Windmill Dynamics: Rotational Symmetry in Combinatorial Geometry — E8 Intelligence Research
{"FINDING:":[0],"The":[1,36,81,122,152],"2011":[2],"IMO":[3],"Problem":[4],"2":[5],"(\\"windmill\\")":[6],"is":[7,50,148,154],"a":[8,16,22,31,48,51,125,156],"combinatorial":[9],"geometry":[10],"problem":[11],"whose":[12],"solution":[13],"hinges":[14],"on":[15,21,87,106,119,161],"discrete":[17],"rotation":[18,153],"group":[19],"acting":[20],"finite":[23],"point":[24,56,68],"set,":[25],"with":[26],"an":[27],"invariant":[28],"line":[29,52,76,101],"through":[30,54,66],"pivot":[32,71,129],"point.":[33],"|":[34],"MATH:":[35],"core":[37],"structure:":[38],"Given":[39],"\\\\(n\\\\)":[40,98,110],"points":[41,86,105],"in":[42],"general":[43],"position":[44],"(no":[45],"three":[46],"collinear),":[47],"\\"windmill\\"":[49],"\\\\(L\\\\)":[53,64,91],"one":[55,120],"\\\\(P\\\\)":[57],"(the":[58,75],"pivot)":[59],"that":[60],"rotates":[61,78],"continuously;":[62],"when":[63],"passes":[65],"another":[67],"\\\\(Q\\\\),":[69],"the":[70,83,95,100,112,132,139,145,162,175],"switches":[72],"to":[73],"\\\\(Q\\\\)":[74],"now":[77],"about":[79],"\\\\(Q\\\\)).":[80],"invariant:":[82],"number":[84],"of":[85,90,159,164,174],"each":[88,107,128],"side":[89],"remains":[92],"constant":[93],"throughout":[94],"process.":[96],"For":[97],"odd,":[99],"always":[102],"has":[103],"\\\\((n-1)/2\\\\)":[104],"side;":[108],"for":[109,138],"even,":[111],"counts":[113],"alternate":[114],"between":[115],"\\\\((n-2)/2\\\\)":[116],"and":[117,141],"\\\\(n/2\\\\)":[118],"side.":[121],"proof":[123],"uses":[124],"parity":[126],"argument:":[127],"switch":[130],"changes":[131],"side-count":[133],"by":[134],"exactly":[135],"\\\\(\\\\pm":[136],"1\\\\)":[137],"old":[140],"new":[142],"pivot,":[143],"but":[144],"total":[146],"imbalance":[147],"preserved":[149],"modulo":[150],"2.":[151],"effectively":[155],"cyclic":[157],"action":[158],"\\\\(\\\\mathbb{Z}_n\\\\)":[160],"set":[163],"\\"ba":[165],"Author:":[166],"Andrew":[167],"Stewart":[168],"Caldin,":[169],"Independent":[170],"Researcher,":[171],"UK.":[172],"Part":[173],"E8":[176],"Intelligence":[177],"Research":[178],"series.":[179],"Platform:":[180],"e8intelligence.com":[181]}
Authors
- Andrew Stewart Caldin
Publication Details
- Journal
- Zenodo (CERN European Organization for Nuclear Research)
- Published
- 2026-09-18
- DOI
- https://doi.org/10.5281/zenodo.22824399
- Primary Topic
- Computability, Logic, AI Algorithms
- Type
- preprint