Cellular Bricks: Modular Building Units

• Cellular bricks are modular units constructed from repeated cell-like motifs whose local interactions—such as DNA hybridization, differential swelling, acoustic resonance, or neural communication—directly determine global structure and function.
• They leverage restricted connectivity and precise geometric design to enable error-controlled self-assembly, reconfigurable metamaterials, and scalable tissue engineering while balancing functional trade-offs.
• Cross-domain implementations of cellular bricks reveal that tuning local binding domains, swelling-induced curvature, resonator dynamics, or decentralized communication is key to optimizing overall performance.

Cellular bricks are modular units whose function emerges from local geometry, local interactions, or repeated cellular motifs. Across the recent literature, the term encompasses at least five distinct research programs: DNA-brick self-assembly, in which uniquely addressable short strands assemble three-dimensional targets through sequence-programmed hybridization; 4D-DLP-printed cardiac bricks, in which foldable hydrogel films roll into cylindrical engineered myocardium modules; LEGO-based phononic crystals, in which snap-on bricks act as reconfigurable resonant stubs; smart robotic bricks, in which cubic modules perform decentralized Neural Cellular Automata inference; and clay metaBricks, in which slit-connected alveoli create coupled thermal and acoustic functionality (Reinhardt et al., 2014, Hosseinabadi, 27 Aug 2025, Celli et al., 2015, Moreno et al., 23 Sep 2025, Lemkalli et al., 2023). The shared theme is modularity with predominantly local coupling, but the operative physics ranges from DNA hybridization and hydrogel swelling to elastic wave dispersion, distributed neural computation, and Helmholtz resonance.

1. Terminological scope and conceptual unification

In the molecular literature, a “brick” is a uniquely designed short DNA strand that hybridizes with exactly four neighbours, enabling three-dimensional structures of almost arbitrary shape with remarkably low error rates. In cardiac tissue engineering, “cardiac bricks” are modular cylindrical tissues formed when a printed PUPEGDA film self-folds around a collagen-cell construct. In metamaterials, LEGO bricks function as vertical stubs on a thin ABS plate, creating a fully reconfigurable phononic crystal. In modular robotics, each brick is a 3-cm cubic unit with local communication, processing, and sensing. In building materials, a clay metaBrick is a hollow brick whose alveoli are modified by narrow slits to form one-degree-of-freedom Helmholtz resonators (Reinhardt et al., 2014, Hosseinabadi, 27 Aug 2025, Celli et al., 2015, Moreno et al., 23 Sep 2025, Lemkalli et al., 2023).

The adjective “cellular” is therefore used in multiple, non-equivalent senses. In the clay metaBrick it refers to an internal cellular motif of cavities and walls. In the smart-brick system it refers to Neural Cellular Automata and to cell-wise local communication. In the DNA-brick and cardiac-brick settings, the relevant commonality is not literal cellularity but a modular architecture in which local interactions determine global form. This suggests that “cellular bricks” is best treated as a cross-domain descriptor for systems built from repeated units whose assembly or function is governed locally rather than centrally.

A common misconception is that all such bricks are intended for autonomous macroscopic self-assembly. The literature is more differentiated. DNA bricks do self-assemble from free monomers. Cardiac bricks self-fold because of differential swelling strain and are then intended for modular packing. LEGO phononic bricks are manually reconfigured for rapid experimental proof of concept. Smart bricks infer global shape class and detect damage in a decentralized way, but the cited work explicitly notes that there is no onboard actuation and that reconfiguration and growth are simulated or externally guided. Clay metaBricks are fixed construction units whose “cellular” behavior is resonant and thermal rather than computational or self-assembling.

2. DNA bricks as a model system for nucleated self-assembly

The DNA-brick work addresses a problem that had seemed straightforward to rationalize: the larger the number of distinct building blocks, the higher the expected error rate for self-assembly. Ke et al. disproved that argument experimentally in 2012, and Reinhardt and Frenkel then studied the phenomenon by Monte Carlo simulation of a highly coarse-grained model of a 998-brick cube (Reinhardt et al., 2014). In that model, each 32-nt strand is reduced to a single rigid particle whose four vertices define the locations of the four binding domains, each domain being 8 bp. Particles are placed on a cubic lattice of spacing $a$; two particles may interact only when they occupy next-nearest-neighbour lattice sites, at separation $\sqrt{3}\,a$, and only if one of the four patches on one brick points directly toward a complementary patch on the other.

The pairwise interaction is written as

$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$

where $\epsilon_{\mathrm{init}}>0$ is a small baseline repulsion of $100\,\mathrm{K}\cdot k_B$, and $\Delta G_{ij}$ is the free-energy change upon hybridisation of the most strongly matching 8 bp domain pair computed using the SantaLucia–Hicks nearest-neighbour parameters, including internal mismatches, dangling-end and terminal A–T penalties, and salt corrections as in Koehler and Peyret. Simulations are carried out in the canonical $(NVT)$ ensemble, typically with $N=998$ distinct bricks in a periodic box of side $62a$. To accelerate sampling of cluster motion, the virtual-move MC algorithm is used. The elementary moves are translation of a single particle or entire bonded cluster by one lattice step and rotation of a particle or cluster into one of 24 distinct tetrahedral orientations. Acceptance follows the Metropolis rule

$$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$

The central thermodynamic result is a nucleation barrier. Defining a correctly bonded cluster of size $\sqrt{3}\,a$0 as a connected set of particles in which every bond between nearest neighbours corresponds to one of the designed complementary patches, and letting $\sqrt{3}\,a$1 be the equilibrium number of such clusters, the reversible work to create a cluster of size $\sqrt{3}\,a$2 is

$\sqrt{3}\,a$3

Using umbrella sampling with the largest-cluster-size as the order parameter, the free-energy profile at $\sqrt{3}\,a$4 exhibits a maximum at $\sqrt{3}\,a$5, beyond which $\sqrt{3}\,a$6 decreases as the cluster grows to the full target size. As $\sqrt{3}\,a$7 increases, $\sqrt{3}\,a$8 grows roughly linearly, making nucleation exponentially rarer; as $\sqrt{3}\,a$9 decreases, $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$0 shrinks and eventually vanishes, leading to unconstrained aggregation of many small clusters.

The kinetics produce a narrow assembly window. Multiple brute-force cooling simulations starting from free monomers showed reproducible correct assembly only between about $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$1 and $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$2. In 14 runs at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$3, 12 yielded the correct cube, corresponding to $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$4; at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$5, $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$6; at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$7, $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$8; and below $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$9 essentially zero correct assembly was observed. The interpretation advanced in the paper is that mis-assembly is disfavoured because a slow nucleation step suppresses multiple simultaneous seeds, after which growth onto a single nucleus is rapid and error-free. Because the model incorporates no aspect of DNA other than binding properties, the authors infer that other low-valence patchy systems, including colloids, may also assemble truly complex structures if specificity and nucleation kinetics are properly designed.

3. Cardiac bricks from 4D-DLP-printed foldable scaffolds

In the cardiac context, cellular bricks are modular cylindrical scaffolds designed to create thicker engineered human myocardium with aligned sarcomeres and synchronous contractility (Hosseinabadi, 27 Aug 2025). The printing platform uses PUPEGDA synthesized by reacting PEG with hexamethylene diisocyanate at $\epsilon_{\mathrm{init}}>0$0 under $\epsilon_{\mathrm{init}}>0$1, terminating with hydroxyethyl acrylate and catalyst, and purifying by IPA precipitation, centrifugation, dialysis, and freeze-drying. The aqueous formulation contains LAP at $\epsilon_{\mathrm{init}}>0$2 and Ru:SPS at $\epsilon_{\mathrm{init}}>0$3. The DLP hardware has a box resolution of $\epsilon_{\mathrm{init}}>0$4 pixels, a theoretical lateral resolution of approximately $\epsilon_{\mathrm{init}}>0$5, an operational lateral resolution of approximately $\epsilon_{\mathrm{init}}>0$6, light intensity of about $\epsilon_{\mathrm{init}}>0$7 at $\epsilon_{\mathrm{init}}>0$8, layer thicknesses of $\epsilon_{\mathrm{init}}>0$9–$100\,\mathrm{K}\cdot k_B$0, and printing speed of about $100\,\mathrm{K}\cdot k_B$1–$100\,\mathrm{K}\cdot k_B$2 per layer. A DENSO robotic arm provides XYZ positioning under a sterile hood for GMP-compatible fabrication.

The planar precursor film can reach $100\,\mathrm{K}\cdot k_B$3. Its groove/ridge patterns have spacing of $100\,\mathrm{K}\cdot k_B$4–$100\,\mathrm{K}\cdot k_B$5 and groove depth of about $100\,\mathrm{K}\cdot k_B$6–$100\,\mathrm{K}\cdot k_B$7; a microporous mesh consists of a $100\,\mathrm{K}\cdot k_B$8 array of $100\,\mathrm{K}\cdot k_B$9–$\Delta G_{ij}$0 pores for mass transport. Self-folding is driven by differential swelling strain $\Delta G_{ij}$1 between patterned layer and bulk, with equilibrium curvature satisfying

$\Delta G_{ij}$2

where $\Delta G_{ij}$3 is the film thickness. For hydrated PUPEGDA, the elastic modulus is given as $\Delta G_{ij}$4–$\Delta G_{ij}$5 and, from tensile testing, Young’s modulus is $\Delta G_{ij}$6. The bending stiffness is

$\Delta G_{ij}$7

with $\Delta G_{ij}$8, and the swelling ratio is

$\Delta G_{ij}$9

with $(NVT)$0–$(NVT)$1 in culture media.

After printing, release into aqueous culture medium induces rapid fluid absorption and tubular self-folding. The final cylinder diameter is in the range $(NVT)$2–$(NVT)$3. Surface topography guides cell alignment: when grooves run circumferentially, sarcomeres align along the long axis of the cylinder. The cell formulation consists of iPSC-derived cardiomyocytes and human foreskin fibroblasts in a $(NVT)$4 ratio, suspended in collagen type I at $(NVT)$5 with ROCK inhibitor at $(NVT)$6. The casting workflow is: cast the CM–HFF–collagen mix into an agarose-coated PDMS/Teflon mold with flexible poles, allow gelation for $(NVT)$7–$(NVT)$8 min at $(NVT)$9, overlay with the printed PUPEGDA film, and release into warm culture medium so that the film self-folds around the tissue to produce a sealed cylindrical brick. Culture proceeds in standard EHM maturation media at $N=998$0 and $N=998$1 for up to $N=998$2–$N=998$3 weeks.

Quantitative metrics indicate that the wall thickness of collagen plus cells is about $N=998$4–$N=998$5, cell density is about $N=998$6, and each brick contains about $N=998$7–$N=998$8 cells. Contractility measured via edge displacement with MUSCLEMOTION yields a typical peak contraction amplitude of approximately $N=998$9–$62a$0 a.u.; using $62a$1 and cross-sectional area $62a$2 gives an estimated force of about $62a$3. Conduction velocity is reported as $62a$4–$62a$5, and Cx43 staining indicates a $62a$6–$62a$7 increase in gap-junction density over randomly organized controls. The sarcomere alignment index, defined as the fraction of $62a$8-actinin-positive striations within $62a$9 of groove direction, exceeds $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$0 after 21 days.

Scale-up and storage are integral to the concept. Fluidic chip roll-to-roll printing with automated robotic handling is reported to give throughput above $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$1 bricks per hour. Cryobanking uses freezing medium containing $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$2 DMSO and $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$3 FBS in maturation media, controlled-rate freezing at $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$4 to $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$5, and storage in liquid $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$6. Post-thaw viability is above $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$7, and functional recovery retains more than $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$8 of beating frequency and contractile amplitude. A plausible implication is that the “brick” here is not merely a scaffold geometry but a proposed unit of biomanufacturing and biobanking.

4. Reconfigurable phononic cellular bricks in metamaterial experiments

The LEGO-based system uses commercial bricks as a fully reconfigurable experimental platform for phononic crystals and metamaterial architectures (Celli et al., 2015). The stud-to-stud unit cell size is $$P_{\mathrm{acc}}=\min\{1,\exp[-\Delta E/(k_B T)]\}.$$9. Cylindrical single-height bricks of approximately $\sqrt{3}\,a$00 and double-height bricks of approximately $\sqrt{3}\,a$01 act as stub resonators. A dense periodic lattice consists of 420 bricks arranged on a roughly $\sqrt{3}\,a$02 stud region, while a coarse periodic lattice occupies every other stud, giving lattice constant $\sqrt{3}\,a$03 and 210 bricks. The host is a thin ABS baseplate of thickness approximately $\sqrt{3}\,a$04 that supports flexural $\sqrt{3}\,a$05 waves with low damping.

The theoretical framing distinguishes Bragg-based and locally resonant bandgaps. For a periodic medium with period $\sqrt{3}\,a$06, a Bloch wave solution

$\sqrt{3}\,a$07

must satisfy

$\sqrt{3}\,a$08

For a one-dimensional chain of identical resonators of mass $\sqrt{3}\,a$09 coupled by springs $\sqrt{3}\,a$10, the discrete dispersion relation is

$\sqrt{3}\,a$11

In the continuous plate limit, flexural-wave dispersion satisfies

$\sqrt{3}\,a$12

giving $\sqrt{3}\,a$13 at low $\sqrt{3}\,a$14. Bragg bandgaps are estimated to open when $\sqrt{3}\,a$15, with approximate center frequency

$\sqrt{3}\,a$16

Each pillar can also be idealized as a mass-spring oscillator with natural resonance

$\sqrt{3}\,a$17

so that excitation near $\sqrt{3}\,a$18 localizes energy in the resonators and produces a sub-wavelength hybridization gap.

Experimentally, the platform uses a Polytec PSV-400-3D scanning laser Doppler vibrometer, scanning the underside of the reflective-painted plate with sampling up to $\sqrt{3}\,a$19, $\sqrt{3}\,a$20 resolution, and 20 averages per point. A Bruel and Kjaer Type 4809 shaker drives a broadband pseudorandom signal from 0 to $\sqrt{3}\,a$21. Frequency-response measurements for the dense 420-brick array show two attenuation bands: a narrow locally resonant band at $\sqrt{3}\,a$22–$\sqrt{3}\,a$23 and a wide Bragg band at $\sqrt{3}\,a$24–$\sqrt{3}\,a$25. Changing the periodicity to $\sqrt{3}\,a$26 shifts the Bragg gap down to approximately $\sqrt{3}\,a$27–$\sqrt{3}\,a$28, whereas the $\sqrt{3}\,a$29 gap remains. In a random brick pattern, the $\sqrt{3}\,a$30 gap survives while the higher gap vanishes. Fine scans on a single central brick show in-phase motion of brick and plate at $\sqrt{3}\,a$31, strong brick tilting with a nearly still plate near $\sqrt{3}\,a$32, and renewed joint motion at $\sqrt{3}\,a$33.

These observations clarify a frequent conceptual confusion in metamaterials: periodicity is not required for the locally resonant gap, but it is essential for the Bragg gap. The platform also demonstrates application-level effects. A snake-like two-cell-wide defect in the dense crystal acts as a waveguide at both $\sqrt{3}\,a$34 and $\sqrt{3}\,a$35, and a “University M” crystal excludes waves from the M-shaped region at $\sqrt{3}\,a$36 and $\sqrt{3}\,a$37, illustrating seismic isolation. Because the bricks snap on by friction and can be rearranged nearly endlessly, the platform serves both rapid proof of concept and didactic visualization of dispersive wave phenomena.

5. Smart cellular bricks for decentralized inference and recovery

The smart-brick system realizes a physical three-dimensional modular architecture in which each brick executes the same learned local update rule without access to global state or positioning information (Moreno et al., 23 Sep 2025). Each module is a 3-cm cubic brick built from six 1.6 mm FR4 PCB faces slotted and soldered into a cube. Five faces are passive PCBs carrying only connectors and power/ground traces. The sixth face carries an ESP32-S2 WROVER microcontroller, a WS2812B RGB LED, a DC-DC converter with input $\sqrt{3}\,a$38–$\sqrt{3}\,a$39 and output $\sqrt{3}\,a$40, and a reverse-voltage protection diode. Mechanical connectors are six 6-pin 2.54 mm headers/sockets, one per face; two pins carry power at $\sqrt{3}\,a$41 nominal, two carry ground, and two provide a bidirectional digital link. Each brick exchanges messages only with up to six face-adjacent neighbors, and missing neighbors default to zero input.

Communication uses a custom PWM-based serial protocol. Each bit is transmitted as a short or long pulse, with a three-unit pulse as header. Each message contains 32-bit floats, one float per channel, repeated five times over a 3 s window. Incomplete or header-less messages are discarded. If no valid message arrives in time, the corresponding neighbor input vector is set to all zeros except for a fixed bias channel equal to 1. The Neural Cellular Automaton state is $\sqrt{3}\,a$42 and evolves according to

$\sqrt{3}\,a$43

with

$\sqrt{3}\,a$44

Here $\sqrt{3}\,a$45 is the alive threshold and $\sqrt{3}\,a$46 in simulation, $\sqrt{3}\,a$47 in hardware. The state vector is

$\sqrt{3}\,a$48

with one alive channel, 20 hidden memory channels, and 7 class-logit channels.

The local network first constructs a tensor $\sqrt{3}\,a$49 using a fixed $\sqrt{3}\,a$50 binary cross mask that picks out the six orthogonal neighbors plus self. The neural network then applies a masked $\sqrt{3}\,a$51 convolution to 84 channels with ReLU, followed by two $\sqrt{3}\,a$52 convolutions, $\sqrt{3}\,a$53 with ReLU and $\sqrt{3}\,a$54 linear with zero initialization, and a tanh-clamp of the 27 update channels. The total parameter count is approximately $\sqrt{3}\,a$55. Training uses per-voxel cross-entropy for shape classification and 7-way cross-entropy for damage localization, backpropagation through an unrolling of $\sqrt{3}\,a$56, gradient clipping, and Adam with learning rate $\sqrt{3}\,a$57. The dataset contains 487 voxelized ShapeNet shapes on a $\sqrt{3}\,a$58 grid: 103 planes, 114 boats, 125 tables, 61 chairs, 10 cars, 6 houses, and 68 guitars. Training is conducted for 4000 iterations with random $\sqrt{3}\,a$59.

Classification performance is high in simulation and demonstrable in hardware. The reported overall simulation accuracy is $\sqrt{3}\,a$60, with class-wise values of $\sqrt{3}\,a$61 for planes, $\sqrt{3}\,a$62 for chairs, $\sqrt{3}\,a$63 for cars, $\sqrt{3}\,a$64 for tables, $\sqrt{3}\,a$65 for houses, $\sqrt{3}\,a$66 for guitars, and $\sqrt{3}\,a$67 for boats. The global class is obtained by majority vote over the local class decisions $\sqrt{3}\,a$68. Hardware experiments on four test shapes, including a plane of 197 bricks and a guitar of 26 bricks, reach 100% consensus in all four cases with convergence in approximately 60 cycles, or 180 s. The authors also report robustness to out-of-distribution shape variations and minimal accuracy loss for most shapes with up to 15% silent or faulty bricks.

Damage detection extends the same decentralized framework. The class-logit channels are replaced by a small 3D-CNN encoder of the target voxelized shape, reduced from $\sqrt{3}\,a$69 to $\sqrt{3}\,a$70 and embedded in $\sqrt{3}\,a$71, with $\sqrt{3}\,a$72 up to 128 and about 10,000 parameters. Cells output a 7-way damage-direction logit corresponding to no damage or one of $\sqrt{3}\,a$73. Reported damage-detection accuracy is above 90% across all shapes. Recovery proceeds by iteratively spawning a new brick in the adjacent empty position whenever an alive cell predicts damage in direction $\sqrt{3}\,a$74, repeating until no cell predicts damage. Recovery accuracy is defined as $\sqrt{3}\,a$75, and increases with hidden dimension. For example, Aircraft rises from 0.48 at $\sqrt{3}\,a$76 to 0.91 at $\sqrt{3}\,a$77, Boat from 0.68 to 0.95, Table from 0.57 to 0.92, and Chair from 0.53 to 0.93. The limitations are explicit: there is no onboard actuation, hardware size and weight limit packing density and speed, and classification of small or rescaled shapes can fail.

6. Clay metaBricks as thermally and acoustically functional building units

The clay metaBrick adapts a standard hollow brick into a metamaterial building element by adding slit-connected cavities (Lemkalli et al., 2023). The unit dimensions are width $\sqrt{3}\,a$78, length $\sqrt{3}\,a$79, and height $\sqrt{3}\,a$80. Inside each brick are three parallel hollow alveoli with wall thickness approximately $\sqrt{3}\,a$81. At the top and bottom faces of each alveolus, a slit of width $\sqrt{3}\,a$82 and length $\sqrt{3}\,a$83 is cut, forming one one-degree-of-freedom Helmholtz resonator per cell. The air-filled cavity volume per alveolus is approximately $\sqrt{3}\,a$84, and the slit aperture area is $\sqrt{3}\,a$85. The brick can therefore be viewed as a periodic repetition of a two-dimensional cross-section consisting of alternating clay walls and slit-connected cavities of air.

The bulk material parameters are specified as follows. Clay has density $\sqrt{3}\,a$86, modulus $\sqrt{3}\,a$87, Poisson ratio $\sqrt{3}\,a$88, thermal conductivity $\sqrt{3}\,a$89, and heat capacity $\sqrt{3}\,a$90. Mortar has $\sqrt{3}\,a$91, $\sqrt{3}\,a$92, $\sqrt{3}\,a$93, $\sqrt{3}\,a$94, and $\sqrt{3}\,a$95. Air has $\sqrt{3}\,a$96, $\sqrt{3}\,a$97, $\sqrt{3}\,a$98, and sound speed $\sqrt{3}\,a$99. A Maxwell–Eucken estimate for effective thermal conductivity at porosity $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$00 gives $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$01. The Helmholtz resonance frequency is

$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$02

with $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$03, yielding approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$04 for the stated geometry. Acoustic transmission is also described by a resonator impedance $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$05 and a transfer-matrix formulation for multilayer walls.

The numerical modeling couples acoustic and elastic fields in 2D through

$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$06

in air and

$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$07

in clay, with incident plane-wave radiation, perfectly matched layers, Floquet–Bloch periodicity, and an unstructured triangular mesh with maximum element size $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$08 at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$09. Thermal modeling uses

$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$10

with a 24 h sinusoidal boundary temperature on the outer face, fixed $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$11 on the inner face, and $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$12.

Experimental validation reports substantially improved acoustic and thermal performance relative to a standard hollow brick wall. In the acoustic test, using a double-steel-box method with logarithmic chirp from 50 to $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$13 at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$14, the metaBrick reaches transmission loss up to $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$15 versus $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$16 for the standard brick over 200–3500 Hz. In the thermal test, one side of the wall is exposed to a $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$17 heat source while the interior remains at $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$18 ambient. After 600 min, the standard-brick interior reaches $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$19, corresponding to a 38% reduction relative to $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$20, whereas the metaBrick interior reaches $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$21, corresponding to a 46% reduction and a $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$22, approximately 8%, extra thermal drop. The R-value estimate rises from approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$23 to approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$24. Under uniaxial compression, the standard brick has strength approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$25 and the metaBrick approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$26, still above the $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$27 threshold for building use.

The design trade-off is explicit. The metaBrick improves thermal insulation and adds approximately $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$28 of transmission loss over $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$29–$$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$30, but loses approximately 33% compressive strength. The authors attribute the gains to increased path length for heat conduction and to Helmholtz-resonator slits that trap and dissipate acoustic energy in the hundreds-of-hertz range. Design guidelines in the summary therefore emphasize tuning $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$31 and $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$32 to place $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$33 in the relevant spectrum while maintaining $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$34.

7. Cross-domain design principles, limits, and research significance

Taken together, these studies suggest several recurring design principles for cellular bricks across otherwise dissimilar physical regimes (Reinhardt et al., 2014, Hosseinabadi, 27 Aug 2025, Celli et al., 2015, Moreno et al., 23 Sep 2025, Lemkalli et al., 2023). First, the operative information is local. DNA bricks use sequence-encoded complementary patches; cardiac bricks use local groove topography, porosity, and swelling mismatch; LEGO phononic crystals use nearest-neighbour periodicity and local resonator dynamics; smart bricks exchange messages only with face-adjacent neighbors; clay metaBricks derive functionality from slit-cavity cells repeated through the structure. Second, restricted connectivity is important. DNA bricks have fixed valence four; smart bricks have up to six face-adjacent communication links; LEGO and clay systems rely on a low-dimensional local neighborhood that makes dispersion and resonance analyzable. Third, geometry is not ancillary but constitutive: tetrahedral patch placement, groove orientation, lattice constant, cube adjacency, and slit-connected alveoli all directly determine global outcome.

The literature also converges on the importance of balancing enhancement against failure modes. In DNA bricks, the nucleation barrier must be moderate: too high and nucleation is exponentially rare, too low and aggregation dominates. In cardiac bricks, curvature, groove amplitude, and porosity must be tuned so that folding, transport, and anisotropy coexist. In LEGO metamaterials, changing lattice constant, stacking mass, or coupling stiffness tunes different bandgap mechanisms rather than one generic attenuation effect. In smart bricks, decentralized inference is robust to missing neighbors and communication faults, yet classification can fail for small or rescaled shapes and the hardware still lacks actuation. In metaBricks, improved transmission loss and R-value come with lower compressive strength.

A further implication is methodological. Each field adopts a reduced local model that is rich enough to recover collective behavior. The DNA-brick model omits all DNA details except binding properties. The cardiac-brick model reduces self-folding to differential swelling strain and standard hydrogel mechanics. The LEGO platform uses simple stub resonators to expose Bragg and locally resonant physics visually. The smart-brick platform compresses decentralized morphology processing into a 28-channel NCA state with approximately 25,899 parameters. The clay metaBrick is captured by coupled acoustic, elastic, and thermal models with experimentally accessible geometry. This suggests that the “brick” abstraction is valuable precisely because it preserves local rules while suppressing superfluous microscopic detail.

For research practice, the main significance of cellular bricks lies in their ability to connect synthesis, fabrication, and theory at the scale of the module. DNA bricks show that high-fidelity assembly can arise in a system of approximately 1000 distinct components. Cardiac bricks reposition scaffold design as modular tissue manufacture with cryobanking and throughput above $$E_{ij}=\epsilon_{\mathrm{init}}+\Delta G_{ij}(T,[\mathrm{Na}^+],[\mathrm{Mg}^{2+}]),$$35 bricks per hour. LEGO metamaterial bricks provide a nearly endless spectrum of rapidly testable topologies. Smart bricks demonstrate a physical realization of decentralized self-recognition and damage detection in three dimensions. Clay metaBricks show that a simple geometric modification can convert a standard construction element into a multifunctional thermal-acoustic metamaterial. The broad lesson is not that all bricks behave alike, but that local module design can be a sufficient control surface for complex global behavior when the interaction rules are appropriately engineered.