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√Ÿâ‰«â„™à«à“ : ¡À—»®√√¬åÕ—μ√“ à«π∑Õß

æÿ¡√’ Õ√√∂√—∞‡ ∂’¬√1

1Õ“®“√¬å ¿“§«‘™“ª√—™≠“·≈–»“ π“ §≥–¡πÿ…¬»“ μ√å·≈– —ß§¡»“ μ√å ¡À“«‘∑¬“≈—¬ ß¢≈“π§√‘π∑√å

 «— ¥’§à– ©∫—∫π’È®–°≈à“«∂÷ß§≥‘μ»“ μ√å´÷Ë߇ªìπ«‘™“∑’Ë¡’

°“√‡√’¬π°“√ Õπ¡“μ—Èß·μàÕπÿ∫“≈ ®π°√–∑—Ëß®π°√–∑—Ëß∂÷ß√–¥—∫

Õÿ¥¡»÷°…“ À≈“¬§πÕ“®√Ÿâ ÷°ª«¥À—«¢÷Èπ¡“∑—π∑’‡¡◊ËÕ‡Õà¬∂÷ß

®”π«πμ—«‡≈¢ ·μà∑’Ë®–°≈à“«μàÕ‰ªπ’ȇªìπ§≥‘μ»“ μ√å∑’˪√“°Ø Õ¬Ÿà√Õ∫Ê μ—«¢Õ߇√“ ‡√‘Ë¡®“°¡πÿ…¬å√Ÿâ®—°°“√π—∫μ—«‡≈¢¡“μ—Èß·μà 37,000 ªï°àÕπ ®“°À≈—°∞“π∑’Ëæ∫„π·Õø√‘°“π—Ëπ§◊Õ√Õ¬∫“°

∫π‰¡â  —ππ‘…∞“π«à“‡ªìπ°“√π—∫«ß‚§®√¥«ß®—π∑√å¢Õß§π„πÕ¥’μ

πÕ°®“°π’È„π‚√¡—π·≈–Õ’¬‘ªμå°Á¡’°“√π—∫®”π«πμ—«‡≈¢

‚¥¬„™â ‘Ëßμà“ßÊ ∑’ËÕ¬Ÿà·«¥≈âÕ¡¡“‡ªìπ —≠≈—°…≥å·∑πμ—«‡≈¢

¥—ßπ’È

 à«πμ—«‡≈¢∑’Ë„™â°—πÕ¬Ÿà„πªí®®ÿ∫—πÀ√◊Õ∑’ˇ√’¬°«à“ ‡≈¢Õ“√∫‘§

‰¥âæ—≤π“¡“®“°°“√π—∫μ—«‡≈¢¢Õß™“«Õ“ª√—∫

al-Banna al-Marrakushiûs form of the numerals

‡∑à“°—∫ 1.6180339887... À√◊Õ (1+ 5)/2 Õ—μ√“ à«π∑Õßπ’È

∂Ÿ°§âπæ∫„πÀπ—ß ◊Õ çElementé ‡≈à¡∑’Ë 6 ¢Õ߬Ÿ§≈‘μ (Euclid)

·μà‡¢“‡√’¬°Õ—μ√“ à«ππ’È«à“ Õ—μ√“ à«π§à“ ÿ¥¢’¥°—∫§à“¡—™¨‘¡

(Exyreme and Meanratio) ‚¥¬· ¥ß«‘∏’°“√·∫à߇ âπμ√ß ÕÕ°‡ªìπÕ—μ√“ à«π¥—ßπ’È (Golden Ratio, 2006)

Õ—μ√“ à«π∑Õß∂Ÿ°π”¡“„™âπ“ππ—∫æ—πªï ·¡â·μà≈’‚Õπ“√å‚¥

¥“«‘π™’ (Leonard da vinci) °Áπ”Õ—μ√“ à«ππ’È¡“„™â„πß“π

»‘≈ª–¢Õ߇¢“ ·≈–‡√’¬°Õ—μ√“ à«ππ’È«à“  —¥ à«π‡∑懮ⓠ(divine proportion) (fibonacci, 2006) „πª≈“¬»μ«√√…∑’Ë 19 ¡’

°“√»÷°…“Õ¬à“ß®√‘ß®—ß®π∂÷ß™à«ßμâπ»μ«√√…∑’Ë 20 ¡’π—°§≥‘μ»“ μ√å

™◊ËÕ Mark Barr ‰¥âπ” —≠≈—°…≥å (Phi,ø“¬) ¡“„™â·∑π Õ—μ√“ à«π∑Õ߇æ◊ËÕ‡ªìπ‡°’¬√μ‘„Àâ·°à Phidias (490-430 BC) ª√–μ‘¡“°√™“«°√’°∑’Ë„™âÕ—μ√“ à«π∑Õß„πß“πª√–μ‘¡“°√√¡

§à“ À√◊ÕÕ—μ√“ à«π∑Õ߇°‘¥®“°°“√»÷°…“ ‘Ëßμà“ßÊ

∑’Ë¡’Õ¬Ÿà„π∏√√¡™“쑇™àπ Õ—μ√“ à«π§«“¡¬“«¢Õß·¢π·≈–ΩÉ“¡◊Õ æ∫«à“¡’Õ—μ√“ à«π∑Õߥ—ßπ’È §«“¡¬“«®“°¢âÕ»Õ°∂÷ߢâÕ¡◊Õ μàÕ§«“¡¬“«®¢âÕ»Õ°∂÷ߪ≈“¬π‘È« ‡∑à“°—∫§à“ Dr.Eddy Levin §âπæ∫«à“Õ—μ√“ à«π§«“¡°«â“ß·≈–§«“¡¬“«¢Õßøíπ  Õß´’ËÀπâ“¢Õß¡πÿ…¬å°Á¡’§à“‡ªìπÕ—μ√“ à«π∑Õ߇™àπ°—π ·¡â·μà º≈ß“π √â“ߢÕß¡πÿ…¬å°Á¡—°®–Õ‘ß°—∫Õ—μ√“ à«π∑Õß ¥—߇™àπ

®“°°“√π—∫®”π«π‡μÁ¡¢Õßμ—«‡≈¢ 0-9 ¡πÿ…¬å‰¥âæ—≤π“

¡“ Ÿà«‘∏’°“√§”π«≥∑“ß§≥‘μ»“ μ√å∑’ËÀ≈“°À≈“¬·≈– ≈—∫

´—∫´âÕπ¬‘Ëߢ÷Èπ¥—߇™àπ ‡√◊ËÕßÕ—μ√“ à«π∑Õß (Golden Ratio)

´÷Ë߇ªìπ®”π«πÕμ√√°¬–Àπ÷Ëß (®”π«πÕμ√√°¬–À¡“¬∂÷ß ‡≈¢

®”π«π∑’Ë· ¥ß‡ªìπ√Ÿª∑»π‘¬¡‰¡à√Ÿâ®∫ª√–‡¿∑‰¡à´È”‰¥â) ∑’Ë¡’§à“

21

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∫√√≥“πÿ°√¡

 ÿ∑—»πå ¬° â“π. 1547. ç‡≈¢æ‘‡»….é «“√ “√°“√»÷°…“«‘∑¬“»“ μ√å §≥‘μ»“ μ√å ·≈–‡∑§‚π‚≈¬’. 128 (¡°√“§¡-°ÿ¡¿“æ—π∏å), 51-54.

Victor J. Katz. 1998. A History of Mathematics : and introduction. The United States of America : Addison- Wesley Educational Pulishrs,Inc.

Base Valued Number 2006. (ÕÕπ‰≈πå).  ◊∫§âπ®“° :

http://www.psinvention.com/zoetic/basenumb.htm [24 ‡¡…“¬π 2549]

Fibonacci. 2006. (ÕÕπ‰≈πå).  ◊∫§âπ®“° : http://nature-life.boxchart.com/fibonacci2.htm [24 ‡¡…“¬π 2549]

Golden Retio. 2006. (ÕÕπ‰≈πå).  ◊∫§âπ®“° : http://encyclopedia.learn.in.th/content.php?encid=49 [24 ‡¡…“¬π 2549]

ß“π ∂“ªíμ¬°√√¡«‘À“√æ“√å∏‘πÕπ

¿“æ‚¡π“≈‘´“ ¢Õß≈’‚Õπ“√å‚¥ ¥“«‘π™’

ªí®®ÿ∫—π„π™’«‘μª√–®”«—π¢Õ߇√“°Á¬—ߧ߇°’ˬ«¢âÕß°—∫

Õ—μ√“ à«π∑ÕßÕ¬Ÿà‡ ¡Õ ‡™àπ Õ—μ√“ à«π¢Õß§«“¡°«â“ß·≈–

§«“¡¬“«¢Õß∫—μ√‡§√¥‘μ, ¢π“¥¢Õß∫—μ√ª√–™“™π, ¢π“¥

¢Õß°√–¥“… A4, ¢π“¥¢ÕßÀπâ“®Õ‚∑√∑—»πå ‡ªìπμâπ Õ—μ√“ à«π

∑Õß À√◊Õ§à“ ‡ªìπ —¥ à«π∑’ˇÀ¡“– ¡∑’Ë ÿ¥À√◊ÕÕ“®∫Õ°‰¥â«à“‡ªìπ  —¥ à«π∑’Ë «¬ß“¡∑’Ë ÿ¥ ©–π—Èπ„π°“√ √â“ß ‘Ëßμà“ßÊ ¢Õß¡πÿ…¬å

®”‡ªìπμâÕßπ”Õ—μ√“ à«ππ’È¡“§‘¥§”π«≥„π°“√ √â“ßÕ¬Ÿà‡ ¡Õ À“°‰¡à‰¥âπ”Õ—μ√“ à«ππ’È¡“„™âÕ“®∑”„Àâº≈ß“π∑’ËÕÕ°¡“¢“¥

§«“¡ ¡ à«π À√◊Õ‰¡à «¬ß“¡®–‡ÀÁπ‰¥â«à“Õ—μ√“ à«π∑ÕßÀ√◊Õ

§à“ ‡ªìπ ‘Ëß∑’ˇ°’ˬ«¢âÕß°—∫¡πÿ…¬å‡√“¡“μ—Èß·μàÕ¥’μ®«∫®π∂÷ßªí®®ÿ∫—π

´÷Ë߉¡àÕ“®À≈’°À𒉥â

Õ—μ√“ à«π√–À«à“ß§«“¡ Ÿß‡Õ’¬ß¢Õßæ’√–¡‘¥μàÕ§«“¡¬“«§√÷Ëß∞“π¢Õß æ’√–¡‘¥°Á‡ªìπÕ—μ√“ à«π∑Õß

ªï∑’Ë 27 ©∫—∫∑’Ë 3 (°.¬. - ∏.§. 2549)

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