
chaturbhuj nirdeshank

chaturthank nirdeshank, jyanna caltrop, tetre kinwa chakovian nirdeshank asehi mhanatat, teey darrell jarmash aanee itaranni vikasit kele hote, kaaran dusare mhanje simplex kinwa tetrahedroncha aadhar polyhedron mhanoon vapar karnari eka samanvay pranali, jeey sope nirdeshank ghete.
bhaumitik vyakhya.
chaar aadhar vahak (parantu aavashyak nahee qee ekak) niyamit tetrahedronchya kendratoon udbhavtat aanee tyachya chaar kopyanmadhye jaatat. tyanche samanvay patte anukrame (1,0,0,0), (0,1,0,0), (0,0,1,0) aanee (0,0,0,1) aahet. heey rotationshivay positive scale kele jaau shakatat (udaa.jee. nakaratmakata) aanee reshiyaritya ektrit karoon paramparik 'xyz' jaaga vyapali jaate, jyamadhye chaar nirdeshankanpaiki kiman eka anavashyak (shunyavar seet kelele) asel.
shaikshanik mahatva.
eka vishisht anuprayog bases tetrahedronchya kadaa ekak mhanoon seet karoo shakato. tetrahedronchi vyakhya khandache ekak mhanoon dekhil kelee jaau shakate (khalee pahaa).
unit volumeche prarambhik tetrahedron dilelya, inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0) aanee cube, octahedron, rombic dodecahedron aanee cuboctahedron khand 3,4,6 aanee 20 sathee purnank nirdeshank pradan karnyasathi chaar chaturbhuj reshiyaritya ekatra kele jaau shakatat.
udaharanarth, a, b, c, d heey anukrame (1,0,0,0), (0,1,0,0), (0,0,1,0) aanee (0,0,0,1) dile tarr, samaan kaath lambi aanee aakarman chaar asalelya ashtabhujakaranche shirobindu a + b, a + c, a + d, b + c, b + d, c + d kinwa {1,1,0,0] chya serve aath kramparivartan asateel. 
{2,1,1,0} chya 12 kramparivartananni (0,0,0,0) varr kendrit asalelya khand 20 cuboctehedron shirobindu paribhashit kelya aahet. heey vahak konatyahi golapasoon tyachya sabhovatalachya 12 shejarila cubic close packingmadhye (see. see. pii.) nirdesh karatat, samkramitamadhye aay. vhee. emm. (isotropic vector matricks) chya samatulya aahet. tyamule see. see. pii. chendu kendranmadhye serve nakaratmak nasalelya purnank nirdeshank asatat.
jarr konee aata yaa khandala 'chaar-aayami' kinwa 'chaar-dishatmak' mhanoon '4d' mhatale tarr aapan r samajoon ghenyasathi pampala prathamik kele aahe. buckminster fullerchi "4dii bhoomiti" kinwa "synergetics". 
yaa american divyatavadi tattvadnyanat, chaar aantarprabhavi eka-trijya chendundware paribhashit kelelya eka kadaa asalelya niyamit tetrahedronla aakarmanache ekak mhanoon ghetale jaate. 
parichit bahirmukh bahupadicha sanch, jyala "ekagra padanukram" mhanatat, too vareel taktyanusar tyachya bhovati asha prakare basavala jaato qee ghanamadhye khand 3, ashtabhuj khand 4, samghan dodecahedron khand 6 aanee ghanghatpatri khand 20 asatat.