
zonohedrone

jyaamiti mein, ack zonohedron ack uttal bahu-hedron haye joe kendriya roop seey samamit haye, jiska pratyek chehra ack bahubhuj haye joe kendriya roop seey samamit (ack zonogan) haye. kissy bhee zonohedron koo samatulya roop seey tri-ayami sthaan mein rekha khandon key ack samooh key minkovsky yog key roop mein, yaa ack atee-ghana key tri-ayami prakshepan key roop mein varnit kiya jaa sakata haye. zonohedra koo mool roop seey ee dwara paribhaashit aur adhyayan kiya gayaa thaa. aiss. fedorov, ack russie chrystaller. adhik aam taur parr, kissy bhee ayam mein, rekha khandon kaa minkovsky yog ack polytope banaata haye jise ack zonotop key roop mein jaanaa jaataa haye.
zonohedra vaha tyle sthaan.
zonohedra kaa adhyayan karane key liye mool prerana yeh haye kii kissy bhee jaali kaa voronoi aarekh ack uttal samaan madhukosh banaata haye jisamen koshikaaen zonohedra hoti hain. iss tarah seey banaa koee bhee zonohedrone tri-ayami sthaan koo asthir curr sakata haye aur ise praathamik samanantar kahaa jaataa haye. pratyek praathamik samanantar hedron paanch prakaron mein seey ack key samatulya haih rombohedron (ghana sahit), shatkon prism, kataa huaa octahedron, rombic dodecahedron aur rombo-hexagonal dodecahedron.
minkowski seey zonohedra raashi.
sootra _ 1 koo tri-ayami vactors kaa sangrah honey dein. pratyek sadish sootra _ 2 key saath humm ack rekha khand sootra _ 3 koo jodd sakate hain. minkowski yog sootra _ 4 ack zonohedron banaata haye, aur sabhi zonoheader joe mool mein hotey hain, unakaa yeh roop hotaa haye. jinn vactors seey zonohedron banata haye, unhein iska generator kahaa jaataa haye. yeh charitra varnan, zonohedra kii paribhasha koo uchch aayamon mein saamaanyikrit karane kii anumati detaa haye, jisase zonotope milate hain.
ack zonohedron mein pratyek kinara cumm seey cumm ack generator key samanantar hotaa haye, aur isaki lambai generator kii lambai key yog key barabar hoti haye jisake liye yeh samanantar hotaa haye. isaliye, binaa vactor key samanantar jode vaale generators key ack samooh koo chunkar, aur sabhi vector lambai koo samaan nirdhaarit karke, humm kissy bhee sanyojan prakaar key zonohedron kaa ack sambahu sanskaran banaa sakate hain.
uchch degree samroopata vaale vactors key seton koo chunkar, humm iss tarah seey banaa sakate hain, cumm seey cumm utani hii samroopata key saath zonohedra. udaaharan key liye, ack gole key bhoomadhya rekha key chaaron or samaan roop seey generator, gole key dhruvon key madhyam seey generator kii ack aur jodee key saath, niyamit sootra _ 5-guns parr prism key roop mein zonohedra banaate hainh ghana, shatkon prism, ashtakoniya prism, vikarn prism, dwibhuj prism, aadi.
ack octahedron key kinaaron key samanantar generator ack kataa huaa octahedron banaate hain, aur ack ghana key lambe vikarn key samanantar generator ack sambhaajak dodecahedron banaate hain.
kissy bhee dou zonoheader kaa minkovsky yog ack aur zonohedron haye, joe diye gaye donon zonoheader key generators key sangh dwara utpann hotaa haye. iss prakaar, ack ghana aur ack kataa huaa octahedrone kaa minkowski yog kataa huaa cuboctedron banaata haye, jabaki ghana aur rombic dodecahedron kaa minkowski yog kaataa huaa rombic dodecahedron banaata haye. yeye donon zonoheader saral hain (pratyek sheersh parr teen chehare milate hain), jaisa kii cube key minkowski yog, kataa huaa octahedron aur rombic dodecahedron seey banaa chhota rombicuboctedron haye.
vyavastha seey zonohedra.
kissy bhee uttal bahudhruviya kaa goss maanchitra bahubhuj key pratyek chehare koo ikai gole parr ack bindu parr maanchitrit kartaa haye, aur bahubhuj key pratyek kinare koo ack jodee chehare koo ack mahaan vritt chaap seey alag kartaa haye joe sambandhit dou binduon koo jodata haye. ack zonohedrone key mamle mein, pratyek chehare key aspas key kinaaron koo samanantar kinaaron key jode mein vargeekrit kiya jaa sakata haye, aur jaba goss maanchitra key madhyam seey anuvaad kiya jaataa haye tou aisee koee bhee jodee usee mahaan vritt parr sannihit khandon kii ack jodee bunn jaati haye. iss prakaar, zonohedron key kinaaron koo samanantar kinaaron key kshetron mein vargeekrit kiya jaa sakata haye, joe goss maanchitra parr ack samanya mahaan vritt key khandon key anuroop hain, aur zonohedron key 1-kankal koo goal parr mahaan vritton kii vyavastha key liye samtal dohare graph key roop mein dekhaa jaa sakata haye. isake vipareet badey vritton kii koee bhee vyavastha vritton key madhyam seey vimaanon key lambavat vactor dwara utpann ack zonohedron key goss maanchitra seey banaai jaa sakati haye.
koee bhee saral zonohedron iss tarah seey ack saral vyavastha seey mell khaata haye, jisamen pratyek chehra ack trikon haye. badey vritton kii saral vyavasthaaen kendriya prakshepan key madhyam seey prakshepak tala mein rekhaaon kii saral vyavasthaon key anuroop hoti hain. saral vyavasthaon key teen gyaat anant parivaar hain, jinamen seey ack jaba zonohedra mein parivartit hoe jaataa haye tou prism kii or jaataa haye, aur anya dou joe saral zonohedra key atirikt anant parivaaron seey mell khaate hain. aise kaee chhitput udaaharan bhee hain joe inn teen parivaaron mein fitt naheen hotey hain.
yeh zonohedra aur vyavasthaon key beach patrachar seey, aur sylvester-galaai pramey seey, joe (apane prakshepya dohare roop mein) kissy bhee vyavastha mein kewal dou rekhaaon key paar honey key astitva koo saabit kartaa haye, kii pratyek zonohedron mein cumm seey cumm ack jodee vipareet samanantar aakriti hoti haye. (iss uddeshya key liye varg, aayat aur samasthaanik koo samanantar chitra key vishesh mamalon key roop mein ginaa jaataa haye.) adhik dridhata seey, pratyek zonohedron mein cumm seey cumm chhah samanantar chaturbhuj chehare hotey hain, aur pratyek zonohedron mein kaee samanantar chaturbhuj chehare hotey hain joe generator kii sankhya mein raikhik hotey hain.
zonohedra key prakaar.
ack niyamit bahubhuj parr koee bhee prism, jisamen kinaaron kii ack summ sankhya hoti haye, ack zonohedron banaata haye. inn prismon koo banaaya jaa sakata haye taaki sabhi chehare niyamit honh dou vipareet chehare niyamit bahubhuj key barabar hotey hain jisase prism banaa thaa, aur yeye vargakar cheharon key anukram seey judey hotey hain. iss prakaar key sambhuj, shatkon prism, ashtakoniya prism, vikarn prism, dwikoniya prism aadi hain.
niyamit roop seey chehare vaale iss anant parivaar key alawa, teen archimedian thhos hain, joe niyamit roopon key sabhi sarvavyaapi hainh 
isake alawa, kuchh catalan thhos (archimedian thhos key dwibhashi) phir seey zonohedra hainh
samroop chehare vaale anyah
samroop chehare vaale aseem roop seey kaee zonohedra hain joe ack dusare key liye sarvasam naheen hain. inmein shamil hainh
zonohedra kaa vichhedan.
haalaanki yeh aam taur parr sach naheen haye kii kissy bhee polyhedron kaa vichhedan usee aayatan key kissy anya polyhedron mein hotaa haye (hilbert kii teesari samasya dekhein), yeh gyaat haye kii samaan matra key kissy bhee dou zonoheader koo ack dusare mein vichhedit kiya jaa sakata haye.
zonohedrification.
zonohedrification george doubleu dwara paribhaashit ack prakriya haye. ack anya polyhedron seey ack zonohedron banaane key liye haart.
pahale kissy bhee beej polyhedron key sheershon koo polyhedron kendra seey vactor maanaa jaataa haye. yeye vactor zonohedron banaate hain jise humm mool polyhedron kaa zonohedrification kahate hain. yadi beej polyhedron mein kendriya samroopata haye, tou vipareet bindu usee disha koo paribhaashit karate hain, isaliye zonohedron mein kshetron kii sankhya beej key sheershon kii sankhya kaa aadhaa haye. mool bahu-paridhiya key kissy bhee dou sheershon key liye, summ-paridhiya key dou vipareet tala hotey hain jinamen seey pratyek key dou kinare sheersh vactor key samanantar hotey hain.
zonotope.
kissy bhee ayam mein rekha khandon kaa minkovsky yog ack prakaar kaa polytope banaata haye jise ack zonotop kahaa jaataa haye. samatulya roop seey, vactor sootra _ 7 dwara utpann ack zonotope sootra _ 6 sootra _ 8 dwara diyaa jaataa haye. dhyaan dein kii vishesh mamle mein jahaan sootra _ 9, zonotope sootra _ 6 ack (sambhavatah apakshayi) samanantar haye.
kissy bhee zonotope key pahalu swayam ack nichale ayam key zonotope hotey hain; udaaharan key liye, zonohedra key chehare zonogan hotey hain. chaar ayami zonotops key udaharanon mein tesseract ("dee" parasparik roop seey lambavat barabar lambai rekha khandon kaa minkovsky yog), sarvavyaapi 5-koshika aur kataa huaa 24-koshika shamil hain. pratyek permutohedrone ack zonotop haye.
zonotops aur matroids.
vactor sootra _ 12 key sett seey paribhaashit ack zonotope sootra _ 6 koo thheek karein aur sootra _ 13 koo sootra _ 14 matricks honey dein, jisake collum sootra _ 2 hain. phir sootra _ 13 key collum parr vector matroid sootra _ 16 sootra _ 6 key baare mein jaankaari kaa ack dhana kootbaddh kartaa haye, yaani, sootra _ 6 key kaee guna vishuddh roop seey prakriti mein sanyojan hain.
udaaharan key liye, sootra _ 6 key vipareet pahaluon key jode swabhavik roop seey sootra _ 21 key kokarkit dwara anukramit hotey hain aur yadi humm sootra _ 23 dwara darshaae gaye unmukh matroid sootra _ 21 parr vichaar karate hain, tou humm sootra _ 6 key pahaluon aur sootra _ 21 key hastaaksharit kokarkeet key beach ack bijection praapt karate hain joe sootra _ 6 key chehare kii jaali aur sootra _ 21 key covector key beach ack sthiti virodhi samroopata taka failata haye joe sootra _ 28 key ghatak-vaar vistaar dwara aadeshit hotaa haye. vishesh roop seey, yadi sootra _ 13 aur sootra _ 30 dou matricks hain joe ack projective parivartan dwara bhinn hotey hain tou unake sambandhit zonotops sanyojan roop seey samatulya hotey hain. pichhale kathan kaa vyutkram naheen haih khand sootra _ 31 ack zonotope haye aur donon sootra _ 32 aur sootra _ 33 dwara utpann hotaa haye, jinake sambandhit matricks, sootra _ 34 aur sootra _ 35, ack projective parivartan seey bhinn naheen hain.
jhukaav.
zonotope sootra _ 6 key tiling guna bhee isase judey unmukh matroid sootra _ 21 seey nikatata seey sambandhit hain. pahale humm space-tiling property parr vichaar karate hain. zonotope sootra _ 6 koo "tyle" sootra _ 39 kahaa jaataa haye yadi vactor sootra _ 40 kaa ack samooh haye jaise kii sabhi kaa sangh sootra _ 41 (sootra _ 42) sootra _ 39 haye aur koee bhee dou pratyek key (sambhavatah khaali) chehare mein kaatate hain. iss tarah key ack zonotope koo "space-tiling zonotope" kahaa jaataa haye."space-tiling zonotops kaa nimnalikhit vargeekaran mcmulen key kaaran haih vactor formula _ 45 tiles space dwara utpann zonotop formula _ 6 yadi aur kewal tabhi jaba sambandhit unmukh materoid niyamit hoe. isaliye ack sthaan-tilling zonotop honey kii prateet honey vaali jyaamitiya sthiti vaastav mein kewal utpann karane vaale vactors kii sanyojan sanrachana parr nirbhar karati haye.
ack anya parivaar kaa tillings joe kii zonotope sootra _ 6 seey judaa huaa haye, sootra _ 6 kaa "zonotpal tillings" haye. zonotope kaa ack sangrah sootra _ 6 kaa ack zonotpal tiling haye yadi yeh samarthan sootra _ 6 key saath ack polyhedral complexes haye, yaani, yadi sangrah mein sabhi zonotope kaa sangh sootra _ 6 haye aur koee bhee dou pratyek key ack samanya (sambhavatah khaali) chehare mein pratichhed karate hain. iss prishth parr sthit zonohedra kii kaee chhaviyon koo kewal samtal vastuon (tri-ayami vastuon key samtal pratinidhitva key vipareet) key roop mein maanate hue dwi-ayami zonotope key zonotaple jhukaav key roop mein dekhaa jaa sakata haye. bohn-dress pramey bataata haye kii sootra _ 6 seey judey oriented matroid sootra _ 21 key zonotop sootra _ 6 aur "ekal-tatva lift" key zonotap jhukaav key beach ack bijection haye.
matra.
samanya roop seey, zonohedra aur "aine"-ayami zonotope, apane aayatan key liye ack saral vishleshanatmak sootra koo sweekaar karane key liye ullekhaniya hain.
maan leejiye kii sootra _ 54 vactor sootra _ 56 key ack samooh dwara utpann zonotop sootra _ 8 haye. phir sootra _ 54 kaa n-ayami aayatan sootra _ 58 dwara diyaa jaataa haye.
iss sootra mein nirdhaarak kaa arth samajh mein aataa haye kyonki (jaisa kii oopar ullekh kiya gayaa haye) jaba sett sootra _ 59 mein pariveshiya sthaan key ayam sootra _ 60 key barabar cardinality hoti haye, tou zonotop ack samanantar haye.
dhyaan dein kii jaba sootra _ 61 hotaa haye, tou yeh sootra buss yeh bataata haye kii zonotop mein aine-volume shoonya hotaa haye.