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Sēo fifhyrne (Niwenglisc: pentagon) oþþe fifhyrne ġesceap is sēo ġesċeap ƿiþ fif hyrneġen.
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Sēo samaflanc fifhyrne hæfde sēo sclæfligruna æf {5} ond innyrdigfealds æf 108°.
Sēo samaflanc fifhyrne hæfde fif rindenes æf ābēodendlīcum gelīċnes, and ymbhrǣdlic gelīċnes æf ordre 5 (þurh 72°, 144°, 216° and 288°). Þē gegendlihts æf sēo ūtbendan gewunlic fifhyrne are in þē golden hlutord to hits rindenes. Giefan hits side lengð hits stēapne (feornes fram ān side to þē wiðerlang hwyrftstān), wīdnes (Niwenglisc: width) (feornes betwēonum tƿā feorrum āgesliten ords, hwic īsgelīcen þē gegendliht lengð ) and ymbhrēodanfex are giefan by:
Sēo oferside æf sēo ūtbendan gewunlic fifhyrne ƿiþ side lengð is giefan by
If þē ymbhrēodanfex æf sēo gewunlic fifhyrne is giefan, hits eċġ lengð is befunden by þē ǣwednes
and hits oferside is
siþþan þē oferside æf þē ymbwrāted hring is þē gewunlic fifhyrne fills nēah 0.7568 æf hits ymbwrāted hring.
Þē oferside æf ænig gewunlic ġesceap is:
ƿǣr P is þē ymbhāt (Niwenglisc: perimeter) æf þē ġesceap, and r is þē onhrēodanfex (gemænelīceg þē apothem). Underbēogenden þē gewunlic fifhyrnes wurðscipes fore P and r giefas þē gerihtlǣcung
ƿiþ sīde lengð t.
Anlīċ to ǣġhwelċ gewunlic ūtbendan ġesceap, þē gewunlic ūtbendan fifhyrne hæfde an inwrāted hring. Þē apothem, hwic is þē hrēodanfex r æf þē inwrāted hring, æf sēo gewunlic fifhyrne is geferian to þē sīde lengð t by
Līciġe ǣġhwelċ gewunlic ūtbendanġesceap, þē gewunlic ūtbendan fifhyrne hæfde sēo ymbwrāted hring. Fore sēo gewunlic fifhyrne ƿiþ æfterfylgendlic ƿēohpunctu A, B, C, D, E, if P is ænig ord on þē ymbhring betwēonum ord B and C, þēn PA + PD = PB + PC + PE.
Fore æn wyllfull ord in þē plane æf sēo gewunlic fifhyrne ƿiþ ymbhrēodanfex , hwæs feornesse to þē gemǣnellǣhtwyrft æf þē gewunlic fifhyrne and hits fif ƿēohpunctu are and gehwādorlīce, ƿē habbað[1]
If are þē feornesse fram þē ƿēohpunctu æf sēo gewunlic fifhyrne to ænig ord on hits ymbhringġesceap , þēn [1]
Þē gewunlic fifhyrne is edlēhendlic ƿiþ færanfæþm and straighteċġ, as 5 is sēo Fermat prime. Sēo gerifl æf ǣrendweges are cnāwen fore bytleden sēo gewunlic fifhyrne. Sum are gedǣled neoþera.
Ān ǣrendweg to bytle sēo gewunlic fifhyrnein sēo giefan hringġesceap is æfwrāted by Ricemund[2] and ġēond gedǣled in Cromwell's Polyhedra.[3]
Þē top scēadwelle geswutelaþ þē edlēhendlic brūcaned in Ricemunden ǣrendweg to cræfte þē sīde of þē inwrāted fifhyrne. Þē hringġesceap geƿissung þē fifhyrne hæfde ord hrēodanfex. Hits middel is befērēd æt ord C and sēo middumord M is gemearcod hālfweg be hits hrēodanfex. Þis ord is tōgæderebunden to þē ytemest swāupp upper þē middel æt ord D. Hyrne CMD is twafealden, and þē twafoldere samētanes þē uppīweard stefn æt ord Q. Sēo wīdnesline þurh Q samētanes þē hringġesceap æt ord P, and strēowa PD is þē þearf sīde æf þē inwrāted fifhyrne.
Tō bestimma þē lengð æf þis side, þē tƿā rihtþrīhyrnes DCM and QCM are geƿeorþod neoþera þē hringġesceap. Brūcende Pīthagoras þeoruma and twa rindenes, þē underwige æf þē lengra þrīhyrnes is befunden as . Sīde h æf þē læssa þrīhyrne þēn is befunden brūcende þē twāfealdhyrne gerihtlǣcung:
ƿǣr cōsīne and sīne æf ϕ are cnāwen fram þē lengra þrīhyrne. Þē geswutelung is:
If DP is sōðlīce þē sīde æf sēo gewunlic fifhyrne, , so DP = 2 cos(54°), QD = DP cos(54°) = 2cos2(54°), and CQ = 1 − 2cos2(54°), hwic īsgelīcen −cos(108°) by þē cōsīne twāfeald gerihtlǣcung. Þis is þē cōsīne æf 72°, hwic īsgelīcen as gewilnode.
Sēo Ceorlēl hringġesceap wæs crǣften as sēo gemetgemostric ǣrendweg to ġefrīgean þē wyrtwela æf sēo fēoworhyrnum efenunga.[4] Þis ǣrendgewritlīce to sēo procedure fore bytleden sēo gewunlic fifhyrne. Þe stōpen are ǣs fallowen:[5]
Steps 6–8 are gemænelīce to þē fylgende feorðunga, getǣled in þē hrinanġemāting:
Sēo gewunlic fifhyrne is towearplic brūcende sēo færanfæþm ond streċċaneċġ, ǣġhwæþer by inscribing ān in sēo giefan hringġesceap or bytleden ān on sēo giefan eċġ. Þis gecynd wæron ǣfwrāted by Ewcleod on his bōc, Ādalen circa 300 BC.[6][7]
Þē gewunlic fifhyrne has Twyflax5 gelīċnes, ordre 10. Siþþan 5 is sēo formarim þǣr is ān underflocc ƿiþ twyflax gelīċnes: Dih1, and 2 cyclic flocc gelīċnese: Z5, and Z1.
Þas 4 gelīċnese cunnan biþ geseon on 4 geondweard gelīċnese on þē fifhyrne. Iohannes Cōnweg ingraves þas by sēo rūna and flocc ordren.[9] Full gelīċnes æf þē gewunlic form is r10 and no gelīċnes is ingraved a1. Đā twyflaxen gemetgemostras syndon dælode be þǣm hwæðer hī faraþ þurh wēohsēoꝛpunga (d fore gegendliht) or eċġen (p fore perpendiculars), and i hwænne ābēodendlīc rindenes bēah þurh bēgen eċġen and wēohpunctu. Hringlic gelīċnese in þē middel stēap are ingraved as g fore hiera midlig gyration ordren.
Gæt wēnġungflocc geliċnes hlēomāþe wēlġað wān oþþe mā dæġēð ān wīnōð ofer unġewōðelicum ƿyrmbēam. Ānlȳċ þæt g5 ƥēodhlēomāþ hæfð nān frīġedēoƿu āc mæġ bēon gesēon swā bewundenum hrēod.
Sēo fīfhyrnedesteorra is sēo gewunlic steorra fifhyrne. Hits Sclæfliruna is {5/2}. Hits rindenes form þē gegendlihts æf sēo gewunlic ūtbendan fifhyrne – in þis gecynd þē rindenes of þē tƿā fifhyrnen are in þē golden hlutord.
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