their internal angle of 108 • is incommensurate with 2π. When positive Gaussian curvature is introduced into the surface, the gaps between the pentagons close while the plane turns into a sphere or radius R. When R = √ 3/2φ, where φ is the Golden Ratio, the pentagons fold into an unfrustrated non-Euclidean crystal: the regular dodecahedron (Fig. 1b) . Similarly, the reference configuration of any regular polyhedron can always be realized as a non-Euclidean crystal in a 3D curved space [5, 8] called honeycomb, or polytope when the number of tiles is finite. Regular honeycombs are classified by the Shläfli notation {p, q, r} where q is the number of p−sided regular polygonal faces around a polyhedron's vertex, and r is the number of polyhedra around an edge. Calling Θ the dihedral angle of a polyhedron, when rΘ < (>)2π, the honeycomb has positive (negative) Gaussian curvature. Important previous work examined properties of 3D non-Euclidean crystals, as well as rich classes of topological defects that relieve stress when the reference configuration is "flattened" to the Euclidean 3D space [8] , such as doubletwist texture of the cholesteric blue phase [9, 10] , metallic glasses [11, 12] , and Frank-Kasper phases [13, 14] .
In this paper, we show how non-Euclidean crystals provide us with sets of "reference metrics"ḡ of the stressfree packing of these polyhedra, and thus offer a starting point to construct an energy of the assembled structures, the minimization of which guides us in the search for self-assembly morphologies. Different from previous theories of curved crystals, the self-assembly processes we consider here are driven by competing attractive and repulsive interactions between the similarly charged inorganic NPs at low volume fractions. Moreover, in NP self-assembly, surface tension is typically low, leading to arrested growth in certain directions [15] , rather than proliferation of topological defects. Thus, instead of forming condensed phases with defects, the NPs have the freedom to adopt low-dimensional morphologies to minimize frustration.
We construct the energy of the assembly by adding up the energy associated with (i) the aforementioned elastic frustration, E elastic , (ii) the boundary of the assembly, E boundary , (iii) electrostatic repulsion, E repulsion , and (iv) binding between the NPs, E bind ,
The elastic energy is E elastic = 1 2 M E elastic √ detḡ d 3 x, where √ detḡ d 3 x is the reference volume element andM is a region in the honeycomb. The (Euclidean) actual FIG. 1: a, Five tetrahedra can almost form a pentamer; Twenty tetrahedra can almost form an icosahedron. b, A 2D example of geometric frustration: pentagons do not tile Euclidean surface, but can form a regular crystal (dodecahedron) on the 2D sphere S 2 . c, Tetrahelices are chiral linear assemblies of tetrahedra. d, Chiral ligands induce a twist between neighboring tetrahedra, and select and handedness of the tetrahelix. When the tetrahelix and the relative twist have the same (opposite) handedness, the distance between 4th-nearest neighbor tetrahedra's centers d (1, 4) increases (reduces), thus lowering (increasing) their electrostatic repulsion. At the same time, the edges of the tetrahelix become more (less) coiled. e, In 3D Euclidean space, two tetrahelices cannot fit side-to-side. f, Two tetrahelices fitting perfectly together in S 3 (a stereographic projectio). g, The base icosahedron of the 600 cell. The 4 toroidal shells are indicated by color gradient. h, Topology of the tetrahelices inside the 600-cell, organized in 3 concentric toroidal shells. Tetrahelices are represented abstractly as triangular prisms with a twist (circular arrow). The triangular faces are identified, so each prism represents a closed ring. Under stereographic projection, the topology of the shells is equivalent to the top right inset. i, The coordinate in Eq. (3) (stereographic projection). j, Cutting a ribbon out of the Clifford Torus (here represented as a square with opposite edges identified). The θ axis is at the same time the long side of the ribbon and the long axis of the R tetrahelices.
metric of the assembly g cannot be equal to the honeycomb's non-Euclidean metricḡ (which describes the ideal, stress-free, distances between the NPs) everywhere, and a strain = (g −ḡ)/2, necessarily develops. Close to any local minimum, E elastic can be expanded in powers of the strain tensor:
where µ, λ are the Lamé coefficients, ν, σ = 1, 2, 3 are the 3 spatial directions and indices are contracted withḡ. The boundary term E boundary = ∂M γ(σ) detḡ(σ)d 2 σ where γ(σ) is the surface tension, σ is a point on the boundary, and detḡ(σ) d 2 σ is the area element. Screened as well as long-ranged electrostatic repulsion, commonly found in NPs are encoded in E repulsion . The binding energy E bind denotes the energy released while the NPs bind, and is proportional to the volume of the assembly.
As mentioned above, we are interested in the case where complex low-dimensional morphologies are adopted by the NPs to minimize the frustration in 3D Euclidean space. The question is then about choosing the appropriate sliceM and solve for the assembled structure. The answer lies in two factors: pathway dependence and low-stress directions in the honeycomb. Any self-assembly follows a pathway through the formation of intermediate structures, such as small clusters, fibers [16] , or sheets, which are precursors to the final configuration. Correspondingly, the reference honeycomb can often be decomposed into sub-structures of representing the precursors. We associate a pathway to every decomposition of the reference honeycomb (or equivalently, subgroups of the honeycomb's global symmetry group). As the assembly progress along the kinetic pathways, the stress builds up, often anisotropically. This anisotropy can guide the selection ofM and eventually lead to the solution of the morphology.
Here we specialize the model to the assembly of charged tetrahedral CdTe NPs binding via chiral surface ligands. These NPs in experiments self-assemble into enantiopure and uniform helices at the scale of microns (Fig. 2b) [15] . In order to characterize the assembly of these NPs, we choose the reference honeycomb to be the 600-cell polytope (Schläfli symbol {3, 3, 5}). This is a periodic tiling of the 3-sphere S 3 (R) (i.e. the surface of a ball in 4D Euclidean space) by regular tetrahedra with the lowest curvature, and thus least stressed in Euclidean space.
How do tetrahedra assemble when they are brought together by attractive interactions? It is well-known that they can form infinite straight 1D helices with no frustration, either Left-handed (L) or Rigth-handed (R), called "tetrahelices" (Fig. 1c and SM, Sec. 1). The chiral ligands induce a small rotation angle between two bound tetrahedral NPs, rather than perfect face-to-face binding. As shown in Fig. 1d , this twist energetically selects tetrahelices with the same handedness as the ligands. Under the spontaneous tendency of tetrahedra to form tetrahelices, even a small chiral bias in the ligands can propagate along the tetrahelix, giving it the same handedness. In the following discussion, we use this chiral symmetry breaking to select the fibration, but ignore the higher order geometric effect of the small twist on the morphology.
Self-assembly of tetrahelices is geometrically frustrated in 3D Euclidean space, because the twist forbids two homochiral tetrahelices to be glued side-to-side (Fig. 1e) , but it can be realized in S 3 (R), where twist is compensated by curvature: when R and the tetrahedron's edge a satisfy R = φa, the tetrahelices appear as closed parallel rings of 30 tetrahedra touching perfectly side to side (Fig. 1f) . Remarkably, 20 such (homochiral) terahelices organized in 3 nested toroidal shells form the 600-cell [17] , and each tetrahelix can be associated with one face of the base icosahedron of the 600-cell (Fig. 1g,h) . The global structure of linked rings has the topology of the Hopf fibration [18, 19] (see SM, Sec. 2). Starting from all R or all L tetrahelices, this construction leads to the same (achiral) 600-cell, so the latter has two chiral decompositions: one contains all L tetrahelices the other all R. They are the reference configuration for self-assembling NPs coated with L or R ligands, respectively.
It is convenient to parameterize the 600-cell with dimensionless coordinates Φ µ = (α, β, θ) ≡ x µ /R on S 3 . The vertices of the 600-cell belong to S 3 (R), embedded in 4D Euclidean space E 4 as:
where X ν are Cartesian coordinates of E 4 , and α ∈ [0, π/2] and β ± θ ∈ [0, 2π[ are angles on S 3 (R). Fixing α = 0, π/2 we find toroidal surfaces parameterized by (β, θ) (the lines α v = 0, π are great circles). The 120 vertices of the 600-cell are recovered by choosing (2α v , β v ) on the v = 1, ..., 12 vertices of the base icosahedron and θ n = (2πn)/10 + δθ n with n = 0, ..., 9. The offset δθ n is 0 at the North and South poles (α v = 0, π) and it is π/10 at the icosahedron's Northern and Southern "Tropical latitudes" 2α N,S ≡ π 2 ∓ arctan 1 2 (see SM Sec. 3). The curvilinear axis θ is aligned with the vertex strands on the tetrahelices (Fig. 1i) . A stereographic projection of Eq. (3) in E 3 shows that the tetrahelices are R. Exchanging θ ↔ β performs a reflection X 3 → −X 3 in E 4 . This reverses the orientation of S 3 (R), exchanging R with L tetrahelices which are now aligned along β.
The reference metricḡ has componentsḡ µν (α) = ∂ µ X· ∂ ν X where ∂ µ = ∂/∂x µ :
The metric depends on α only, so the surfaces Σ α (α = const.) are flat tori, as shown by Bianchi in 1894 [20] .
Thin shell self-assembly The next step is to choose the sliceM which represents the low dimensional assembly (with the thickness h much smaller than width W and length L) at low surface tension. The (one-tetrahedron-thick) shell between two special toroidal surfaces Σ N,S located at α = α N,S (with the mid-surface at the Clifford torus α = π/4) provides such a natural reference sliceM , because Σ N,S are smooth physical boundaries between neighboring shells of tetrahelices (Fig. 2a) . Σ N,S are special also because they are made of geodesics of S 3 , thus they contain "unfrustrated directions" for the tetrahelices to grow along. We now use Eq. (4) in Eq. (1) to compute the effective energy of the thin assembly, and minimize it to find the morphology of the assembly. Since tetrahelices in 3D Euclidean space are open chains and not closed rings, we should interpret the 600cell order only as a local reference configuration: the global topology of the tetrahelices is not fixed by the topology of the reference 600-cell. This is different from e.g. models of the cholesteric blue phase, which used the global topology of the 3-sphere [9] to find the frustrated ground state. Therefore, we cut the shell along the θ and β directions into an open rectangular prism (where θ, β are no longer bounded by [0, 2π[, Fig. 1j ). We expand g ij (α) around the mid-surfacē
where we defined the parameter t ≡ R(α−π/4) along the thickness direction, similarly to a thin shell in elasticity [21] . The reference first and second fundamental forms
The second fundamental form is off-diagonal, so it favors pure twist around the axis defined by the tetrahelix direction. The twist between the contact surfaces of the tetrahelices packed in the shell generates stress between the upper and lower surfaces of the sheet. This geometric frustration manifests in the fact thatā andb are incompatible in a surface embedded in Euclidean space for which detb/ detā = −1/R 2 , while K(ā) = 0, violating Gauss' Theorema Egregium detb/ detā = K(ā).
We minimize the energy of the shell to find the actual first and second fundamental forms a and b. We first consider the elasticity part,
where nowM 0 is the Clifford torus and dA = R 2 √ detā d 2 Φ is the area element of the mid-surface. The out-of-plane bending energy E bend
and the in-plane stretching energy, E stretch elastic depends similarly on a −ā with κ replaced by k, with stiffness
where h is the thickness of the sheet, Y, ν are the in-plane Young's modulus and Poisson ratio (see SM, Sec. 6).
The minimization of E shell elastic has analytic solutions in two limits [22] [23] [24] [25] [26] : "wide" ribbons (W √ Rh) are stretching-dominated, so a ā, and the solution is a cylindrical helical ribbon. "Narrow" ribbons (W √ Rh) are bending-dominated, so b b , and the solution is a helicoid. This crossover comes from the competition between the bending energy, E bend ∼ LW 5 κ 4 , and the stretching energy, E stretch ∼ h 2 · LW κ 2 [22] .
Interestingly, the assembled ribbons observed in Ref. [15] have W > √ Rh, which may lead to the conclusion that they belong to the wide ribbon limit. However, the observed morphologies are much closer to helicoids. As we analyze below, this is due to the bending stiffening effect of the electrostatic repulsion.
Bending stiffening from electrostatic repulsion In all realistic cases, NPs carry some charge from spontaneous ionization of their surface and adsorptions of charged species from the media. Tetrahedral CdTe NPs in this experiment are negatively charged. The electrostatic repulsion, screened by ions in the solution, effectively stiffens the bending rigidity and enlarges the bending-dominated regime to much wider ribbons. Assuming a uniformly charged shell with total charge q and charge density ρ = q/hW L, the potential at a point R(σ) on the sheet is
where d(σ, σ ) = |R(σ) − R(σ )| is the 3D Euclidean distance between two points on the sheet, ε is the dielectric constant, and ξ is the Debye screening length, which depend on the solvent. We will study the regime h ξ W , where repulsion has a significant effect on the bending stiffness, but is still a short range force.
The electrostatic energy density E rep = hρ φ(σ) can be written as an effective bending energy (SM, Sec. 8, 9) :
We neglected corrections to the stretching energy because in the thin sheet limit, the stretching and bending elas- 
where
where
is the electric self-energy of a patch of size ξ on the ribbon. Similar corrections to the elastic moduli were studied for charged fluid membranes in ionic solutions in [27] [28] [29] [30] . The correction tob affects its traceless part,b 0 ij =b ij − Tr(b)δ ij /2, which obtains an overall factor −1 up to numerical factors of order 1, while the trace part is unaffected (see SM, Sec. 9). Thus, repulsion has two effects: it increases the bending rigidity [Eq. (13) ] and lowers the reference curvature [Eq. (15) ]. These two effects are related, as Q/κ ∼ , so in the limit of strong repulsion, 1 and bending rigidity is dominated by repulsion.
An important consequence of the correction is that the reference radius R of S 3 (R) is enlarged into R. In this strong repulsion regime, the characteristic length scale for the bend-stretch crossover, √ Rh, can exceed the physical width (W √ Rh), bringing a ribbon into the bending-dominated regime even when W > √ Rh. Equation (15) gives the critical charge density ρ above which the self-assembly is bending-dominated is ρ c = W ξ −3/2 Y /a.
Without loss of generality, we describe an assembly of R tetrahelices, with long axis aligned with θ. In the reference configuration, they are packed side by side across the direction β. The ligand's R-chirality favors the formation of long R-strands of tetrahedra while inter-helices bonds are more frustrated, so we conjecture that the longest dimension L of the thin-shell is parallel to the R-helices. We therefore cut a rectangular region out of the Clifford torus with the long side L parallel to θ and the short side W parallel to β (Fig. 1j) . In the limit of L W the actual metric a needs to be independent of θ, so stress grows only in the width direction, minimizing the energy. In the repulsion-controlled bending-dominated regime, we impose the constraint b =b and find the actual 2D metric a ij by minimizing E stretch elastic :
The embedding of the mid-surface, under free boundary conditions, is an R helicoid (Fig. 2c , SM Sec. 7)
R(β, θ) = R (sinh β cos θ, sinh β sin θ, β) .
Importantly, this R helicoid has the same chirality as the tetrahelices and the ligands. This prediction is consistent with the structures seen in [15] , where high R ligand concentration systematically leads to R sheets with nearly perfect enantioselectivity. Similarly, for L ligands, we exchange β ↔ θ in Eq. (3) which reverses the handedness of the tetrahelices, but leavesb invariant. Imposing ∂ β a ij = 0 and minimizing E stretch , we find that now the helicoidal solution must be L (Fig. 2c ). The pitch of the helicoid is given by
Hence the repulsion between NPs is a crucial design parameter: the pitch can be tuned by the charge density ρ on the NPs, the screening length ξ via the control of ion concentration in the solution, and the solution's dielectric constant ε (see Methods). It is worth noting that we started from a rectangular sheet of length L and width W , but the resulting pitch p is independent of L and W . This is true for the repulsion-controlled bending-dominated regime we discussed above. In the intermediate regime where stretching and bending energies are comparable, p may depend on W , as discussed for the purely elastic case in Ref. [22] .
In this experiment, the width of the helicoid is determined by the competition between the binding energy E bind ∼ − bind × (hW L) and the (repulsion corrected) elastic energy of the ribbon,
At equilibrium, ∂(E bind + E elastic+repulsion )/∂W = 0, giving
Because stress does not accumulate along the long axis of the helicoid, the length L is controlled by the kinetic processes of the assembly. In experiments, twisted sheets were produced in a mixture of water and methanol, while concentration of Cadmium ions was used to control the kinetic rate of the assembly. As discussed in Methods, ξ and ε depend on the concentration of ions and the water/methanol ratio. The predicted dependence of the pitch on ξ, h, ε [Eq. (18) ] agrees qualitatively with the experimental data (Fig. 2d) .
Additionally, the measured thickness h of the ribbons is much greater than a single tetrahedra (∼ 5nm in size). This may indicate that the assembled helices consist of multiple layers of stacked single-tetrahedral-thick helicoids. Because the helicoids are minimal surfaces, this stacking does not lead to additional stress. As a result, we expect the aforementioned predictions of the morphology still holds in the multi-layer case, as verified via the agreement in Fig. 2d . The theory described above provides a strategy to experimentally tune the pitch of chiral helices by adjusting charge density, solvent properties, and curvature of the reference metric, offering control of a range of physical properties. By measuring the circular dichroism (CD) spectra of self-assembled helicoids in dispersions, we find that water/methanol ratio induces different chiroptical responses at a range of wavelengths via its control of the pitch (Fig. 3bc and Methods). We also numerically simulated circularly polarized light interacting with self-assembled helicoids with geometric parameters in the range produced from the experiments (varying the pitch in the range 200nm−1600nm) (Fig. 3a) , finding a monotonic increase of the CD peak with the pitch (Fig. 3d) , the linear part (at smaller pitch) of which agreeing with experiment ( Fig. 3e) . Importantly, the amplitude of the CD spectra is much higher than for typical biological molecules and the maximum located in the near-infrared part of the spectrum suitable for biomedical imaging, remote sensing, and information technologies.
We have presented a non-Euclidean self-assembly theory for polyhedral NPs, showing that the geometrically frustrated self-assembly of tetrahedral NPs subject to chiral binding and electrostatic repulsion gives helicoidal structures in agreement with experimental observations.
Although this theory focuses on the equilibrium morphology, it also provides insight into the assembly's kinetic pathways. In particular, the translational symmetry of the 2D reference metricā means that the assembly of these sheets is scalable: smaller pieces of the sheet can merge and form a larger sheet, giving the self-assembly process a high yield of the target structure. If instead the in-plane metric was not translationally invariant, the local NP coordination would be different in different locations, and the sheet could only grow from one nucleation seed, which is much slower. The scalability of the metric greatly increases the yield, which was also observed experimentally, and provides an important measure when this theory is applied to a new new self-assembly system.
In addition, as observed in previous studies of geometrically frustrated systems, topological defects such as disclinations may arise, easing the stress at the expense of losing local attraction. Here, similarly, extra tetrahelices can be inserted as disclination lines in the 600-cell, increasing its radius R and decreasing its curvature. Under this consideration, the proposed continuum model, at a lower reference curvature, can also be viewed as a continuum limit of tetrahedra assemblies with a continuous distribution of disclinations. We expect this to be a more realistic model of the experimentally observed morphologies, given their mesoscale size.
We would like to also point out interesting relations between this work and the self-assembly of amphiphilic molecules and peptides into chiral ribbons [31] [32] [33] . Although the elastic energy of these molecular assemblies shares similarities with our theory, the origin of the twist comes from chiral bonding between the molecules, which is intrinsically different from the geometrically frustrated polyhedral tiling we consider here.
Generalization of this theory to more varieties of NPs, which exhibit tilings in either spherical or hyperbolic space, opens a suite of intriguing new questions for future study. The new morphologies that will emerge, may lead to novel materials capabilities. Besides chiral optical response in Fig. 3 , the engineering of self-assembled structures in non-Euclidean space is applicable to realization of metamaterials with unique mechanical, acoustic, catalytic, and biological properties.
Materials. L-Cysteine (L-Cys) hydrochloride monohydrate, hydrochloric acid (HCl) sodium hydroxide (NaOH), sulfuric acid (H 2 SO 4 , 98%) and methanol were purchased from Sigma-Aldrich. Cadmium perchlorate hexahydrate (Cd(ClO 4 ) 2 · 6H 2 O) was obtained from Alfa-Aesar. Aluminum telluride (Al 2 Te 3 ) was purchase from Materion Advanced Chemicals. All chemicals were used as received. Ultrapure deionized water (18.2 MΩ) was used for all solution preparations.
Synthesis of CdTe NPs. The synthesis of CdTe NPs were according to previous publications [34] with appropriate modifications. Briefly, 0.985 g Cd(ClO 4 )2 · 6H 2 O and 0.99 g Cysteine hydrochloride monohydrate were dissolved in 100 mL deionized water. The pH of the solution was adjusted to 11.2 with 1.0 M NaOH. The obtained solution was transferred into a 250 mL three-neck round-bottomed flask and connected to a 50 mL threeneck round-bottomed flask by tubes. The system was quickly purged with nitrogen for 30 min to remove all the oxygen in the glasses and solution. Then 0.10 g Al 2 Te 3 was added into the small flask and purged another 30 min to remove any possible oxygen in the system. 10 mL 0.50 M H 2 SO 4 was quickly injected into the small flask to react with Al 2 Te 3 to generate H 2 Te gas, which was slowly purged into the reaction solution of cadmium precursor by nitrogen flow. The reaction solution was refluxed at 100 • C for 8 h to obtain CdTe NPs in a size of 4.5 ± 0.42 nm. The as-synthesized NPs need to be wrapped with Al foil and aged as least three days before further assembly behavior.
Self-assembly of CdTe NPs. The self-assembly of CdTe NPs into helices with a series of pitch lengths was referred to our recent publications [15, 35] with appropriated modifications. Firstly, 500 µL CdTe NPs with aging time beyond 3 days were mixed with 20 µL 0.10 M Cd(ClO4)2. The pH value of the mixed solution was adjusted to 8.0 with 1.0 M HCl. Then different volumes of methanol from 500 to 5000 µL were respectively added into the 500 µL above solution to initiate the selfassembly of CdTe NPs. The obtained turbid solution was incubated at room temperature under light irradiation for 3 days to assembly NPs into helices. Afterwards, the assembled helices were centrifuged at a speed of 6000 rpm for 3 min and dispersed in water to wash unassembled NPs by another 2 times' centrifugations in the same conditions as above. The obtained helix was finally dispersed in water for further measurements and characterizations.
Characterization. CD and extinction spectra were acquired using J-1700 CD spectrophotometers with a PMT detector and an InGaAs NIR detector. All the spectra were measured in a quartz cuvette with a light path of 10 mm. The zeta-potential were measured by Zetasizer Nano ZSP (Malvern Instruments Ltd., GB). SEM images were taken by FEI Nova 200 Nanolab Dual Beam SEM with an acceleration voltage of 5 kV and a current of 0.4 nA. For counting the geometrical parameters of the helices, the middle region of the helices was used for analysis and more than 50 helices were counted.
Calculation of Debye screening length. The Debye screening length, ξ, was calculated using
where e is the elementary electric charge, N A is the Avogadro's number, z i is the charge number (valence) of i th component, c i is the molar concentration of i th component, ε r is the relative electric permittivity of the electrolyte, ε 0 is the electric permittivity of vacuum, k is the Boltzmann constant, and T is the absolute temperature. For 500 mL CdTe NPs solution before mixing with methanol, the ions in the solution were consisted of Na + (0.1635 M), Cl − (0.0851 M), Cd 2+ (0.0275 M) and ClO 4 − (0.0315 M). After mixing with different volume of methanol, the concentration of each ion was diluted to different extents to get a series of Debye screening lengths.
Dielectric constant of water/methanol mixtures. The dielectric constants of water/methanol mixtures were according to Ref. [36] , which summarized a polynomial formula for the dielectric constant of methanol/water mixtures with the percentage of water in the mixtures based on a series of reported dielectric values:
where x the molar fraction of water in methanol/water mixtures. FDTD Simulations. The CD spectra for nanohelices with variable pitch lengths were simulated with commercial software package Lumerical FDTD Solutions. The size of nanohelices used for simulation were according to SEM images of the assembly of L-CdTe under the water/methanol ratio of 1 : 3, which generated a righthanded ribbon with a length, width, thickness and pitch of around 1200, 300, 100 and 600 nm. The pitch was varied from 200 to 1600 nm while kept other geometric parameters the same. To study the pitch effect on CD peak positions, the nanohelix was illuminated by left/right-handed circularly polarized light (CPL) consisted by two total-field scattered-field (TFSF) sources with the same k−vector but with a phase difference of ±90 • for left/right-handed CPL, respectively. Two analysis groups consisted of a box of power monitor were used to calculate the absorption and scattering intensity, respectively. The CD spectra were recorded as the difference of the extinction under left/right-handed CPL. The simulation wavelengths were set in the range of 300 to 2000 nm. The refractive index for CdTe was obtained from the Sopra Material Database. The refractive index of water backgrounds was 1.33. The mesh size was 10 nm. The orientation of nanohelices were considered in the simulation. The nanohelices were placed in a parallel, perpendicular and 4π-averaged orientations [32, 35] in comparison with the k-vector of incident photons, which show nanohelices under perpendicular orientation have a similar peak CD and extinction peak position (Fig. 3) . In this section we discuss the geometry of tetrahelices, which are straight one-dimensional crystals of tetrahedra (Fig. S1) .
In a tetrahelix, neighboring tetrahedra attach to one another with perfect face-to-face binding. It is also called a Bernal spiral [S1, S2]. We can generate a tetrahelix starting from a tetrahedron whose vertices are given by 4 vectors {e 0 , e 1 , e 2 , e 3 }:
Where the set with positive (negative) coordinates generates a Right(Left)-handed tetrahelix. Consider the face generated by 3 vertices F = e 1 , e 2 , e 3 , we define its center c 123 = 1 3 (e 1 + e 2 + e 3 ) and its outward normal unit vector
The neighboring tetrahedron has vertices {e 1 , e 2 , e 3 , e 4 } where
The tetrahelix is generated by repeated application of the procedure described above. Given the tetrahedron
Every iteration rotates the normal vector N i+1,i+2,i+3 so the tetrahedra twist around the central axis of the helix. The set (S1) generates a Right-handed tetrahelix under (S4) . The tetrahelix can be inscribed in a cylinder: in this sense we can say that it is a straight array of tetrahedra. We identify the axis of the tetrahelix with the axis of the circumscribed cylinder. The tetrahelix breaks mirror symmetry, so it exists in a Right-handed and a Left-handed version. Two tetrahelices cannot perfectly bind face to face in Euclidean space, as it is illustrated in Fig. 1e in the main text. This implies that a structure assembled from tetrahelices is frustrated in flat space: we must introduce stress to fit the tetrahelices together. Hence, a crystal of tetrahelices must contain residual stress in flat space.
In this section we discuss the internal structure and the symmetry group of the 600-cell, which is a crystalline tiling of tetrahedra in curved space.
Five tetrahedra can be glued face-to-face in 3D Euclidean space E 3 around a common edge e, forming a pentamer (Fig. 1a in the main text) . Since the dihedral angle of a tetrahedron is Θ = arccos 1/3 ∼ 70.5 • , the tetrahedra occupy a dihedral angle of 5Θ ∼ 352.5 • , leaving an empty wedge bounded by two triangular faces (say F 1 , F 2 ) with a deficit angle ∆Θ = 2π − 5Θ ∼ 7.5 • . Since the face-to-face attachement fails to propagate across E 3 , the cluster is geometrically frustrated in flat space.
Topologically, the missing wedge can be closed by identifying F 1 with F 2 . This operation concentrates positive Gaussian curvature (∆Θ > 0) around the edge e [S3] , because a vector rotates by an angle ∆Θ when parallel transported around e. Repeating this construction around every edge, we obtain a regular network of edges carrying positive Gaussian curvature. Around every edge we find 5 regular tetrahedra, and around every vertex there is a perfect icosahedron made of 20 regular tetrahedra. If the structure could exist in flat 3D space, this icosahedral order would give the densest (100% volume fraction) local packing of tetrahedra. The structure is a piecewise-flat regular tessellation of the 3-dimensional sphere called the 600-cell or polytope {3, 3, 5} (in Shläfli notation) [S4] , in analogy with polygonal tessellations of the 2D sphere S 2 . The regular arrangement of 600 tetrahedra realizes a non-Euclidean crystal [S5] of finite size. It contains N 0 = 120 0-dimensional (0D) vertices, N 2 = 720 1D egdes, N 3 = 1200 2D faces and N 3 = 600 3D tetrahedral cells. In 3 dimensions, Euler's formula generalizes to the alternating sum:
were P is a polytope. Recalling all orientable 3D manifold (like S 3 (R)) have zero Euler characteristic (χ = 0) we can verify that {3, 3, 5} satisfies Eq. (S5). Unlike the icosahedral net on S 2 , the edges beteween the fibers' vertices are projected onto disjoint great circles in S 3 , so the net is also a well-behaved coordinate grid on S 3 .
Geometrically, in the same way as polyhedra (2D polytopes) can be inscribed in a 2D sphere, the 600-cell is inscribed in the 3-dimensional sphere S 3 (R), i.e. the surface of a 4D ball of radius R in E 4 . Introducing Cartesian coordinates X µ (µ = 1, ..., 4) in 4-dimensional Euclidean space E 4 and choosing the origin at the center of S 3 (R), the vertices of the 600-cell belong to the hypersurface
The radius R of the circumscribed sphere S 3 (R) is uniquely determined by the edge length a via
where φ = (1 + √ 5)/2 is the Golden Ratio. The 600-cell can be viewed as a tessellation of S 3 (R), i.e. the 3D version of the triangulation of S 2 . The edges of the 600-cell are straight segments in the flat geometry of E 4 . They are entirely contained inside the 4D ball enclosed by S 3 (R), and they intersect S 3 (R) only at the vertices. The tetrahedral NPs are represented by the flat tetrahedra inscribed in S 3 (R). Figure S2 illustrate the projection using an analogy with 2D polyhedra embedded in 3D space, which allows for a direct visualization.
In this subsection we construct the reference crystal of tetrahelices inside the 600-cell as a subgroup of the 600-cell's full symmetry group.
The 600-cell can be decomposed into various combinations of substructures. Each regular decomposition is a subgroup of the 600-cell's full symmetry group, and it corresponds to a pathway of the self-assembly. In the main text, we argued that the observed chiral thin sheet assembles from closely packed tetrahelices. The 600-cell can be organized in 20 close-packed tetrahelices measuring 30 tetrahedra in length [S6] . We briefly describe the symmetry group H 4 ⊂ SO(4) of the 600-cell, and we show how the crystal of tetrahelices is associated to a specific subgroup of H 4 called the binary icosahedral group.
1. The Binary Icosahedral Group. If we inscribe the 600-cell in the unit 3-sphere S 3 by rescaling the edge length, its 120 vertices with coordinates (X 0 , X 1 , X 2 , X 3 ) in R 4 can be represented as unit quaternions:
where the last relation defines a unit quaternion and the symbols i, j, k satisfy Hamilton's relations
Under quaternion multiplication, the 600-cell's vertices form a nonabelian group, called the Binary Icosahedral Group denoted by 2I. 2I is generated by two generators a, b, as indicated in the group's presentation S10) or, in short, 5, 3, 2 . All the 120 group elements are generated by forming products of the generators a, b. In particular, the identity [i.e. the pure real quaternion e = 1 ≡ 1 + 0i + 0j + 0k ≡ (1, 0, 0, 0)] is written (ab) 4 = a 6 = b 10 = 1. Thus, a and b generate cyclic subgroups of order 6 and 10, respectively. This property is key to identify the interlocked tetrahelices and to show that they form a discrete Hopf fibration inside the 600-cell. We now pause the analysis of the binary icosahedral group to introduce the full symmetry group of the 600-cell.
The group H 4 . The 600-cell is invariant under the Coxeter group H 4 (also denoted [3, 3, 5] ) containing 14400 elements. The group is made of two copies of the binary icosahedral group:
where L, R stand for the Left, Right copies of 2I. Each copy of 2I has 120 elements so H 4 has order 120 2 = 14400. Fix a vertex q of the 600-cell, and two elements l ∈ (2I) L (called a left screw motion) and r ∈ (2I) R (called a right screw motion). The group H 4 acts on q as q → lqr. The operations (−l)qr and lq(−r) are equivalent so they are identified using the the quotient by Z 2 .
3. Discrete Hopf fibration of H 4 . Having introduced both 2I and H 4 , we now show that topologically H 4 is a fiber bundle, with a base of 12 elements and fibers of 10 elements (12×10=120). These fibers are the strands of 10 vertices between neighboring tetrahelices. Since H 4 contains Left and Right copies of 2I, this gives two distinct decompositions of the 600-cell into sets of Left or Right tetrahelices.
Fix the identity into (2I) R and consider the action of (2I) L × 1 onto the vertices of the 600-cell, i.e. left screw motions q → lq1. If we fix a vertex q 0 (which is also elements of the group (2I) L !) and we apply the element b ∈ (2I) L 10 times on q 0 , we come back to q 0 : we generated a closed ring of 10 vertices containing q 0 . A simple example is the "North Pole" 1 = (1, 0, 0, 0) of the 600-cell. By applying b, we generate the closed ring R 1
Two more simple examples are the rings R a and R a 2 generated by applying b to a and a 2 from the Left. Notice the three rings R 1 , R a ,R a 2 cannot intersect, because they contain different powers of a, and the exponent of a m cannot be changed by applying b from the Left. We can find a set of 12 quaternions (i.e. vertices of the 600-cell) in 2I that give rise to non-intersecting rings. We call this set the Base B of the group (2I) L , and write its elements using the generators:
The 12 elements of B form 12 vertices of an abstract icosahedron, called the icosahedral base. By applying b from the left to each element of B we generate the 120 vertices of the 600-cell. Each point on the base generates a closed fiber of 10 points ( Figure S3 ). Each vertex has been generated exactly once, so we have partitioned the 600-cell into a set of 12 non-intersecting rings of 10 vertices each. Thus, the symmetry group of the 600-cell can be written as a fiber bundle with base B and fibers generated by b. Finally, we take the stereographic projection of these rings into R 3 , and find that the rings are linked pairwise with Hopf link 1 ( Figure S4 ). Filling in the edges, we find a set of Left-handed tetrahelices between the fibers. Each fiber of 10 vertices is shadred by 5 tetrahelices. The tetrahelices are closed rings of 30 tetrahedra [S6] . There are 20 tetrahelices, corresponding to the 20 faces of the base icosahedron, giving the 600 = 20 × 30 tetrahedral cells of the the 600-cell. This means that the action of left screw motions partitions the 600-cell into a crystal of 20 Left-handed tetrahelices linked once into a discrete Hopf fibration. Due to the curvature of the 600-cell, the vertices of the helices belong to great circles (geodesics) on the circumscribed 3-sphere, so the tetrahelices are 'straight' (and closed) in the 600-cell. The same construction is valid for Right screw motions of the form 1 × r, giving rise to a crystal of Right-handed tetrahelices. If we cut any tetrahelix out of the 600-cell and we straighten it in flat space, it looks like the sequence (S4) (either Left-or Right-handed) with i = 29.
The discrete Hopf fibration by an icosahedron and a decagon is a useful tool to visualize the packing of tetrahelices in the 600-cell. For illustrative purposes, we align the icosahedron's 5-fold symmetry axis along the z−direction. Let γ be the polar angle: γ = 0, π are the North and South poles of the icosahedron and the 10 remaining vertices belong to two latitudes at γ N,S = π 2 ∓ arctan 1 2 .
(S14)
We start from γ = 0 and describe the order of the tetrahelices as we travel from the North to the South Pole in the base icosahedron. At γ = 0 there is a chain of 10 vertices, shared between 5 tetrahelices (one for each triangle on the base between γ = 0 and γ = γ N ). This means that close to the North pole, there are 5 tetrahelices bundled perfectly around a common long edge. The North pole maps to the axis of the bundle, while the 5 vertices at γ S map to the exposed corners of the 5 tetrahelices. The bundle has the topology of a solid torus, because the tetrahelices are closed in rings of 10 tetrahedra. The 5 edges at γ N map to the outer surface of the torus, which contains 20 × 5 = 100 triangular tiles. For γ N < γ ≤ γ S , we encounter 10 tetrahelices arranged into a solid torus that wraps around the 'polar bundle' (Fig.  1h in the main text) . In fact, there are 5 tetrahelices that coat the outer surface of the polar bundle, thus making a star-shaped torus with a cylindrical cavity inside and a zig-zag outer surface. The other 5 fit into the zig-zag surface thus completing the solid hollow torus. The surface exposed to the South pole has again 100 triangular tiles. The South pole bundle is identical to the one at the North pole, and wraps around the hollow torus, covering its exposed surface. The 600-cell is thus made of 3 nested solid tori, separated by 2 surfaces at γ N,S . Importantly, there are two ways of constructing the discrete Hopf fibration, depending on whether the tetrahelices wrap around each other with a right-handed or a left-handed rotation.
The crystal of tetrahelices constructed in section II A can be inscribed in a set of linked great circles in S 3 (R). In the next sections, we will use this continuum structure to build the elastic energy of the self-assembly.
The topology of the tetrahelices in the 600-cell is the discrete version of the Hopf fibration, a remarkable structure discovered by H. Hopf in 1931 [S7] while studying the topology of the 2-sphere. The Hopf fibration is associated to a set of maps {h : S 3 → S 2 } that reveal the non-trivial 3rd homotopy group of the 2-sphere S 2 . The pull-back h −1 (p) of a point p ∈ S 2 is a circle in S 3 . Remarkably, for all pairs of distinct points p = q ∈ S 2 the circles (fibers) h −1 (p) and h −1 (q) are disjoint and linked once. This shows that the 3-sphere S 3 is a topologically interesting fiber bundle, where S 2 is the base and the fibers are circles (S 1 ) linked pairwise.
In this section we introduce a continuous coordinate for the 600-cell on S 3 . To do this, we project the discrete Hopf fibration of the 600-cell onto S 3 (R), which produces a continuum Hopf fibration by great circles of S 3 (R). Since each decagon is inscribed into a great circle, the the straight edges are projected on the great circles. Moreover, the toroidal decomposition of the 600-cell maps into a continuous foliation of the 3-sphere by concentric 2D tori. We use the disjoint great circles contained in the 2D tori and the direction orthogonal to the toroidal surfaces as curvilinear coordinate axis on S 3 (R). The parametrization of this grid of lines in S 3 (R) is
where α ∈ [0, π/2] and β, θ ∈ [0, 2π[ are the polar angles on the unit 3-sphere S 3 . In addition to the polar angles Θ µ = (α, β, γ) we introduce coordinates with dimension of length x µ = RΘ µ . The curvilinear axis along θ correspond to a crystal of Left-handed tetrahelices, while those along θ are aligned with Right-handed tetrahelices. At every fixed α ∈]0, π/2[ we find a 2-dimensional surface parametrized by β, θ. The coordinate system Eq. (S15) divides S 3 in a stack of concentric toroidal surfaces in a natural way (see Appendix B for the flatness of the tori). In other words, the vector fields t α , t β , t θ (t µ = ∂ µ X) are associated to a foliation of S 3 by toroidal surfaces (called leaves). The vector t α is normal to these toroidal surfaces. We will explain later how we use the foliation to find the reference geometry of the upper, mid-and lower surfaces of a thin shell of tetrahelices. Generically, the existence of a 2D shell-like solution for the self-assembly is related to the existence of well-behaved foliations of the reference manifold, since the leaves provide natural boundary and mid-surfaces for the shell. In this parametrization, the vertices of a tetrahelix are contained in the great circle X(θ) on S 3 given by
where the pair (α * , β * ) belongs to the vertex of an icosahedron. The coordinates Eq. (S15) are related to the Hopf fibration in a simple way. The icosahedral base B mentioned in section II A becomes the Base 2D sphere S 2 of the Hopf fibration (section II C) with latitude-longitude coordinates (2α, β). The decagonal fibers become 1D circles S 1 parameterized by θ.
It is worth noting that the 600-cell is faceted: the interior of the tetrahedra is flat, while edges carry positive Gaussian curvature. The coordinate we introduced above is a coordinate of the 3-sphere S 3 (R) that circumscribes the 600-cell. This is analogous to using the spherical coordinate of the 2-sphere for the circumscribed icosahedron: all vertices lie on the 2-sphere, but the interior of the triangles are flat and not on the 2-sphere. As an elasticity theory we build here for the self-assembly problem, this continuous coordinate ignores the flatness of the tetrahedra themselves, and approximate the geometric frustration by a homogeneous curvature.
We view this continuous theory as an approximation of the actual material, where hard tetrahedra are connected by soft ligands. By using this continuum theory, we homogenize the discrete materials into a continuous media, where the hard tetrahedra plus the ligands are approximated as spherical tetrahedra.
This approximation is increasingly accurate in the limit of a/R → 0 where a is the edge length of the tetrahedra and R is the radius of the 3-sphere. Although the crystalline 600-cell is not in this limit (a/R = φ −1 ∼ 0.618), we expect a continuous distribution of disclinations which greatly reduce this ratio. Sadoc showed that it is possible to preserve the topology of the fiber bundle and elongate the fibers at the same time, if one chooses specific disclinations that change the icosahedral base into a non-regular polyhedron [S6] . Since the α−helices are less dense than a tetrahedral self-assembly, in [S6] the tetrahelices were not closely packed. The study of topological defects and the low-energy hierarchical structures they form lead to a number of beautiful results in glasses [S2, S8] , amorphous solids [S11] (where a hierarchy of disclinated polytopes was found) and Frank-Kasper phases [S5, S12] as well as 2D crystals [S9] . The radius of S 3 (R) effectively increases by introducing a density of defects, so the curvature of the non-Euclidean crystal proportional to 1/R 2 decreases, making it more compatible with flat space.
We now characterize the reference distances in the crystal of tetrahelices with the metric induced by Eq. (S15). The 4D tangent vectors to S 3 (R) are
where the partial derivatives are taken with respect to the angles Θ µ = (α, β, γ) and the factor of R −2 makes the tangent vectors dimensionless. The components of the metric tensor in the coordinates (S15) areḡ µν = t µ · t ν :
The vector t α is orthogonal to t β , t θ . By computing the Ricci scalar from Eq. (S18), we verify that the Gaussian curvature of
The metric is block-diagonal (ḡ αβ =ḡ αθ = 0) so the α−axis is everywhere orthogonal to the β− and θ−axis. Notice thatḡ ij depends only on the coordinate α. Fixing α =const gives a 2D toroidal surface in S 3 . The metric is constant on this surface. The line element is
which can also be written as
where Θ i = (β, θ). The 2-dimensional metric induced on the toroidal surfaces at fixed α is and it is constant once we fix α. We recall that we inscribed the icosahedral base of the 600-cell in the base 2-sphere, and 2α is the latitude on the base S 2 . We will use 2α to label both the latitudes of the icosahedral base and of the continuous S 2 base. For α N,S = π/4 ∓ 1/2 arctan(1/2) on the icosahedral base, we find two groups of 50 vertices arranged into a grid formed by the great circles along β and θ (see Fig. S5 ). The angle between the red and black lines is arccos(1/ √ 5). The surfaces Σ α N,S are separated by an angular distance ∆α = 2(α S − α N ). The surface α = π/4 which is equidistant from Σ α N,S is the Clifford torus and does not contain any vertex of the 600-cell. The coordinate grid (β, θ) is orthogonal on the surface of the Clifford torus (see Fig. S5 ). The Clifford torus is the mid-surface of a shell whose upper and lower boundaries are Σ α N,S .
The determinant ofḡ is detḡ = sin 2 2α and the reference volume element is
The determinant ofḡ vanishes at α = 0, π/2 so the coordinate system (α, β, θ) is singular at the North and South pole of the base icosahedron. The singularities are analogous to the spherical coordinate system on the 2-dimensional sphere S 2 : at the North and South pole the tangent vectors t β and t θ are parallel. In fact, at α = 0 the coordinates parametrize a 1-dimensional circle X = R(cos θ, sin θ, 0, 0) rather than a 2D surface. At the South Pole (α = π/2), the embedding becomes X = R(0, 0, cos(θ + β), sin(θ + β)) that is again a great circle C π/2 covered twice by β + θ ∈ [0, 4π]. We fixed β = 0 to avoid the double-covering of C π/2 . Figure 5 shows the grid formed by the curvilinear axis θ (red lines) and θ (black lines) on the surface of a torus. In order to draw the tetrahelices, we should connect these vertices with the ones on the neighboring toroidal shell. The tetrahelices that are aligned with the red lines are Right-handed, and those that are aligned with the black lines are Left-handed. Thus the 600-cell is composed by two bundles of tetrahelices linked pairwise: one contains only Right-handed helices, the other only Left-handed helices (see e.g. FIG. 1f in the manuscript) .
Under the exchange of β with θ, the parametrization (S15) transforms as
while the reference metric tensorḡ µν = ∂ µ X · ∂ ν X remains invariant, because it is quadratic in the components of X.
In the experiments it is observed that the ligands and the self-assembled tetrahelices are always homochiral, so chiral symmetry is explicitly broken. Quasi-1D tetrahelices can grow unfrustrated along their axis. We model this by choosing one coordinate direction to be aligned with the tetrahelices (e.g. we choose θ for R-helices) and we impose that the actual metric of the self-assembly is invariant under translations along θ (see section VII). Finally, we look for thin-shell minimizers of the elastic energy. We find that the shell's mid-surface is a Right-handed helicoid. There is no Left-handed solution, consistently with the fact that the asymmetry between β and θ breaks chiral symmetry. The elastic energy of the shell depends only on width (β direction) hence the helicoid will have a large length-to-width aspect ratio (see FIG. 2 c, Right in the manuscript) . We discuss the transformation relating Right-handed with Left-handed solutions (for the case where the assembly starts from L tetrahelices) in subsection VII B. 
Any value of β = n(2π)/5 with n = 0, 4 can be absorbed in a shift of θ, by redefining k = k + 2n.
In this section we discuss the general elastic energy of materials with a reference metricḡ that is incompatible with the 3D Euclidean space. For any curved crystals such as the 600-cell, since the components of the Riemmann tensor don't vanish due to the Gaussian curvature K G (ḡ) = 0, the 3D reference configuration cannot be embedded isometrically in flat Euclidean space E 3 . The 3D (non-linear) strain tensor, which describes the unavoidable deviation of the actual metric from the reference metric, is defined by
where the actual metric g µν depends explicitly on the embedded shape R of the self-assembly:
The reference metricḡ µν for a close-packed self-assembly of tetrahedra is the metric of the 600-cell. For a crystal of tetrahelices, the reference metric is conveniently written as Eq. (S18). To lowest order in strain invariants, the non-linear elastic energy of the self-assembled crystal is a quadratic function of the strain:
where λ and µ are the Lamé parameters, and indices are raised and lowered with the reference metricḡ. The integration ranges over a portionM cut out from S 3 that represents the reference configuration of the finite size assembly. It will prove convenient to rewrite Eq. (S28) introducing the elastic tensor A constructed from the reference metric as
where the Y is the Young's modulus and ν is the Poisson's ratio. They are related to the Lamé parameters via
With these definitions and (S27), the elastic energy (S28) is rewritten as
In this section we discuss the expansion of the elastic energy discussed in the previous section for a thin sheet. As we discussed above, the geometric frustration of the elastic energy leads to stresses as the crystal grows in the 3D Euclidean space. This suggests that self-assembly favors thin sheets over thick bulk structures, which balances the surface tension and the elastic energy.
In order to find the actual configuration of a thin self-assembly, we must minimize the elastic energy over a thin regionM of S 3 (R). As discussed in the main text, we select a toroidal surface X 0 (β, θ) ≡ X(α 0 , β, θ) at a fixed α = α 0 , and construct a sequence of toroidal surfaces around it varying α ∈ [α 0 − h/(2R), α 0 + h/(2R)]. This region is the reference configuration of a 3-dimensional self-assembly of dimensions h × L × W . When the scales h W, L are well separated, we call it a thin sheet. Although the coordinates β, θ ∈ [0, 2π] in the 3-sphere, they are not bounded by these limits in this sheet, as they now have the topology of an open sheet, not a torus, as we discussed in the main text.
The material is not uniform across its thickness, so it is analogous to an elastic thin shell rather than a thin plate [S10] . We define the thickness parameter
with dimensions of length, and expandḡ ij (α) around α = α 0
The constant tensorsā ij andb ij are the first and second fundamental form of the mid-surface:
where N 0 is the normal vector to the X 0 mid-surface. The coordinate expression ofā ij ,b ij is
which is positive for α 0 ∈ [0, π/2]. The area element is zero at the poles of the base 2-sphere, where 2α 0 = 0, π. The geometric frustration manifests in the fact thatā andb are incompatible in a surface embedded in Euclidean space because detb/ detā = −1/R 2 , while K(ā) = 0, violating Gauss' Theorema Egregium We assume that the flat-space configuration is a layered structure, similarly to the reference configuration. In particular, we assume that we can define a vector t α that is orthogonal to a sequence of 2-dimensional slices in the actual material, so that we can write the actual metric as
with a ij , b ij , c ij = b m i b mj defined on the mid-surface. Since αα ∝ g αα −ḡ αα = 0, the sum in Eq. (S31) is restricted to the indices i, j = {β, θ}. The elastic energy is E = E dαd 2 Φ where E is given by Eq. (S31) , d 2 Φ ≡ dβdθ and where the elastic energy density is
The elastic tensor, with indices restricted to the directions on the surface is
The reduced energy density of the thin shell is obtained by fixing α = α 0 , expanding in t = α − α 0 and integrating over t:
The pre-factor in front of the integral is a convention. The total elastic energy of the thin shell is [S10]
Since the integration in Eq. (S41) is symmetric around t = 0, the terms of E (α 0 + t, β, θ) linear in t integrate to zero. Thus, we expand the elastic tensor, the strain tensor and the area element in Eq. (S39) to second order in t.
Let the inverse metric have the series expansionḡ ij = A ij − 2t B ij + t 2 C ij . We can fix the coefficients A, B, C by substitutingḡ ij and Eq. (S33) in the identityḡ ikḡ kj = δ i j and matching order by order in t. The result is
Usingc ij =b m ib mj , we can simplify Eq. (S43) in the form
Then the products in Eq. (S44) are expanded as
We substitute Eq. (S44) in Eq. (S40) keeping terms up to o(t 2 ), writing schematically
where A ijkl 0 is just Eq. (S44) withḡ replaced byā.
The area element is proportional to detḡ ij . We expand the determinant using Eq. (S33):
where we used a compact notation M * N ≡ 1 2 ε ik ε jlM ijNkl . It is clear from its definition that the symbol is symmetric: M * N = N * M . By explicit computation, we can express M * N as
Applying Eq. (S48) to Eq. (S47) we find
Taking the square root of Eq. (S49) and expanding to o(t 2 ), the final expression of the area element is
In order to simplify the notation, we write Eq. (S50) schematically
Notice that for the special choice of the Clifford Torus α 0 = π/4, the second fundamental form is traceless, i.e. S 1 = 0 and S 2 = −2.
Since the strain tensor contains the differences (a ij −ā ij ), (b ij −b ij ), (c ij −c ij ), we introduce the simplified notation ∆M ij = M ij −M ij for any matrix M ij . Using Eq. (S33) and Eq. (S38) the difference (g ij −ḡ ij ) is written
Using the expansions Eq. (S46), Eq. (S51) and notation Eq. (S52) the 3-dimensional energy density is rewritten as
We distribute the products, keeping up to o(t 2 ) and neglecting all terms that are linear in t, because the integration range is symmetric around t = 0. The result is
The term proportional to (∆a) 2 is the in-plane stretching energy of the shell. First, we define the reduced strain tensor in the plane of the shell
We neglect the terms of o(t 2 ) compared to A 0 in Eq. (S54a), integrate over t from −h/R to h/R according to Eq. (S41) and find the stretching energy density
The terms that scale with the square of the thickness depend on the shell's extrinsic curvature b ij . After integration over t, we find a bending energy density that scales with the cube of the thickness. The first term is the bending energy encountered in the elasticity of elastic plates:
The remaining terms in Eq. (S54b) and Eq. (S54c) are corrections to the bending energy of a plate, arising from the fact that the reference metric depends on the coordinate t along the thickness. They represent non-positive definite couplings between the in-plane strain and the out-of-plane bending of the material. We now make two observations. First, the Clifford torus is a special reference mid-surface, because S 1 = 0. Second, in the bending dominated regime, ∆b ij = 0 and the energy density reduces to a simple correction to the bending energy Eq. (S57). Using the symmetry A ijkl 0 = A klij 0 , and recalling c kl = b m k b ml , we rewrite the second term of Eq. (S54b) as
In conclusion, the couplings between stretching and bending affect neither the bendingdominated nor the stretching dominated solutions. In order to study the pure stretching and pure bending limits, we can work with the following expression for the elastic energy of a shell: it is the sum of in-plane stretching energy (linear in the thickness h) and out-of-plane bending energy (cubic in the thickness h)
The curvature of S 3 , introduced to realize a perfect crystal of tetrahelices, implies that the metric tensor Eq. (S18) depends on the thickness α. This gives rise to spontaneous reference metricā and reference extrinsic curvatureb in the elastic energy of the thin sheet Eq. (S59). The bending modulus associated to deformations normal to the surface is κ ≡ h 3 Y /12(1 − ν 2 ) with dimensions of energy.
In this section we study the solutions to the elastic equilibrium problem δE shell = 0, taking E bend = 0 so δE shell = δE stretch . This is accurate in the limit W √ Rh. As we discuss in the main text and SM Sec. VIII, when the repulsion is included, this regime widens to W √ Rh where is a dimensionless constant characterizing the ratio between the repulsion and the elasticity strength.
A. Solving for the actual metric and the morphology As mentioned in the main text, we take the mid-surface of the shell on the Clifford torus at α = π/4. The reference 1st and 2nd fundamental forms of the shell are Eq. (S35) evaluated at α = π/4:
The coordinate θ runs along great circles circumscribed to right-handed tetrahelices. As we discussed in the main text, we take the actual metric a to be translationally invariant along θ direction, which is the direction along these tetrahelices. The metric Ansatz is then a function of β:
In the bending-dominated limit, the bending energy vanishes iff b =b, whereb is written in Eq. (S61). Then, a(β) is the 1st fundamental form of a surface R(β, θ) with normal vector N and second fundamental form N · ∂ i ∂ j R =b ij iff it satisfies the Gaussi-Codazzi-Peterson-Mainardi (GCPM) equations [S13]
and the Gauss equation
where Γ i jk are the Christoffell symbols:
and where we defined x 1 = β and x 2 = θ to keep the notation simple. The explicit expressions of the Christoffel symbols are:
where we indicated partial differentiation with a comma symbol. Using the expression ofb and Eq. (S66) in Eq. (S63), the GCPM equations reduce to
From the first equation, we either have E(β) = E 0 = const or F = 0. In the former case, we cannot satisfy the second equation so the only non-trivial solution is F = 0, leading to a diagonal metric. Then, from the second equation we find EG ,β = GE ,β , i.e.
Using Eq. (S75) in Eq. (S72), we can compute the stress
and impose a vanishing stress at the boundary (free BC) σ 11 (W ) = 0 to fix the parameter c 1 . The result is
Since W/ √ Rh is small, we formally expand a in W and keep only the lowest order:
where the linear correction is − cosh β sinh βδ ij . We could solve the Weingarten equations to find the surface corresponding to the actual metric Eq. (S78) and the off-diagonal 2nd fundamental form in Eq. (S61). More simply, we observe that the mean curvature vanishes (H ∼ Tr(b) = 0) so the solution must be a minimal surface. Moreover, the second fundamental form is off-diagonal, so the surface has pure torsion and zero bending: it is a ruled surface generated by the rotation of a straight line around a curve. Finally, the Gaussian curvature of the surface is negative: detb/det a = −1/(R 2 cosh 4 β). The solution is an helicoid with embedding R(β, θ) = R(sinh θ cos β, sinh θ sin β, β) .
The solution is a Right-handed helicoid. We can check that the first fundamental form a ij = ∂ i R · ∂ j R is Eq. (S78). The second fundamental form of the helicoid is b ij = N · ∂ i ∂ j R. By explicit calculation of the unit normal vector
we can verify that b =b. Notice that the positive sign in front of the third component of the embedding R 3 = β in Eq. (S79) determines the handedness of the solution. To summarize:
• We start from a crystal of right-handed tetrahelices, which are aligned with the θ direction of the coordinates
• The actual metric is taken to be translationally invariant along θ, the long direction of the ribbons, due to the chiral symmetry breaking of the tetrahelices self-assembly pathway and energy minimization in the limit of L W
• We minimize the elastic energy in the bending-dominated limit with free boundary conditions, finding an helicoid of the same handedness as the tetrahelices.
In the previous subsection we studied how right-handed tetrahelices may self-assemble into a right-handed thin shell. We can analyze a thin shell of left-handed tetrahelices by interchanging β with θ. The reference metricḡ (and thereforeā andb) is invariant under this transformation. Let us call S R the right-handed helicoid and S L the helicoid obtained by exchanging β and θ. Let (a R , b R ) and (a L , b L ) be their respective 1st and 2nd fundamental forms. If the normal to S R is N R (see Eq. (S80)) then the normal to S L is −N R . The reason is that the order of the tangent vectors ∂ β R and ∂ θ R is exchanged under θ ↔ β. Consequently, the 2nd fundamental form of S L (b L ij = N L · ∂ i ∂ j R L ) is −b R . The solution must satisfy b L =b L . Butb L =b R . In order to compensate for the minus sign, we have to reflect the coordinates of E 3 , e.g. by reflecting R 3 → −R 3 , so the embedding is a Left-handed helicoid:
R L (β, θ) = R(sinh β cos θ, sinh β sin θ, −θ) .
(S81) This predicts that tetrahelices and self-assembled thin shells must have the same handedness. The right-and lefthanded solutions have the same energy. The symmetry is broken by the chiral ligands: right-(left-) handed ligands can induce the formation of enantiopure right-(left-) handed tetrahelices, and eventually select the chirality of the thin shell. This corresponence between chiralities is indeed observed in the experiments. Notice that we defined the handedness of the helicoid with respect to its long axis. The helicoid twists with the opposite sense of rotation around a short axis (perpendicular to the length direction).
In this section we derive the energy associated with electrostatic repulsion of the assembled sheets. The energy of charged elastic surface is a sum of Euclidean invariants of the surface's position X and its derivatives ∂X, plus its electrostatic energy E[X(σ)] = κ 2 Σ d 2 σ ∆X(σ) · ∆X(σ) (S82)
where V is the electrostatic potential. The first is the bending term, that depends on the Laplacian (∆X) 2 = H 2 (H is the mean curvature). The second term is the stretching energy, written in terms of the Lamé parameters µ, λ and the intrinsic metric of the surface a ij = ∂ i X · ∂ j X. The strain tensor is
whereā ij is the reference metric of the surface in S 3 . The last term is the electrostatic interaction. It depends on the Euclidean 3-dimensional distance |X(σ) − X(σ )| between points σ, σ in the surface. Supposing that the surface is uniformly charged with charge Q and charge density ρ = Q/Area[Σ] The potential energy from electrostatic short-range repulsion E rep has the following expression
This term yields a correction to the bending energy, because the double integral is performed over the embedded shape, so it depends on the extrinsic curvature of the surface. Making this contribution manifest in a generic geometry is difficult. We can extract some information from the double integral of Eq. (S86) in a simplified geometry. If the membrane is nearly flat, we can describe it using the Monge parametrization. To this end, we take a plane Π external to Σ, we choose two orthogonal coordinates (x, y) on Π, and a z direction, orthogonal to Π. The surface Σ is embedded as X(x, y) = (x, y, f (x, y)) (S87)
For notational convenience, we define the displacement D = X(x) − X(x ). The potential diverges as 1/|D| as (x , y ) is taken closer and closer to (x, y), and it is localized within a radius ξ around (x , y ) for every fixed (x , y ). Next, we organize the integral Eq. (S86) into: (i) an integral for the 3d potential φ(x) = ρ d 2 x e −|D|/ξ /|D| and (ii) an integral over the sources ρ 2 d 2 xφ(x). Writing x 0 = (x 0 , y 0 ) and x = (x, y), the distance between two points on the membrane is
The potential is localized, so we take x → x 0 and expand f :
we define
We have one such expansion for each (x 0 , y 0 ). We can integrate the elementary potential around a fixed (x 0 , y 0 ):
I(x 0 , y 0 ) = hρ 4π In the text, we further assume that the curvature is small compared to the size W of the membrane. The dashed line represents a geodesic between the points (x0, y0) and (x, y) on the membrane, while D is their separation in flat 3 dimensional space. ξ represents the localization scale of the short-range electrostatic interaction (the inverse of an effective photon mass).
using the expansion above. The metric is
where ∂f = (f x , f y ). Taking the derivative of Eq. (S89) wrt x k ,
and using it into Eq. (S93) we find
Within the disk of radius ξ, we approximate the surface with a paraboloid, and we write the 3d distance as
where we suppressed the boldface to ease the notation. Collecting (δx) 2 = r 2 , we can write d(x, x 0 ) = |δx|
we have used the fact that all gradients at any point of the surface are small, as well as the curvature. So this is an expansion in derivatives of f . The derivatives ∂ i f (x 0 ) and ∂ 2 ij f (x 0 ) are related to the local second fundamental form of the surface. By integrating over x, y we want to recover the expansion in the local b ij (x 0 ) with effective coefficients. These coefficients will be the correction to the bending rigidity.
We now expand the integrand of Eq. (S91) for small values of derivatives of f . We keep only up to quadratic terms in ∂f and ∂ 2 f . Expanding the determinant, we write schematically
Similarly we expand the distance function as:
d(x, x 0 ) = |δx|(1 + B) (S100) with B = 1 2|δx| 2
(∂ i f δx i ) 2 + 1 4 (∂ 2 ij f δx i δx j ) 2 + (∂ i f δx i )(∂ 2 ij f δx i δx j ) ã (S101)
The full expressions are more complicated and involve products of the first and second derivatives. At lowest order in gradients, the bending energy is ν(T rb) 2 + (1 − ν)T r(b 2 )