{"id":1725,"date":"2023-10-21T17:50:21","date_gmt":"2023-10-21T17:50:21","guid":{"rendered":"https:\/\/flatarmal.is\/?p=1725"},"modified":"2023-10-22T12:25:25","modified_gmt":"2023-10-22T12:25:25","slug":"soguhornid-arkimedes-fra-syrakusu","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=1725","title":{"rendered":"S\u00d6GUHORNI\u00d0: Ark\u00edmedes fr\u00e1 S\u00fdrak\u00fasu"},"content":{"rendered":"\n

Krist\u00edn Bjarnad\u00f3ttir. <\/p>\n\n\n\n

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Ark\u00edmedes hefur veri\u00f0 talinn mesti st\u00e6r\u00f0fr\u00e6\u00f0ingur fornaldar, e\u00f0a jafnvel allra t\u00edma. Hann f\u00e6ddist \u00ed borginni S\u00fdrak\u00fasu \u00e1 Sikiley \u00e1ri\u00f0 287 f. Kr. Hann mun hafa stunda\u00f0 n\u00e1m \u00ed Alexandr\u00edu \u00ed Egyptalandi, en sn\u00fai\u00f0 aftur til S\u00fdrak\u00fasu \u00fear sem hann l\u00e9st \u00e1ri\u00f0 212 f. Kr. \u00c1 t\u00edmum Ark\u00edmedesar voru h\u00e1\u00f0ar styrjaldir milli R\u00f3mverja, sem bjuggu \u00e1 mi\u00f0jum \u00cdtal\u00eduskaganum, og P\u00fanverja, sem bjuggu sunnan vi\u00f0 Mi\u00f0jar\u00f0arhaf \u00ed borginni Kar\u00feag\u00f3, n\u00fa \u00ed T\u00fanis. Sikiley, sy\u00f0sta ey \u00cdtal\u00edu, var mi\u00f0ja vegu milli \u00feessara borgr\u00edkja og str\u00ed\u00f0sa\u00f0ilar b\u00f6r\u00f0ust me\u00f0al annars um yfirr\u00e1\u00f0 yfir henni \u00ed \u00feremur styrj\u00f6ldum sem nefnd voru p\u00fanversku str\u00ed\u00f0in.<\/p>\n\n\n\n

S\u00ed\u00f0vetrar \u00e1ri\u00f0 2023 settist pistilritari a\u00f0 \u00ed S\u00fdrak\u00fasu. Erindi\u00f0 var a\u00f0 nj\u00f3ta h\u00e1lkulausrar \u00fativistar og a\u00f0 kynnast Ark\u00edmedesi betur. Fyrstu kynnin vi\u00f0 Ark\u00edmedes h\u00f6f\u00f0u raunar veri\u00f0 vi\u00f0 l\u00f6gm\u00e1l hans, Ark\u00edmedesarl\u00f6gm\u00e1li\u00f0 um uppdrif \u00ed v\u00f6kva. L\u00f6gm\u00e1l Ark\u00edmedesar var kennt \u00ed e\u00f0lisfr\u00e6\u00f0i til landspr\u00f3fs \u00e1 \u00e1runum 1946\u20131976. Fyrsta skrefi\u00f0 var a\u00f0 skilja hva\u00f0 or\u00f0i\u00f0 \u201euppdrif\u201c \u00fe\u00fdddi. Uppdrif merkir a\u00f0 hlutur sem liggur \u00ed v\u00f6kva l\u00e9ttist jafnmiki\u00f0 og v\u00f6kvinn vegur sem hluturinn ry\u00f0ur fr\u00e1 s\u00e9r.<\/p>\n<\/div>\n\n\n\n

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Pistilritari me\u00f0 vini s\u00ednum Ark\u00edmedesi.<\/strong><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n
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Uppdrif \u00ed v\u00f6kva<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\u00deegar \u00feyngd hlutar er j\u00f6fn \u00feyngd v\u00f6kvans sem hann ry\u00f0ur fr\u00e1 s\u00e9r, \u00fe\u00e1 marar hluturinn \u00ed kafi. Hlutur fer a\u00f0 flj\u00f3ta \u00feegar hann er l\u00e9ttari en v\u00f6kvi me\u00f0 sama r\u00famtak. L\u00f6gm\u00e1li\u00f0 er undirsta\u00f0a skipasiglinga. \u00dea\u00f0 eru mikilv\u00e6g fr\u00e6\u00f0i a\u00f0 finna \u00fat hve miklu m\u00e1 stafla \u00e1 skip svo a\u00f0 \u00fea\u00f0 flj\u00f3ti tryggilega en s\u00f6kkvi ekki. \u00dea\u00f0 er fremur einfalt ef skipi\u00f0 er bundi\u00f0 \u00ed h\u00f6fn, en \u00e1 h\u00f6fum \u00fati \u00fearf a\u00f0 gera r\u00e1\u00f0 fyrir a\u00f0 skipi\u00f0 velti og taki jafnvel \u00e1 sig sj\u00f3.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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L\u00f6gm\u00e1l Ark\u00edmedesar m\u00e1 einnig nota til a\u00f0 kanna hvort m\u00e1lmur er falsa\u00f0ur. Fr\u00e6g er sagan af H\u00eder\u00f3 II konungi S\u00fdrak\u00fasu sem r\u00edkti \u00fear \u00e1 \u00e1runum 275\u2013215 f. Kr. (Konungsr\u00edkin voru sm\u00e1 \u00ed \u00fe\u00e1 daga). Konungur afhenti gullsmi\u00f0i gull til a\u00f0 sm\u00ed\u00f0a \u00far \u00fev\u00ed k\u00f3r\u00f3nu fyrir sig. Konung gruna\u00f0i a\u00f0 gulli\u00f0 hafi veri\u00f0 blanda\u00f0 silfri og kalla\u00f0i \u00e1 Ark\u00edmedes s\u00e9r til hj\u00e1lpar.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Ark\u00edmedes t\u00f3k mola af silfri og annan af gulli sem b\u00e1\u00f0ir v\u00f3gu jafnmiki\u00f0 og k\u00f3r\u00f3nan. Hann s\u00f6kkti hvorum mola \u00e1samt k\u00f3r\u00f3nunni \u00ed fleytifullt kar af vatni og m\u00e6ldi hve miki\u00f0 vatn flyti \u00fat af fyrir hvern grip. Silfur er l\u00e9ttara en gull, og silfurmolinn ruddi meira vatni fr\u00e1 s\u00e9r en gullmolinn. Og k\u00f3r\u00f3nan skila\u00f0i meira vatni en gullmolinn. Af \u00feessu \u00e1lykta\u00f0i Ark\u00edmedes a\u00f0 m\u00e1lmurinn \u00ed k\u00f3r\u00f3nunni v\u00e6ri svikinn, gulli\u00f0 hafi veri\u00f0 blanda\u00f0 silfri. Sagan segir a\u00f0 Ark\u00edmedes hafi hlaupi\u00f0 \u00fat \u00e1 g\u00f6tu \u00feegar hann f\u00e9kk \u00feessa hugmynd eftir a\u00f0 hafa stigi\u00f0 ofan \u00ed ba\u00f0, og hr\u00f3pa\u00f0 \u201eEureka, eureka!\u201c, \u201e\u00c9g hef fundi\u00f0 \u00fea\u00f0, \u00e9g hef fundi\u00f0 \u00fea\u00f0!\u201c<\/p>\n<\/div>\n<\/div>\n\n\n\n

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D\u00e6mi\u00f0 um k\u00f3r\u00f3nu H\u00eder\u00f3s konungs hefur gengi\u00f0 aftur \u00ed kennslub\u00f3kum \u00f6ldum saman. Algebra<\/em> eftir \u00d3laf Dan\u00edelsson var l\u00edka lesin til landspr\u00f3fs um mi\u00f0ja s\u00ed\u00f0ustu \u00f6ld. \u00dear er \u00feetta d\u00e6mi a\u00f0 finna:<\/p>\n\n\n\n

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Gullk\u00f3r\u00f3na H\u00eder\u00f3s konungs \u00ed S\u00fdrak\u00fasu v\u00f3 20 \u201em\u00ednur\u201c, en var bl\u00f6ndu\u00f0 silfri. Ark\u00edmedes fann, a\u00f0 \u00feegar hann v\u00f3 hana \u00ed vatni, l\u00e9ttist h\u00fan um 1\/10 af \u00feyngd sinni. Hve miki\u00f0 var af hvorum m\u00e1lmanna \u00ed k\u00f3r\u00f3nunni? Gull l\u00e9ttist um 1\/20, \u00feegar \u00fea\u00f0 er vegi\u00f0 \u00ed vatni, en silfur um 1\/10.<\/p>\n<\/div>\n\n\n\n

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P\u00ed \u2013 hlutfall umm\u00e1ls og \u00feverm\u00e1ls hrings<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Uppdrif \u00ed v\u00f6kva var engan veginn meginvi\u00f0fangsefni Ark\u00edmedesar. Hann reikna\u00f0i til d\u00e6mis \u00fat hlutfalli\u00f0 p\u00ed, \u03c0, milli umm\u00e1ls og \u00feverm\u00e1ls hrings me\u00f0 meiri n\u00e1kv\u00e6mni en \u00e1\u00f0ur haf\u00f0i \u00feekkst.<\/p>\n\n\n\n

Talan \u03c0 er stundum nefnd fasti Ark\u00edmedesar. Ark\u00edmedes nota\u00f0i n\u00e1lgunara\u00f0fer\u00f0 til \u00feess a\u00f0 finna gildi \u03c0. Hann reikna\u00f0i umm\u00e1l reglulegra marghyrninga me\u00f0 \u00e6 fleiri hornum \u00feannig a\u00f0 l\u00f6gun \u00feeirra n\u00e1lga\u00f0ist hring \u00e6 meir. A\u00f0fer\u00f0inni var beitt \u00ed yfir 1000 \u00e1r.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n

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Ark\u00edmedes reikna\u00f0i efri og ne\u00f0ri m\u00f6rk \u03c0 me\u00f0 \u00fev\u00ed a\u00f0 reikna umm\u00e1l reglulegra sexhyrninga innan \u00ed og utan um hring og tv\u00f6falda s\u00ed\u00f0an fj\u00f6lda hli\u00f0anna \u00fear til hli\u00f0arnar voru or\u00f0nar 96. Ark\u00edmedes s\u00fdndi fram \u00e1 a\u00f0 gildi \u03c0 l\u00e6gi \u00e1 bili milli tveggja brota: 223\/71 < \u03c0 < 22\/7. Tugabrot voru \u00fe\u00e1 ekki komin til s\u00f6gunnar en \u00feetta jafngildir a\u00f0 \u03c0 s\u00e9 \u00e1 bilinu 3,1408 < \u03c0 < 3,1429. \u00de\u00e1 eru \u00fer\u00edr t\u00f6lustafir r\u00e9ttir. Fyrstu fimm r\u00e9ttir aukastafir eru 3,14159, en broti\u00f0 22\/7 var gjarnan nota\u00f0 \u00ed kennslub\u00f3kum fram a\u00f0 \u00f6ld vasareikna.<\/p>\n<\/div>\n\n\n\n

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Pistilritari g\u00e6gist milli leggja <\/strong><\/em>
l\u00edkans af p\u00ed \u00ed S\u00fdrak\u00fasu.<\/strong><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n
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Um \u00e1ri\u00f0 480 e. Kr. n\u00e1\u00f0i k\u00ednverski st\u00e6r\u00f0fr\u00e6\u00f0ingurinn Zu Chongzhi sj\u00f6 r\u00e9ttum t\u00f6lust\u00f6fum me\u00f0 12.288-hli\u00f0a marghyrningum. \u00dea\u00f0 var n\u00e1kv\u00e6masta gildi\u00f0 \u00e1 \u03c0 sem \u00feekkt var n\u00e6stu 800 \u00e1rin.<\/p>\n\n\n\n

\u00c1ri\u00f0 1699 n\u00e1\u00f0ist 71 t\u00f6lustafur me\u00f0 \u00f3endanlegum r\u00f6\u00f0um. Hra\u00f0virk reiknirit komu fram \u00e1ri\u00f0 1914 \u00feegar indverski st\u00e6r\u00f0fr\u00e6\u00f0ingurinn Srinivasa Ramanujan birti tugi af n\u00fdjum form\u00falum fyrir \u03c0. \u00de\u00e6r voru a\u00f0d\u00e1unarver\u00f0ar fyrir gl\u00e6sileika, st\u00e6r\u00f0fr\u00e6\u00f0ilega d\u00fdpt og hra\u00f0a samleitni. S\u00ed\u00f0an hafa or\u00f0i\u00f0 til form\u00falur \u00fear sem fj\u00f6ldi t\u00f6lustafa skiptir billj\u00f3num.<\/p>\n\n\n

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Fleiri s\u00edgildar st\u00e6r\u00f0fr\u00e6\u00f0ireglur<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Ark\u00edmedes fann fleiri s\u00edgildar st\u00e6r\u00f0fr\u00e6\u00f0ireglur. Hann s\u00fdndi til d\u00e6mis fram \u00e1 a\u00f0 r\u00famm\u00e1l k\u00falu v\u00e6ri 2\/3 hlutar af r\u00famm\u00e1li s\u00edvalnings me\u00f0 sama \u00feverm\u00e1l og s\u00f6mu h\u00e6\u00f0. S\u00f6mulei\u00f0is er r\u00famm\u00e1l keilu me\u00f0 sama \u00feverm\u00e1l og s\u00f6mu h\u00e6\u00f0 og s\u00edvalningurinn einn \u00feri\u00f0ji af r\u00famm\u00e1li s\u00edvalningsins. Me\u00f0 \u00f6\u00f0rum or\u00f0um eru r\u00famm\u00e1lshlutf\u00f6ll<\/p>\n\n\n\n

keilu, k\u00falu og s\u00edvalnings<\/p>\n\n\n\n

1 : 2 : 3<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Ef h\u00e6\u00f0 keilu og s\u00edvalnings er 2r eins og k\u00falunnar \u00fe\u00e1 er<\/p>\n\n\n\n

r\u00famm\u00e1l keilu: 2\/3 \ud835\udf0b\ud835\udc5f3<\/sup>, r\u00famm\u00e1l k\u00falu: 4\/3 \ud835\udf0b\ud835\udc5f3<\/sup>, r\u00famm\u00e1l s\u00edvalnings: 6\/3 \ud835\udf0b\ud835\udc5f3<\/sup><\/p>\n\n\n\n

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Eftir Ark\u00edmedes eru var\u00f0veitt \u00fdmis rit um st\u00e6r\u00f0fr\u00e6\u00f0i, me\u00f0al annarra rit um keilur og k\u00falur. Riti\u00f0 Talning sandkorna<\/em> segir fr\u00e1 s\u00f3lmi\u00f0jukenningu Aristarkosar fr\u00e1 Samos og mun \u00fea\u00f0 vera ein besta heimildin fyrir s\u00f3lmi\u00f0jukenningunni. \u00dear kemur fram a\u00f0 Ark\u00edmedes mi\u00f0ar vi\u00f0 a\u00f0 heimurinn s\u00e9 k\u00fala \u2013 hvelfing fastastjarnanna \u2013 sem hann \u00e1leit hafa t\u00edu\u00fe\u00fasundfalt \u00feverm\u00e1l jar\u00f0ar. S\u00ed\u00f0an taldi hann hve m\u00f6rg sandkorn \u00feyrfti til a\u00f0 fylla \u00feann heim upp.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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V\u00e9lar og hagn\u00fdt t\u00e6ki<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Ark\u00edmedes gat s\u00e9r miki\u00f0 fr\u00e6g\u00f0aror\u00f0 \u2013 og vakti \u00f3tta \u00f3vinanna \u2013 fyrir v\u00edgv\u00e9lar sem hann fann upp til a\u00f0 granda \u00f3vinunum. Margar \u00feeirra bygg\u00f0u \u00e1 vogarstangarreglunni, me\u00f0al annarra voru kranar sem lyftu \u00feungu fargi, jafnvel heilum skipum. Samkv\u00e6mt vogarstangarreglunni getur l\u00edtill kraftur \u00e1 l\u00f6ngum armi jafnast \u00e1 vi\u00f0 st\u00f3ran kraft \u00e1 stuttum armi. Fr\u00e6g eru umm\u00e6li Ark\u00edmedesar: \u201eLj\u00e1i\u00f0 m\u00e9r fastan punkt til a\u00f0 standa \u00e1 og \u00fe\u00e1 skal \u00e9g hreyfa j\u00f6r\u00f0ina.\u201c<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Hann fann l\u00edka upp hagn\u00fdt t\u00e6ki. Einna fr\u00e6gust uppfinning Ark\u00edmedesar er skr\u00fafan sem kennd er vi\u00f0 hann og var notu\u00f0 til a\u00f0 d\u00e6la upp vatni.<\/p>\n\n\n\n

Sk\u00fdringarmynd af skr\u00fafu Arkimedesar.<\/em><\/strong><\/p>\n<\/div>\n\n\n\n

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Dau\u00f0i Ark\u00edmedesar<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Andl\u00e1t Ark\u00edmedesar bar a\u00f0 me\u00f0 s\u00f6gulegum h\u00e6tti \u00ed annarri p\u00fanversku styrj\u00f6ldinni. Sagan segir a\u00f0 hann hafi seti\u00f0 \u00ed fj\u00f6runni vi\u00f0 S\u00fdrak\u00fasu og teikna\u00f0 hringa \u00ed sandinn er r\u00f3mverskan hermann bar a\u00f0. Hann skipa\u00f0i Arkimedesi a\u00f0 fylgja s\u00e9r til r\u00f3mverska hersh\u00f6f\u00f0ingjans Marcellusar sem haf\u00f0i m\u00e6lt fyrir um a\u00f0 Ark\u00edmedesi skyldi hl\u00edft. \u201eEkki rugla hringunum m\u00ednum\u201c er haft eftir Ark\u00edmedesi. En herma\u00f0urinn br\u00e1 \u00fe\u00e1 brandi og hj\u00f3 hann. Sagt er a\u00f0 ekkert hafi Marcellus teki\u00f0 jafn n\u00e6rri s\u00e9r og dau\u00f0a Ark\u00edmedesar.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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Heimildir og \u00edtarefni:<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Geir \u00de. \u00de\u00f3rarinsson. (2011, 22. desember). Hver var Ark\u00edmedes og hvert var hans framlag til v\u00edsindanna? <\/em>V\u00edsindavefurinn. http:\/\/visindavefur.is\/svar.php?id=61534<\/a>.<\/p>\n\n\n\n

Giovanni di Pasquale (2018). Archimedes in Syracuse<\/em>. Giunti Editore S.p.A.<\/p>\n\n\n\n

Gu\u00f0mundur Arnlaugsson (1991). Gull\u00f6ld gr\u00edskrar st\u00e6r\u00f0fr\u00e6\u00f0i. \u00cd Sigur\u00f0ur A. Magn\u00fasson, Kristj\u00e1n \u00c1rnason, \u00deorsteinn \u00deorsteinsson og Gu\u00f0mundur J. Gu\u00f0mundsson (ritstj\u00f3rar), Grikkland \u00e1r og s\u00ed\u00f0<\/em> (bls. 135\u201358). Hi\u00f0 \u00edslenzka b\u00f3kmenntaf\u00e9lag.<\/p>\n\n\n\n

J\u00f3n \u00deorvar\u00f0arson (2005). Og \u00e9g skal hreyfa j\u00f6r\u00f0ina<\/em>.<\/em> ST\u00c6 ehf.<\/p>\n\n\n\n

Krist\u00edn Bjarnad\u00f3ttir. (2020, 25. n\u00f3vember). Hva\u00f0a a\u00f0fer\u00f0um er beitt til a\u00f0 finna aukastafi p\u00ed? <\/em>V\u00edsindavefurinn. http:\/\/visindavefur.is\/svar.php?id=71214<\/a>.<\/p>\n\n\n\n

\u00d3lafur Dan\u00edelsson (1951). Kennslub\u00f3k \u00ed algebru<\/em>. \u00cdsafoldarprentsmi\u00f0ja.<\/p>\n\n\n\n

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Krist\u00edn Bjarnad\u00f3ttir pr\u00f3fessor emerita vi\u00f0 H\u00e1sk\u00f3la \u00cdslands<\/strong><\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Krist\u00edn Bjarnad\u00f3ttir. Ark\u00edmedes hefur veri\u00f0 talinn mesti st\u00e6r\u00f0fr\u00e6\u00f0ingur fornaldar, e\u00f0a jafnvel allra t\u00edma. Hann f\u00e6ddist \u00ed borginni S\u00fdrak\u00fasu \u00e1 Sikiley \u00e1ri\u00f0 287 f. Kr. Hann mun hafa stunda\u00f0 n\u00e1m \u00ed Alexandr\u00edu \u00ed Egyptalandi, en sn\u00fai\u00f0 aftur til S\u00fdrak\u00fasu \u00fear sem hann l\u00e9st \u00e1ri\u00f0 212 f. Kr. \u00c1 t\u00edmum Ark\u00edmedesar voru h\u00e1\u00f0ar styrjaldir milli R\u00f3mverja, sem…<\/p>\n","protected":false},"author":1,"featured_media":1803,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1725","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1725","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1725"}],"version-history":[{"count":56,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1725\/revisions"}],"predecessor-version":[{"id":1824,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/1725\/revisions\/1824"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/1803"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1725"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1725"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}