# Crystal periodic table and sunlight — Element Projection Lab 0.13.0 Theory revision 1.74. The atlas now has **118 element entries, 67 compiled primitive crystal cells, and 11 measured-input bulk optical reconstructions**. It includes a rotatable offline table, a full PNG/SVG poster, a 27-cell repeat view, and a separate mathematical 4D-shadow view. ## The geometric minimum The rendered object is a primitive repeating cell with its atomic basis and periodic boundary conditions. It is not a smallest stable free-standing nanocrystal. Lattice vectors form a real 3-by-3 matrix A; atomic positions are `r = (n + f) A`, where n is an integer translation and f a fractional basis coordinate. A full 3D crystal requires the repeated environment, even when the primitive basis contains one atom. Among the 67 compiled structures, 37 have one basis atom, 29 have two and one (samarium) has three. Boundary images are counted with sharing weights: eight corner markers in gold's primitive parallelepiped contribute one atom total. The conventional FCC cube instead has four atoms. Neither count proves a one- or four-atom isolated cluster has bulk gold properties. The 51 unavailable geometries comprise 10 explicitly missing bases, 6 unresolved phase/basis assignments, 5 isolated molecular/atomic source entries, 29 absent reference entries and 1 unsupported structure. These are missing-data states, not claims that the elements cannot crystallize. In particular, hydrogen and oxygen are not silently converted from molecular references into monatomic crystals. Diamond is the selected carbon allotrope; noble-gas cells and solid mercury require conditions different from ambient room temperature. The official ASE snapshot supplies rounded reference parameters, attributed upstream to Ashcroft–Mermin. AFLOW corroborates prototype vectors and bases. Datum-specific temperatures, pressures and errors are not supplied by this compilation. The full provenance and omissions are in `CRYSTAL_SOURCE_NOTES.md`. ## Exposing the bulk material to light The optical branch reconstructs reflected spectra from published complex indices n+i k for Au, Ag, Cu, Ti, V, Cr, Mn, Fe, Co, Ni and Pd. At normal incidence from air: `R(lambda) = ((n-1)^2 + k^2) / ((n+1)^2 + k^2)`. The reflected spectrum is multiplied by a 5772 K Planck solar approximation or official CIE D65 daylight. Official CIE 1931 two-degree functions integrate 380–780 nm at 1 nm spacing; Bradford adaptation and sRGB encoding supply the display swatch. Unadapted XYZ, unclipped linear RGB and a clipping flag are retained. Gold clips slightly in sRGB, so its hex value is not an exact or intrinsic material property. | Material | Primitive structure / atoms | Reflectance at 450 nm | Reflectance at 650 nm | Adapted solar display | | --- | --- | ---: | ---: | --- | | Gold | FCC / 1 | 0.408194 | 0.956522 | #FFDEA3 | | Silver | FCC / 1 | 0.980236 | 0.989834 | #FEFDFC | | Copper | FCC / 1 | 0.538258 | 0.935218 | #F8CFBF | Their normalized primitive Gram matrices are identical; their measured-input reflectances are not. Thus **FCC topology alone does not determine color**. The numerical lattice constants differ, so this is explicitly a test of the normalized shape, not a claim that every coordinate in angstroms is identical. No lattice parameter or optical coefficient was fitted to a target color. This is an effective, clean, smooth, optically thick planar surface calculation. The original Johnson–Christy inputs are film measurements; the transition-metal samples are polycrystalline. The scalar response does not provide the orientation-dependent tensor of every pure single crystal. Surface oxidation, roughness, finite size, thin-film interference, defects, atmosphere, solar lines, fluorescence and observer variability remain outside the calculation. A single cell is not assigned an experimentally established optical color. ## A scale check on geometric interference For FCC, the largest spacing associated with a nonzero reciprocal-lattice vector is d111=a/sqrt(3). First-order free-space Bragg diffraction requires lambda=2 d sin(theta), hence lambda<=2d. The compiled maximum wavelengths are 0.471118 nm for Au, 0.472273 nm for Ag and 0.416847 nm for Cu; visible light in this calculation starts at 380 nm. Visible gold color is therefore not ordinary first-order diffraction from these atomic FCC planes. This does not remove the role of periodic structure in electronic bands, nor rule out visible structural color from much larger photonic patterns. ## The 4D shadow is an explicit display convention Embed each reference coordinate as (x,y,z,0), apply an orthogonal x–w rotation, and project back to (x,y,z): `(x,y,z,0) -> (x cos(theta), y, z, x sin(theta))`. The four-dimensional norm is preserved; the displayed three-dimensional norm need not be. At 90 degrees, two points differing only in x have the same 3D shadow. This exactly defines what is drawn. It does not infer a hidden physical motion, and the rotation does not alter the independent bulk optical response. ## Verification, reproducibility and limits All 218 unit tests pass: 192 previous tests and 26 new geometry, optics and view tests. The optics tests include 2,156 independently evaluated scalar Fresnel comparisons over the 539 original n,k rows and four incidence angles. All 67 supplied bases pass a finite redundant-translation check and boundary sharing sums; a deliberately nonprimitive conventional BCC basis is rejected as redundant. The Canvas/interaction smoke test passes 633 checks, including all element selections and 320/736-pixel drawing sizes. The complete browser CSS layout has not been validated because a local browser binary is absent; Canvas previews and the static poster were visually inspected. `outputs/independent_crystal_audit.json` records a separate scalar source-to-output replay without production-module imports. The release builder requires that audit to pass and verifies old-file preservation before packaging. Its exact counts and numerical discrepancies are recorded there, rather than treating code agreement as independent experimental confirmation. The independent replay verifies all 118 identities, 67 primitive cells, 31 nontrivial possible extra translations, 4,411 reflectances, 8,822 interpolated n/k values and 22 source/observer colors. Maximum reflectance difference is 6.66e-16 and maximum color-component difference is 5.22e-15. It also verifies 15 raw optical-source hashes and four checksums reported in official CIE metadata. The omitted ASE Fe/Co/Ni magnetic fields are explicitly recorded; they are not geometric inputs. Original input files, source checksums, code, numerical outputs, the previous paper and all earlier results are preserved. The new tabulated CIE observer is a separately versioned optical layer and does not overwrite the older frozen Au/Ag analysis. A pre-release optical test replaced an unrealistically exact zero comparison by a floating-point tolerance (residual about 5.9e-32); no physical model or source data changed for that fix. These are compiled reference geometries, calculations using external measured optical inputs, written geometric identities and finite software checks. They are not new measurements, held-out empirical predictions, or evidence for an extra spatial dimension. Neutral display marks mean unknown color, not grey material. The earlier ATLAS benchmark exclusion and recast block remain intact. ## Sources and next experiment - [IUCr primitive cell](https://dictionary.iucr.org/Primitive_cell). - [ASE reference data](https://docs.ase-lib.org/ase/data.html), pinned to commit c77931fd38c5fdaa560bc204488fad5e4c50f4c8, 18 September 2026. - [AFLOW FCC geometry](https://aflow.org/p/A_cF4_225_a-001/), with other phase prototypes and source citations listed in `CRYSTAL_SOURCE_NOTES.md`. - [Johnson–Christy, 15 December 1972](https://doi.org/10.1103/PhysRevB.6.4370) and [15 June 1974](https://doi.org/10.1103/PhysRevB.9.5056). - [CIE observer](https://cie.co.at/datatable/cie-1931-colour-matching-functions-2-degree-observer) and [CIE D65](https://cie.co.at/datatable/cie-standard-illuminant-d65), with official tables, metadata, checksums and licenses retained. Next bounded experiment: choose one specific phase and surface, specify an electronic Hamiltonian/response calculation, freeze its dielectric prediction, then compare with an independent reflectance dataset. A model that takes n,k from measurements has reconstructed color; it has not yet predicted n,k from our geometric hypothesis. Missing molecular crystal bases can be filled only from complete phase-specific structures, with their own validation.