A diffraction bench, 1982 to 1987
Rotate a reconstruction of Shechtman's forbidden spots, test whether a periodic lattice can own them, then build a six-index icosahedral model and diffract its higher-dimensional cut. The photograph starts the case; source-checked electron, X-ray, NMR, and peak-shift records close the loophole.
A bright point in a diffraction pattern is not a picture of an atom. It is a direction in which waves returned in step. A field of sharp points is therefore an object you can interrogate: rotate it, index it, and ask what kind of order could have put the points there.
Layer 1, the forbidden turn
The plate below is a geometric reconstruction of the published tenfold symmetry, not a digitisation of measured peak positions. Every noncentral point is live geometry. Rotate the cyan copy and the browser pairs each point to its nearest gold point.
The central beam is drawn but excluded from the residual. At least ten noncentral spots are always retained, so an empty plate and a central-dot-only match cannot pass.
computed liveZero is unavailable because it agrees by definition.
The tenth-turn overlay is real geometry, but it does not yet decide the structure. An inversion-symmetric diffraction pattern viewed along a fivefold axis appears tenfold. More importantly, several periodic domains can be rotated into a sharp composite with the same turn. Choose the generic foil above and press align. It passes.
Layer 2, the six-coordinate answer
Now make a model set. Each integer sextuple m ∈ Z⁶ is projected into two perpendicular three-dimensional spaces. Keep it only when its internal-space image falls inside the window. The other image becomes a site. The six-dimensional lattice is a representation, not six physical directions in the alloy.
Left: a fivefold view of the accepted three-dimensional sites. Right: 2,025 direct Fourier sums across a reciprocal plane, with the six-index module positions marked by rings. Nothing is fetched and no intensity is stored.
direct sumMove the acceptance window. Sites enter and leave by a rule in the unseen companion space. The full sampled plane reorganises around isolated high intensities rather than being drawn only at expected peak positions. Turn up display noise if you want to see how the strong positions persist. Blur and noise affect only the drawing. They never enter the site selection, index fit, or Fourier sum.
Switch the labels to the three-generator trial. It searches every integer coefficient from -2 to 2 for each of the 84 generated module positions. The failure is a statement about this displayed screen trial, not a new fit to the historical data. The physical claim is the one in the papers: the original peaks could not be indexed to a Bravais lattice, while later single-crystal records admitted six-index icosahedral indexing.
This panel is the live audit. The green values are recomputed after every control change. The source-traced facts and the model-derived values are kept in different boxes.
Offline reproduction: node research/quasicrystal-diffraction/verify-quasicrystal-diffraction.mjs
The record, in order
The evidence did not arrive in one exposure. Each line below says what the cited object added. Recognition comes last because a prize is not a measurement.
NIST's institutional history reports the date, composition, sample number, and notebook annotation shown above. It also reports that Shechtman looked for twins that afternoon. This is a discovery record, not a peer-reviewed diffraction analysis.
Shechtman, Blech, Gratias, and Cahn reported Al-14-at.%-Mn with sharp diffraction spots, icosahedral point-group symmetry inconsistent with lattice translations, and no Bravais-lattice indexing. The paper reports microdiffraction from different volume elements and dark-field checks across grains.
Pauling published a specific periodic proposal: a cubic cell about 26.7 angstroms wide with about 1,120 atoms, assembled as 20 directed twin domains. Those are Pauling's published model values. The generic foil in Layer 1 does not implement that structure.
Cahn, Shechtman, and Gratias described how diffraction establishes icosahedral quasiperiodicity, analysed several single-crystal electron patterns, and connected the indexing to six-dimensional cut-and-project crystallography.
Kortan, Chen, and Waszczak reported Laue X-ray diffraction from a millimetre-size Al-Li-Cu single quasicrystal, direct icosahedral point symmetry, and peaks indexable with six indices. Bennett and colleagues found the NMR spectrum of the icosahedral Al-Mn phase differed from the cubic G phase used in Pauling's proposal; they noted twinning should scarcely alter NMR away from boundaries.
Budai and colleagues measured systematic phason-strain shifts in oriented icosahedral Al-Mn and stated that the results ruled out Pauling's icosatwin model. This page does not reproduce their numerical peak shifts because the checked record did not provide a coordinate table.
The IUCr executive report says the committee and its terms were approved in April 1991. It records the working description of a crystal by an essentially discrete diffraction diagram and an aperiodic crystal by absent three-dimensional lattice periodicity. This was not a single 1992 vote performed by this page.
The Royal Swedish Academy of Sciences announced the sole Chemistry prize to Dan Shechtman for the discovery of quasicrystals. The award recognises the evidence above. It does not substitute for it.
Diffraction from a real density is inversion symmetric in the kinematic approximation: a peak at q is accompanied by one at -q. A fivefold physical axis can therefore make ten arms in the flat diffraction image. This page calls the image tenfold and the corresponding solid axis fivefold.
A periodic three-dimensional reciprocal lattice needs three integer coefficients. The icosahedral reciprocal module uses six icosahedrally related basis directions. The browser constructs a six by six orthogonal decomposition whose absolute determinant is one, then projects the same integer sextuple into physical and internal three-spaces. The accepted physical points and candidate reciprocal vectors are different projections of integer data.
The window is chosen for legibility and to expose the cut-and-project mechanism with one continuous control. Different windows change local decoration and intensities. The page claims only that this particular mathematical model is aperiodic in the cited infinite construction and has sharp module diffraction. It does not claim to place the atoms of sample 1725.