Penrose Tilings: 5-Fold Symmetry via Golden Ratio Projection from 5D Lattice — E8 Intelligence Research

{"FINDING:":[0],"Penrose":[1,96],"tilings":[2,7],"are":[3,27],"non-periodic,":[4],"5-fold":[5,77,143],"symmetric":[6,78],"of":[8,33,169],"the":[9,19,158,170],"plane,":[10],"generated":[11],"by":[12,118],"inflation/deflation":[13],"rules":[14],"whose":[15,104],"scaling":[16],"factor":[17,51],"is":[18,87,145],"golden":[20],"ratio":[21,44],"squared":[22],"(φ²":[23],"=":[24,46,52,54,59,61],"2.618),":[25],"and":[26,92,102],"equivalent":[28],"to":[29],"a":[30,34,72,76],"2D":[31,73],"slice":[32],"5D":[35],"cubic":[36],"lattice":[37,39],"(root":[38],"A₄).":[40],"|":[41,130],"MATH:":[42],"Golden":[43],"φ":[45,91],"(1+√5)/2":[47],"≈":[48,56,63],"1.618;":[49],"inflation":[50],"φ²":[53],"(3+√5)/2":[55],"2.618;":[57],"reciprocal":[58,121],"φ⁻²":[60],"(3−√5)/2":[62],"0.382.":[64],"The":[65,83,94,108,142],"tiling's":[66,109],"vertices":[67],"project":[68],"from":[69],"Z⁵":[70,119],"onto":[71],"plane":[74],"via":[75],"projection":[79,159],"(de":[80],"Bruijn's":[81],"pentagrid).":[82],"algebraic":[84],"number":[85],"field":[86],"ℚ(√5),":[88],"with":[89],"units":[90],"φ⁻¹.":[93],"two":[95],"rhombi":[97],"have":[98],"acute":[99],"angles":[100],"36°":[101],"72°,":[103],"sines/cosines":[105],"involve":[106],"√5.":[107],"Fourier":[110],"transform":[111],"has":[112],"Bragg":[113],"peaks":[114],"at":[115],"positions":[116],"indexed":[117],"(5D":[120],"lattice),":[122],"not":[123],"Z²":[124],"—":[125],"hence":[126],"\\"quasicrystal\\"":[127],"(Shechtman,":[128],"1982).":[129],"CONNECTION:":[131],"Directly":[132],"encodes":[133],"0.382":[134],"(φ⁻²),":[135],"0.618":[136],"(φ⁻¹),":[137],"1.618":[138],"(φ),":[139],"2.618":[140],"(φ²).":[141],"symmetry":[144],"forbidden":[146],"in":[147,156],"periodic":[148],"2D/3D":[149],"crystals":[150],"(crystallographic":[151],"restriction":[152],"theorem)":[153],"but":[154],"allowed":[155],"5D;":[157],"y":[160],"Author:":[161],"Andrew":[162],"Stewart":[163],"Caldin,":[164],"Independent":[165],"Researcher,":[166],"UK.":[167],"Part":[168],"E8":[171],"Intelligence":[172],"Research":[173],"series.":[174],"Platform:":[175],"e8intelligence.com":[176]}