene (DNPE-PEO-b-PS) template yielded SDD-structured titania whose high dielectric contrast was calculated to open a CBG^[42]. In these co-assembly systems, epitaxial intergrowth relationships between SDD and SG, SDP and SDD, and lamellar and SDD structures were observed^[41,199,200], and a transition from a 2D cylindrical phase through SG to SDD was found, with SG as a metastable intermediate phase resembling the growth of SG in butterfly wing scales rather than a node-rearrangement transition (no five-fold junctions during the cylinder-to-G transition were observed)^[201,202]. Before 2021, SD self-assembly had been reported only in thermotropic liquid crystals^[142]. Sheng et al. achieved the first direct SD synthesis in block copolymers using a triblock copolymer [poly(ethylene oxide)-b-polystyrene-b-poly(L-lactide) (PEO-b-PS-b-PLLA)] with TEOS and explored its epitaxial intergrowth relationships with DD and SG, and the curvature-driven transition^[203]. DD and SD share crystallographic directions (such as $$ \langle 111\rangle $$ and $$ \langle 001\rangle $$), while SD and SG connect side by side via the common {121} lattice plane, with unit cell parameters satisfying a_SD:a_SG ≈ 1:1.2, close to the theoretical 1.225; this lattice matching lowers the transition energy barrier, allowing metastable SD to exist locally with DD or SG support^[203]. Experiments confirmed that at the boundary between SD and SG, the four-fold nodes gradually transform into three-fold nodes, although SD remained a minor phase with SDD dominant^[203]. Subsequently, Wang et al. reported a pure SD titania with a CBG^[26] [Figure 7] - the first one-pot synthesis of pure SD in block copolymers. PS-b-PEO (f_PEO = 6.9%) was used as a template to form the D structure in a tetrahydrofuran/hydrochloric acid mixed solvent under thermodynamic control, with PS forming the D surface, and PEO forming two sets of channels for titania deposition^[26]. High hydrochloric acid content accelerated TIA hydrolysis and condensation, giving dense nucleation sites that filled both channels (DD); sufficiently low acid content gave sparse nucleation sites, allowing titania to grow in only one set of PEO channels (SD). Photonic bandgap calculations showed a CBG with a bandwidth of 11.54%, but reaching the visible range would require expanding the unit cell parameter to 240 nm^[26]. Nonetheless, TiO₂ undergoes pronounced volume shrinkage during the calcination-crystallization process, posing a considerable challenge.

Karpov et al.^[204] and Djeghdi et al.^[205] used polyisoprene-b-polystyrene-b-poly(glycidyl methacrylate) (PI-b-PS-b-PGMA) to form the D^A structure and prepared the corresponding SD by selectively removing the PI networks and backfilling the voids with gold. Using ptychographic X-ray computed tomography (PXCT), they reconstructed the 3D gold network; three orthogonal slices matched the theoretical SD model, and vertical shrinkage during film drying made the actual unit cell approximately monoclinic rather than cubic. Defects with topological charges of +1/2 (comet-like) and -1/2 (trefoil-like) were identified at the boundaries between two mismatched grains and at the intersection of three grains, respectively^[204]. They originated from inter-grain strain, emerged simultaneously, and jointly balanced the topological charge and strain, maintaining long-range order through local disorder. This study demonstrates that topology-driven self-assembly is an important mechanism for structure formation in soft matter, connecting the topological physics of biological and synthetic systems^[204-206].

Cooperative self-assembly of small-molecule surfactants with amphiphilic block copolymers introduces competing electrostatic condensation and volumetric expansion, offering high flexibility for material design and curvature control. For instance, adding stearyltrimethylammonium bromide to the self-assembly of polystyrene-b-poly(acrylic acid) (PS-b-PAA) produced saddle-shaped hierarchical lamellar phases [Figure 8A-D₃], providing direct evidence for the lamellar-to-bicontinuous phase (cubic G) evolutionary pathway^[207]. The saddle-shaped lamella exhibits predominantly negative Gaussian curvature (98% of the surface) and near-zero mean curvature [Figure 8E-H₃], yet retains lamellar stacking, revealing the intermediate phase between lamellar and hyperbolic geometries. Lower cationic surfactant additions introduced an unexpected shifted tG phase: although the cationic surfactant attracts and compresses the PAA chains (reducing the interfacial curvature), the insertion of surfactant micelles into the PAA microdomains increases the hydrophilic block volume (volumetric expansion effect), increasing the interfacial curvature. The balance between these two effects thermodynamically drives the formation of the unique shifted tG structure^[208].

For increasingly complex self-assembly systems, data-driven approaches offer new possibilities for phase behavior prediction. For architecturally complex miktoarm star block copolymers (PEO-s-PS₂), numerous factors influence their self-assembled structures, making manual structure-parameter analysis difficult. Cui et al. constructed a multilayer perceptron (MLP) that learned the self-assembly behavior of PEO-s-PS₂ from 802 experimental data points, successfully predicting multi-dimensional synthesis-field diagrams including hyperbolic surface structures^[209]. The shape of PEO-s-PS₂ resembles that of phospholipid molecules, and its phase behavior provides an ideal model for understanding biological membrane formation^[209]. While this data-driven approach enables phase prediction, it does not by itself establish causal mechanisms, which still require independent experimental and theoretical validation^[210-212]. Overall, these studies reveal the epitaxial intergrowth transition among TPHS structures, discover various low-symmetry shifted structures (SDD, SDP and shifted tG) and single-network structures (SG, SD). They further combine thermodynamic control with kinetic and curvature regulation and deep learning to achieve phase behavior prediction in complex self-assembly systems.

The TPHS phases formed by curved lipid bilayers feature two non-intersecting aqueous channels, enabling partitioned drug loading: hydrophilic drugs reside in the aqueous channels, lipophilic drugs in the lipid bilayer, and amphiphilic drugs at the lipid-water interface. Cubosome encapsulation efficiencies can exceed 90% for both small molecules and macromolecules, owing to their extensive internal surface area and tunable lipid composition. The thermodynamic stability of cubosomes preserves structural integrity under physiological conditions. Depending on the formulation, drug release from cubosomes may follow zero-order kinetics or Higuchi diffusion-controlled models, and can be modulated through lipid composition, surface modification or responsive moieties^[17,18,236]. Jennings et al. constructed a library of 96 lipidoids comprising 8 hydrophilic polyamines and 12 hydrophobic acrylates^[246]. They demonstrated two applications: stimuli-responsive thin films formed by interfacial protonation-driven self-assembly that undergo reversible DG ↔ lamellar phase transitions induced by acid, and colloidally stable cubosomes produced by solvent evaporation that retain the internal DG structure and exhibit enhanced membrane fusion, enabling efficient genetic material delivery^[246]. Chen et al. developed a cubosome-based transdermal system co-loaded with doxorubicin (DOX) and indocyanine green (ICG) (DOX-ICG-cubo) for chemo-photothermal melanoma therapy^[247]. The DD-structured cubosomes had a particle size of ~ 101.90 nm and encapsulation efficiencies exceeding 90% for both DOX and ICG, with favorable physiological stability, pH/thermo-dual-responsive release and robust photothermal stability. This system showed synergistic cytotoxicity against B16F10 melanoma cells upon laser irradiation (drug-interaction coefficient < 1) and increased DOX skin permeation to 923.89 ng/cm² (enhanced 1.95 times by microneedle pretreatment). In a B16F10 tumor-bearing mouse model, DOX-ICG-cubo gel combined with 808 nm laser irradiation achieved a 97.59% tumor inhibition rate (complete elimination in 3 of 6 mice), outperforming chemotherapy alone (81.41%) and photothermal therapy alone (92.92%), with no obvious toxicity to major organs^[247].

TPHS structures feature a 3D continuous and highly ordered pore channel network, which facilitates mass transfer and diffusion of reactants/products and inhibits pore blockage. Their high specific surface area and curved geometry expose more active sites, while the large mesopore size helps expel gas bubbles, synergistically enhancing catalytic activity, selectivity and stability^{[13,217,248-251]}. Wang et al. utilized crystalline ordered mesoporous boron phosphate (BPO₄) with an SDD structure as a solid acid catalyst^[218]. In the acetalization of furfural with methanol, it achieved > 70% conversion at room temperature within 24 h, with 99.9% selectivity to 2-(dimethoxymethyl)furan, and was reused for 5 cycles without significant loss of activity [Figure 9]. In propane oxidative dehydrogenation, the SDD-structured BPO₄ exhibited higher reaction rates and olefin yields than ordered macroporous BPO₄, reaching 8.70 g_olefin·g_cat^-1·h^-1 at 540 °C with olefin selectivity above 90%^[218]. Li et al. used SDD-structured SiO₂ as a hard template for vacuum high-temperature melamine polymerization followed by SiO₂ etching, to replicate an inverse-phased SDD-structured mesoporous graphitic carbon nitride (g-C₃N₄)^[217]. Under visible light (λ > 400 nm), it exhibited excellent photocatalytic water splitting with a hydrogen-evolution rate of 6,831 μmol·g^-1·h^-1 - 5 times that of the DG structure and 16 times that of the nonporous control sample, with 90% activity retained after 4 cycles, attributed to enhanced mass transport, bubble release and charge transfer^[217]. In electrocatalysis, nitrogen-doped hollow G carbon-Pt (HGC-Pt) catalysts exhibited an electrochemically active surface area 3.6 times that of solid carbon counterparts and a mass activity 2.7 times that of commercial Pt/C, owing to enhanced mass transport and reactant accessibility from the hollow structure^[28].

Li et al. also introduced ordered bicontinuous mesostructures into covalent organic frameworks (COFs) and metal-organic frameworks (MOFs) using TPHS structures templated from block copolymer self-assembly^[240]. Filling MOF precursors into DP-structured cubosome templates, followed by coordination and template removal, yielded SP-structured MOFs (SP-MOFs), including SP-ZIF-8, SP-ZIF-67 and SP-HKUST-1. When SP-ZIF-8 was used as a bioreactor, the trypsin loading reached 325 μg·mg^-1, achieving 37% sequence coverage and 45 matched peptides in digestion of bovine serum albumin (BSA) within 30 min - significantly outperforming free enzyme and conventional ZIF-8 - while maintaining high activity after 5 cycles^[240]. The ordered bicontinuous structure also enhances light utilization through multiple reflections. Using SDD-structured silica as a hard template, Liu et al. synthesized SDD-COFs with hierarchical pore structures, overcoming the pore-blockage limitation of microporous or small-mesoporous COFs^[239]. Benefiting from enhanced mass transport, improved light utilization and promoted photogenerated carrier separation, SDD-COFs performed excellently in visible-light-driven photocatalytic NO removal. A vinylene-linked COF (SDD-COF-3) achieved a NO conversion rate of 51.4%, far exceeding its non-mesoporous counterpart and the classical photocatalyst g-C₃N₄, with excellent cycling stability. This provides a strategy for improving the accessibility of internal active sites in COFs and developing efficient photocatalysts^[239].

The periodic channels of TPHSs enhance light harvesting through multiple reflections. The interconnected channels and frameworks reduce ion/electron charge-transfer resistance, markedly improving electrode reaction kinetics. The saddle-shaped non-positive-Gaussian-curvature surface lowers the free energy of catalytic intermediates (density functional theory), and enhances the interfacial electric field, promoting ion adsorption (finite element simulations). The highly symmetric periodic structure also alleviates charge-discharge mechanical stress, extending device cycle life^[11,13,252]. Tang et al.^[214,244], Xiang et al.^[215], and Li et al.^[253] explored electrochemical energy storage using DP-structured cubosomes from SEOs as templates, which have one set of open channels and another enclosed by the copolymer bilayer. Filling the open channels with various precursors and removing the template yields SP-structured bicontinuous mesoporous materials. An SP polypyrrole (SP-PPy) network served as the cathode host for Na-I₂ batteries: the bicontinuous structure facilitates iodine loading and electrolyte infiltration, and positively charged polaron/bipolaron groups adsorb polyiodide ions (I₃^-) through electrostatic interactions, suppressing the I₃^- shuttle effect. This cathode achieved a record-high specific capacity of 235 mA·h·g^-1 at 0.5 C^[215]. SP-PBA, a sodium-ion cathode from slow crystallization of Prussian blue analogs (PBAs) via HCl vapor diffusion, showed enhanced sodium-ion diffusion kinetics and uniform stress distribution during Na⁺ intercalation, achieving 79.3% capacity retention after 3,000 charge-discharge cycles^[214]. Polymerization of 3-aminophenol with formaldehyde within the open channels, followed by argon pyrolysis, produced SP-structured hard carbon (SP-HC) as a sodium-ion anode. Its saddle-shaped curvature increased carbon-framework defect density [observed by high-resolution TEM (HRTEM), Raman spectroscopy, X-ray diffraction (XRD), X-ray photoelectron spectroscopy and electron paramagnetic resonance; proposed based on these results], providing more Na⁺ adsorption sites, while the expanded 4.0 Å interlayer spacing enabled nearly barrier-free Na⁺ intercalation. SP-HC achieved a capacity of 172 mA·h·g^-1 at 10 A·g^-1 (full charge within 1 minute) and maintained a capacity of 80 mA·h·g^-1 after 100,000 cycles, a record for hard-carbon sodium-ion anodes^[244]. A marine-sponge-inspired carbonaceous sulfur host (bimeso-CoNC) featuring an SP structure and Co-N₄ single-atom sites was synthesized by filling the template with a Co-doped ZIF-8 precursor and pyrolyzing under argon. Its mesoporous channels (~ 31 nm) ensured rapid Li⁺ transport and high sulfur loading, while the Co-N₄ single-atom sites suppressed the polysulfide shuttle effect and accelerated sulfur conversion. The bimeso-CoNC/S cathode maintained 570 mA·h·g^-1 at 6 C and an areal capacity of 11.6 mA·h·cm^-2 at 13.5 mg·cm^-2, with only 0.039% capacity decay per cycle over 1,200 cycles^[253].

Werner et al. departed from conventional layered lithium-ion battery architectures by sequentially constructing three spatially continuous functional components separated by an electrolyte layer^[254]. An oligomeric phenol-formaldehyde resol (the carbon precursor for the anode) was co-assembled with PI-b-PS-b-PEO into a core-shell DG structure. After inert calcination, a gyroidal carbon framework remained. Poly(phenylene oxide) (PPO) was electropolymerized onto the carbon as a solid-state electrolyte, and a sulfur/poly(3,4-ethylenedioxythiophene) (PEDOT) composite was infiltrated as the cathode, forming a complete lithium-ion battery whose areal capacity is theoretically 4,700 times that of a planar thin-film battery with the same critical dimensions^[254]. Digital light processing (DLP) 3D printing was used to fabricate a monolithic G-structured solid oxide cell, eliminating the metallic interconnects and seals that typically account for over 75% of stack weight. Its specific power and volumetric power density exceeded 1 W·g^-1 and 3 W·cm^-3, respectively (nearly an order of magnitude above planar stacks), with excellent stability during hundreds of hours of operation, reversible cycling, and thermal cycling - a new pathway for weight/volume-sensitive applications such as aerospace and portable power^[255].

Owing to pore channels that enable rapid mass transport and full utilization of active sites, TPHS structures are also widely employed in sensors to enhance signal response^{[7,224-226,256]}. Pang et al. utilized the P structure as the mechanical scaffold of a triboelectric nanogenerator (TENG), designing a linear thickness gradient that forms three layers of different wall thicknesses along the z-axis^[225]. The P structure serves as the positive triboelectric layer and supporting scaffold, and its layer-by-layer compression deformation endows the device with three distinguishable electrical signal modes corresponding to long-distance, medium-distance and emergency braking states, so the braking condition can be identified in real time from the output voltage peak and duration - a novel sensing solution for autonomous-driving braking safety^[225]. Inspired by the periodic porous membrane and breathing of the mammalian respiratory system, Chen et al. synthesized DP-structured cubosome templates from SEO, impregnated them with SnCl₄·5H₂O solution, and calcined the products to obtain SP-structured mesoporous SnO₂, which was deposited onto microelectromechanical systems (MEMS) chips as the sensing layer^[7] [Figure 10A]. At low temperatures, gas is adsorbed onto the SnO₂ surface; upon rapid heating, the reaction rate increases sharply while gas diffusion lags, dropping the surface gas concentration to the reaction equilibrium concentration and determining the overshoot-peak amplitude. SnO₂ is an n-type semiconductor, and surface-adsorbed oxygen extracts electrons from the conduction band to form an electron depletion layer (high resistance). When reducing gases react with the adsorbed oxygen, electrons are released back to the conduction band, thinning the depletion layer and decreasing the resistance (increasing current). Different gases exhibit distinct current change magnitudes and rates due to differences in transferred-electron number and reaction activation energies [Figure 10B-D]. Pulse heating combined with a convolutional neural network identified six types of gases from the shapes of the overshoot peaks with 99.8% accuracy^[7] [Figure 10E].

TPHS structures are advantageous for integrated compact heat exchange and catalytic reforming owing to their extremely high specific surface area, the boundary layer disruption effect of their complex internal channels and two mutually interconnected channels that can separately accommodate the heat-supplying and reacting fluids - making TPHSs an ideal platform combining compactness, efficiency and design flexibility in thermo-fluid-reaction multi-physics coupled systems^{[235,257-260]}. Zhao et al. embedded the G structure into the cooling tube of a metal hydride hydrogen storage reactor to enhance heat dissipation^[261]. Computational fluid dynamics simulations showed that the tortuous channels disturb the cooling fluid, inducing repeated flow impingement, flow-direction alterations and vortex formation, thereby generating turbulence without specialized processes. The solid-G reactor reduced the time to reach 1.2 wt.% hydrogen storage capacity by 23.40% and the peak average bed temperature rise by 9.49% relative to the conventional reactor^[261]. Chen et al. compared serpentine channels with G, D and I-WP cold plates for electronic chip cooling^[262]. The D structure performed best: under a heat flux of 140 W·cm^-2, the maximum chip-surface temperature was reduced by 14.73 °C compared to the serpentine channel, owing to its larger heat transfer surface area and stronger secondary flows. They then designed a graded D structure with the unit cell size varying linearly along the height direction. Its equivalent convective heat transfer coefficient was 8.10%-10.30% higher than that of the uniform structure, with only a 3.81% higher pressure drop. With pumping power below 2 W, heat dissipation reached 256.9 W·cm^-2, an overall improvement of 3.58%-6.24%^[262].

Materials with TPHS structures possess a 3D periodic distribution of the dielectric constant that enables a photonic bandgap analogous to the electronic bandgap in semiconductors, forbidding light propagation within specific frequency ranges - the core mechanism of photonic crystals^[263-265]. Certain TPHS structures, such as SG and SD, can open CBGs or wide-angle photonic bandgaps when the dielectric contrast and fill fraction are appropriate. When intrinsically chiral (e.g., SG), the structure manipulates the polarization state of light, while dielectric contrast and fill fraction allow control of its dispersion. The bandgap position and width are tailored by adjusting structural parameters such as volume fraction, unit cell size, and network shift^[1,266,267]. These properties are fundamentally enabled by the geometric topology of TPHSs, which dictates how electromagnetic waves are scattered, dispersed and polarized.

The SD structure has been called the “holy grail” of photonic crystals^[26]. Bandgap calculations on shifted tG structures indicate that such distortions also provide an effective route to photonic bandgaps^[208]. Cui et al. combined Fourier analysis with deep learning to explore photonic bandgaps across six space groups, confirming the superiority of SD and SG while discovering that the lcs topology with space group Ia$$ \overline{3} $$d approaches SD performance, warranting further investigation^[29]. Flavell et al. simulated the programmed self-assembly of SG colloidal photonic crystals using two types of patchy spheres - both self-assembling into SG of either chiral enantiomer^[238]. Bandgap calculations show that both the SG structures and their inverse structures support CBGs, requiring refractive index contrasts of approximately 2.7 and 2.2, respectively. Circular dichroism and transmission spectra verify their selective response to circularly polarized light, providing theoretical guidance for chiral photonic crystals and topological photonic devices^[238]. Achieving visible bandgaps has long been challenging. Zhang et al. employed two-photon polymerization lithography and high-temperature annealing to fabricate SD titania photonic crystals with a patterning resolution reaching 180 nm and a refractive index of 2.4 to 2.6, whose reflectance within the photonic bandgap reaches ~ 100% in the visible range^[268]. Jo et al. utilized high-molecular-weight (M_n = 680 kDa) polystyrene-b-poly(methyl methacrylate) (PS-b-PMMA) to prepare non-affinely distorted DG films with a lateral unit cell size of 335.7 nm via SVA, opening a visible-regime bandgap (center wavelength 562 nm)^[269]. The optical chirality of SG originates from its intrinsic geometrical chirality and is more robust to local defects than that of negative-index metamaterials. Vignolini et al. fabricated SG gold metamaterial films with ~ 50 nm unit cells: on twinned domains along the [110] projection, reflectance varied sinusoidally over 525-630 nm with rotating linear polarization (linear dichroism)^[270]. Tilting by 35° and rotating around the [110] axis maximized the LH/RH circular-polarization transmission difference along the [111] chiral axis, experimentally verifying the optical chirality of SG^[270]. The negative refraction of DG metallic metamaterials originates from the metal-insulator-metal (MIM) waveguide mode of their dual-network structure rather than geometrical chirality - supported by the mirror-symmetric SD also exhibiting a negative refraction band. This negative refraction band arises from band folding at the Brillouin zone boundary, with the refractive index continuously tunable by lattice size^[271]. In topological photonics, DG and SG serve as ideal platforms for topological band degeneracies including Weyl points and line nodes. Lu et al. theoretically designed DG-based photonic crystals with broken parity and time-reversal symmetries, predicting frequency-isolated Weyl points whose topologically nontrivial surface states manifest as Fermi arcs connecting Weyl points of opposite chirality^[272,273]. Weyl points in DG were experimentally verified at microwave frequencies through angle-resolved transmission measurements. Space group theory revealed that SG can open a bandgap between the second and third bands, whereas DG cannot, and that the glide-Z₂ topological invariant in DG is determined by the parity of the number of bands below the gap. Topological one-way fibers based on helical modulation of DG achieved one-way transmission protected by the second Chern number, creating new opportunities throughout photonics^[272-279].

The generation and characteristics of phononic bandgaps depend on intrinsic material properties (including density and component volume fractions), and are significantly influenced by structural parameters such as symmetry, connectivity and geometric configuration^[280-285]. Periodic minimal surfaces have recently been introduced into acoustic/elastic wave manipulation. Integrating acoustic metamaterial design with topological concepts provides a new pathway for lightweight and multifunctional acoustic devices^[237,286]. Wu et al.^[287] combined the micro-perforated plate (MPP) with a TPHS sandwich structure and investigated its sound transmission loss and absorption. Experiments and simulations demonstrate that MPPs provide significant sound-absorption capability within specific bands, while improving sound insulation through enhanced acoustic energy dissipation^[287]. Further optimization of the G-type MPP-TPHS sandwich structure reveals that non-uniform perforation broadens the absorption bandwidth and improves overall acoustic performance, guiding the design of lightweight sandwich structures integrating load-bearing with sound insulation and absorption^[287]. Among the 16 bicontinuous cubic network structures, the I-WP structure was identified as the champion phononic bandgap structure with the largest complete phononic bandgap (width 0.41) at its optimal volume fraction and strut thickness for a bismuth telluride network in an epoxy matrix, by finite difference frequency domain calculations^[237]. Chen et al. designed and experimentally validated a 3D I-WP acoustic crystal, achieving a double-zero index, where the normalized effective mass density and effective compressibility simultaneously equal zero^[286]. Exciting a four-fold degenerate cone formed by the degeneracy of one monopolar mode and three dipolar modes at the Dirac point frequency produces acoustic void behavior. Experiments confirmed that the acoustic crystal maintains a planar wavefront in a 90° bent waveguide and achieves acoustic cloaking when an internal obstacle is introduced^[286].

TPHSs attract significant attention in mechanical metamaterials for their lightweight nature and outstanding mechanical properties. The non-positive-Gaussian-curvature regions distribute loads and reduce stress concentrations, enhancing load-bearing capacity (supported by both experimental and computational studies). Under impact or compression, their lattices undergo significant deformation, absorbing and dissipating substantial energy. Because sheet-based TPHS structures exhibit relatively homogeneous stress distributions, their fatigue performance generally exceeds that of conventional lattices at equivalent porosity and under well-controlled printing conditions, whereas printing defects and internal porosity can introduce local stress concentrations and degrade this advantage. Their geometry and mechanical properties can be tailored by adjusting the parameters of their governing structural equations^[288-295]. Zhang et al. systematically investigated the mechanical properties of 3D-printed TPHS structures, proposing a transition from finite regulation of known TPHSs to the infinite design capability of arbitrary TPHSs^[30,241-243]. Inspired by crystal twinning, they introduced twin boundaries into G and D structures. Blocking stress transfer paths and introducing structural symmetry transformed the shear bands from wide and irregular distributions into symmetric ones, avoiding catastrophic failure at low strains in D^[243]. In G structures, the chirality and number of SG networks significantly affect mechanical properties: racemic structures exhibited peak-stress improvements of up to 51.14% over homochiral structures, because the alternating LH/RH networks create alternating and opposing stress transfer paths, leading to more uniform stress distribution^[241]. Eight bifurcating TPHS structures with non-cubic symmetry generally outperformed the classical G, D and P structures: tD-tDelta-L (loaded along the left side of tD-tDelta) showed a 26.5% improvement over D-[100] (loaded along the [100] direction of D), and tP-oPa-L (loaded along the left side of tP-oPa) showed a 41% improvement over P-[100] (loaded along the [100] direction of P). The bifurcating structures maintained minimal-surface characteristics under non-cubic deformation, exhibiting more flexible deformation adaptability^[242]. They also proposed a free modeling method based on Fourier synthesis that constructs arbitrary periodic smooth surfaces by superimposing Fourier components corresponding to different Bragg reflections [Figure 11A and B], overcoming the limited scope of conventional level-set equations. It achieved an 86.21% increase in peak stress for a Fourier-designed P-derived structure (N-c₁₁₁-0.478) over the conventional P surface (N-c₁₀₀-fit) [Figure 11C-E] and a 98.02% improvement in energy absorption through optimized wall thickness distribution (P-position2) over the uniform P structure (P-uniform) [Figure 11F-K], and enabled non-equilibrium TPHS structures assembled via mortise-tenon joints^[30]. These works establish a paradigm for TPHS mechanical metamaterials, transitioning from finite structures to infinite possibilities.