
saadhaarana 4-polytope

ganithashaasthrathil, oru saadhaarana 4-polytope oru saadhaarana naalu-dimensional polytoppaanu. avaa trimaanangalil saadhaarana polyhedrayude naalu dimensional analogukalum rund alavukalil saadhaarana polyganukalumaanu.
aarr konveksukalum patthu staar regular 4-polythoppukalum undu, ithu motham pathinaar nalkunnu.
charithram.
pathonpathaam noottaandinte madhyathil swis ganithashaasthranjanaaya ludwig schlaflyanu convex regular 4-polythoppukale aadyamaayi vivarichathu. atharam aarr kanakkukal krithyamaayi undennu addeham kandethi.
grand 120-sell, grettu stelletted 120-sell, grand 600-sell, grettu grand stelletted 120-sell enningane naalu saadhaarana nakshatra 4-polythoppukalum schlafly kandethi. koshangalilo sheershakachithrangalilo (seero-hole torri "f" − "e" + "v" 2) euler swabhaavam parajayappetta roopangal addeham anuvadikkaathathinaal addeham backie aarr ozhivaakki. grettu dodecahedron (5,), cheria stelletted dodecahedron (5) thudangiya koshangalum sheershaka roopangalum ithil ninnu ozhivaakkunnu.
edmund hess (1843-1903) 1883-ll thante jarmman pusthakamaaya "ainlaittung inn dye lehre voonn derr kugelteylung mitt besonderer beruxichtigang ihrar anvendung ouf dye theori derr gleachflachijan undu derr gleacheckijen polidar" enna pusthakatthil poornna pattika prasidheekarichu.
nirmmaanam.
oru saadhaarana 4-polytope formulayude nilanilpp parimithappeduthunnathu athinte koshangalum dihedral angle parimitiyum roopappeduthunna saadhaarana polyheadra formulayude nilanilppaan.
koshangal othucherunnuvennu urappakki oru adanja 3-uparithalam roopappeduthunnu.
vivarichirikkunna aarr convex, patthu staar polytopukal maathramaan eee parimithikalkkulla eaka pariharam.
saadhuvaaya cellukalum (p, q), sheershaka samkhyakalum (q, r) ullathum dihedral testtil vijayikkunathum ennaal parimithamaaya samkhyakal srishtikkunnathil paraajayappedunnathumaaya naalu nonkonveks schlufly chinnangal (p, q, r) undu {3,3}, {4,3,}, 3,4,3,}.
saadhaarana convex 4-polythoppukal.
trimaanathilulla platonic kharapadaarthangalude chaturmana analogukalum rund alavilulla convex saadhaarana bahubhujangalumaan saadhaarana convex 4-polythoppukal.
aaril anjennam bandhapetta anchu platonic kharapadaarthangalude vyakthamaaya saadrushyangalaanu. aaraamatheth, 24-sellinu, moonnu alavukalil saadhaarana analog illa. ennirunnaalum, kramarahithamaya oru jodi kharapadaarthangal nilavilundu, cuboctahedrone athinte eratta rombic dodecahedrone, avaa 24-koshathinu bhagikamaaya samaanathakalaan (pooraka reethikalil). avaye orumichu 24-sellinte trimana analogaayi kaanaan kazhiyum.
oro convex regular 4-polytopinum ore tharatthilum valippathilumulla platonic kharapadaarthangalaaya trimana "cellukal" undu. iva pathivayi athath mukhangalil (mukhaamukham) gadippichirikkunnu.
swathukkal.
avayude trimana analogukal pole, convex regular 4-polythoppukalkku athe vyaasathil 4-dimensional ulladakkathinte (hypervolium) alavu enna nilayil valuppam anusarichu swaabhaavikamaayi kramappeduthaam. eee shreniyile oro valia polytope athinte mungaamiyekkaal "raundar" aanu, ore vyaasathinullil kooduthal ulladakkam ulkkollunnu. 4-simplexes (5-sell) aanu ettavum cheria cases, 120-sell aanu ettavum valuth. sankeernnatha (konfigareshan maatriksukal allengil kevalamaya konukalude ennam thaarathamyam cheythukondu alakkunnathu pole) athe kramam pinthudarunnu.
thaazhepparayunna pattika aarr convex regular 4-polithoppukalude chila gunangal pattikappedutthunnu. eee 4-polithoppukalude samamiti grouppukalellaam koksettar grouppukalaanu, aaa lekhanathil vivarichirikkunna notteshanil nalkiyirikkunnu. groupinte perinu sheshamulla nambaraanu groupinte kramam.
simplexes, orthoplex, tesseract, octaplex allengil polyoctahedron (pioo), tetraplex allengil polytetrahedron (pidi), dodecaplex allengil polydodecahedron (pidi) ennee perukal joonn konve vaadichu.
norman jonson enn-sell, allengil pentaakoron, hexadecacoron, tesseract allengil okdaakoron, ikosittetraakkoron, hexacosicoron, hecatonic cosacoron (allengil dodecacondakoron) ennee perukal vaadichu, "polycoron" enna padam 3 dii polyhedronumaayi 4 dii saadrishyavum 2 dii pooligoon, greak verukalil ninnu "poolii" ("niravadhi"), "choroz" ("murii" allengil "idam") ennivayil ninnu prakadippikkunnu.
ellah 4-polythoppukaludeyum euler swabhaavam poojyamaanu, yoolarinte polyhedral formulayude 4-dimensional analog namukkunda
ivide "n" k "ennathu polytoppile" k "-mukhangalude ennathe soochippikkunnu (oru sheershakam 0-mukham, oru attam 1-mukham muthalaayava).).
thannirikkunna eathenkilum 4-polytoppinte topology nirvachikkunnath athinte betti nambarukalum tortion coffificientukalum aanu.
konfigareshanukal pole.
oru saadhaarana 4-polytoppine athinte ghadaka gadakangalude ennam adangiya oru konfigareshan matricks aayi poornnamaayum vivarikkaam. varikalum nirakalum shringangal, arikukal, mukhangal, cellukal ennivayumaayi poruthappedunnu. diagonal samkhyakal (mukalil idathuninnu thaazhe valathottu) muzhuvan 4-polytopilum oro moolakavum ethrayaanu sambhavikkunnathenn parayunnu. nirayude moolakangalil ethra ennam variyude moolakatthilo allengil athil sambhavikkunnuvennu noonn diagonal samkhyakal parayunnu. udaaharanathinu, oro attatthum 2 konukalundu (oro attatthinum 2 konukalundu), koodaathe eathenkilum saadhaarana 4-polittoppil 2 koshangal oro mukhathum (oro mukhavum 2 koshangalil pedunnu) kandumuttunnu. matricks 180 digri karakkunnathiloode duval polytoppinte krameekaranam nedaanaakum.
drishyavalkkaranam.
thaazhepparayunna pattika eee 4-polithoppukalude chila dvimaana pravachanangal kaanikkunnu. mattu vividha drishyavalkkaranangal chuvadeyulla baahya linkukalil kaanam. koksettar-dinkin rekhaachithra graafukalum schlufly chinnathinu thaazhe nalkiyittundu.
saadhaarana nakshathrangal (schlufly-hess) 4-polythoppukal.
 
10 saadhaarana svayam-verthirikkunna nakshatra polycorakalude (naalu-dimensional polythoppukal) poornna koottamaanu schlufly-hess 4-polythoppukal. avarude kandethalukalude bahumaanaarthamaanu avaykku eee peru nalkiyirikkunnath ludwig schlafly, edmund hess. oronnineyum prathinidheekarikkunnath oru sclafly chihnamaanu ("p", "q", "r") athil oru samkhyayundu. athinaal avaa saadhaarana nonkonveks keppler-poinsot polyhedraykku samaanamaan, avaa pentagraaminodu samaanamaan.
perukal.
keppler-poinsot polyhedraykku keyliyude perukal vipuleekarichukondu joonn konveyaanu ivide nalkia perukal nalkiyathu "stelletted", "grettu" ennivaykkoppam addeham oru "grand" modifier kootticherkkunnu. konve eee pravarthanaparamaaya nirvachanangal vaagdaanam cheythu
3 saadhaarana seld 4-polythoppukalil ninnulla 10 roopangalkku joonn konve peru nalkunnu pt = polytetrahedron {3,3,5} (oru tetrahedral 600-sell), pi = polycoshedrone {3,5,} (oru icosahedral 120-sell), pd = polydodecahedron {5,3,3} (oru dodecahedral 120-sell), prefix modifierukalode "g", "a", "s" ennivaykku grettu, (Ag) grand, stelletted. avasana steleshan, "grettu grand stelletted polydodecahedron" avayellaam "gaspidae" aayi adangiyirikkunnu.
samamiti.
patthu polichorakalkkum [3,3,5] (h4) hexacosicoric samamiti undu. avaa 6 anubandha gorsaatt tetrahedra russional-oordar simetri grouppukalil ninnaan srishtikkappedunnathu [3,5,5/2], [5,5/2,5], [5,3,5 2], [5/2,5,5/2], [5,5/2,3], [3,3,5 2].
oro groupinum 2 staar-polycora undu, svayam erattayaaya rund grouppukal ozhike, onnu maathram. athinaal patthu saadhaarana nakshatra polycorakalil 4 eratta jodikalum 2 svayam eratta roopangalumundu.
swathukkal.
kurippu
koshangal (polyheadra), avayude mukhangal (polygonukal), "polygonal edju figureukal", "polyhedral vertex figureukal" enniva avayude scluffly chinnangalaal thirichariyappedunnu.

