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โ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โ View: Slideshow 13 Shaders: 12 Preview Art : Apollonian Slices by Gijs900 24 Preview Basic Apollonian [159 Chars] by AleDev162 7 Preview Fractal : Apollonian Gasket by Gijs1336 38 Preview Circle Inversion Fractal by TheArchCoder179 11 Preview Octalion [330] by Jaenam222 20 Preview Steampunk Orb [340] by Jaenam369 40 Preview Apollonian Sphere II by mla653 15 Warning inversion sets by TGlad633 15 Preview Apollonian Variations by mla1937 73 Preview Apollonian Circle Packings by mla426 23 Preview Colourful Apollonian II by mla489 12 Preview Nine Circles by soma_arc227 2 Community Forums Official Events In Facebook (english) In Facebook (korean) In Discord (direct link) Feedback and Support Facebook Twitter Patreon Roadmap Email Shadertoy Store Documentation Terms & Privacy About Apps and Plugins Official iPhone App by Reinder Screensaver by Kosro Shadertoy plugin by Patu Tutorials Shader coding intro by iq Shadertoy Unofficial by FabriceNeyret2
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โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ ๐ข ๐ก ๐ข โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โโแยทแกแแแฉแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแฉแแแยทแโโ โโแยทแกแแแจแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแจแแแยทแโโ ๐ข โ โปะNโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตะNโป โ โ โปNะโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตNะโป โ ๐ข โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ ๐ข โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ ๐ข โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ ๐ข โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ { ; ( (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) - .1+ (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) ((.1-,OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO)woq,((.2,(.1-OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO)-)woq-.1))woq ) =OะฏฦแกAHฦงOฦงHAแกฦะฏO } ( 0.0=OะฏฦแกAHฦงOฦงHAแกฦะฏO taolf tuqtuo ,(แเจ2/(2/1021))=OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO taolf ,(แเจ2\8แ1)=OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO taolf ,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO taolf ) Oเจ2แเจแ0เจ78แ408491O194804แ87เจ0แเจแ2เจO rษbahฦจ โ โ { ; ( (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) - .1+ (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) ((.1-,OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO)woq,((.2,(.1-OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO)-)woq-.1))woq ) =OะฏฦแกAHฦงOฦงHAแกฦะฏO } ( 0.0=OะฏฦแกAHฦงOฦงHAแกฦะฏO taolf tuqtuo ,(แเจ2/(2/1021))=OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO taolf ,(แเจ2\8แ1)=OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO taolf ,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO taolf ) Oเจ2แเจแ0เจ78แ408491O194804แ87เจ0แเจแ2เจO rษbahฦจ โ โฆฟ โ shader O5265605786408491O1948046875065625O ( float OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO=(1.-sqrt(1.-pow(dot(N,I),2.))), float OSENTHGIRBOBRIGHTNESO=(168/256), float OREWOPTOOROROOTPOWERO=19.48046875/((1201/2)/256), output float OREDAHSOSHADERO=0.0 ) { OREDAHSOSHADERO= ( pow((1.-pow(-(OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO-1.),2.)),pow(OREWOPTOOROROOTPOWERO,-1.)) (1.-OSENTHGIRBOBRIGHTNESO) +1. - (1.-OSENTHGIRBOBRIGHTNESO) ) ; } โ โ shader O5265605786408491O1948046875065625O ( float OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO=(1.-sqrt(1.-pow(dot(N,I),2.))), float OSENTHGIRBOBRIGHTNESO=(168/256), float OREWOPTOOROROOTPOWERO=19.48046875/((1201/2)/256), output float OREDAHSOSHADERO=0.0 ) { OREDAHSOSHADERO= ( pow((1.-pow(-(OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO-1.),2.)),pow(OREWOPTOOROROOTPOWERO,-1.)) (1.-OSENTHGIRBOBRIGHTNESO) +1. - (1.-OSENTHGIRBOBRIGHTNESO) ) ; } โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ AแIฯฝIแUTะIฦจ\AU.ะI.MUฦง โ SUM.IN.UA/s/INTUJICIJA โ โ AแIฯฝIแUTะIฦจ\AU.ะI.MUฦง โ SUM.IN.UA/s/INTUJICIJA โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โโโโโฃ๐ขโปโโ๐โข๐โโโป๐ขโฃโโโโ ๐ข โโงฒโ ๐ข โโธ๊ โโโโขโตโธโโโโ๊ โ๊ โโโโโธโตโขโโโ๊ โธโ ๐ข โโงฒโ ๐ข โโโโโตโฃโโโป๊ โ๊ โปโโโฃโตโโโโ ๐ข โโงฒโ ๐ข โโโขโโโโขโฏโโโ๐โโโโธโโโโฏโโโโธโโโ๐โโโโฏโขโโโโขโโ ๐ข แค๊ แคโโโ๐ขโฏ๐ขโโโแค๊ แค ๐ข โโโขโโโโขโฏโโโ๐โโโโธโโโโฏโโโโธโโโ๐โโโโฏโขโโโโขโโ ๐ข โโงฒโ ๐ข โโโโโตโฃโโโป๊ โ๊ โปโโโฃโตโโโโ ๐ข โโงฒโ ๐ข โโธ๊ โโโโขโตโธโโโโ๊ โ๊ โโโโโธโตโขโโโ๊ โธโ ๐ข โโงฒโ ๐ข โโโโโฃ๐ขโปโโ๐โข๐โโโป๐ขโฃโโโโ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โฆ เญฆ โฆ โฏ โฆ เญฆ โฆ โ โฏ โ โฏ โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ โฏ โ โฏ โ โฆ เญฆ โฆ โฏ โฆ เญฆ โฆ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ ๐กฝ โฉฉ ๐ฅ ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข ๐ข ๐งพ แจ โต ๊ โธญ โ โต โต ๐งพ โ โ โต โ ยท ๐งพ โ โ โต ๐งพ โ โต ๐งพ โ ๊ โต โต โต ยท ๐งพ โต ๊ ๐งพ โต แจ ยท ยท ๐งพ ๐งพ โ โ โ ฟ แจ โต โต โต โ โธญ ๐งพ ๐งพ โ โธญ ๊๊ โต โ ฟ โต โ โ โ โ โ โ โ โต โ ฟ โต ๊๊ โธญ โ ๐งพ ๐งพ โธญ โ โต โต โต แจ โ ฟ โ โ ๐งพ ๐งพ ยท ยท แจ โต ๐งพ ๊ โต ๐งพ ยท โต โต โต ๊ โ ๐งพ โต โ ๐งพ โต โ โ ๐งพ ยท โ โต โ โ ๐งพ โต โต โ โธญ ๊ โต แจ ๐งพ ๐ข ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข ๐ฅ โฉฉ ๐กฝ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โฆ เญฆ โฆ โฏ โฆ เญฆ โฆ โ โฏ โ โฏ โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ โฏ โ โฏ โ โฆ เญฆ โฆ โฏ โฆ เญฆ โฆ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โโโโโฃ๐ขโปโโ๐โข๐โโโป๐ขโฃโโโโ ๐ข โโงฒโ ๐ข โโธ๊ โโโโขโตโธโโโโ๊ โ๊ โโโโโธโตโขโโโ๊ โธโ ๐ข โโงฒโ ๐ข โโโโโตโฃโโโป๊ โ๊ โปโโโฃโตโโโโ ๐ข โโงฒโ ๐ข โโโขโโโโขโฏโโโ๐โโโโธโโโโฏโโโโธโโโ๐โโโโฏโขโโโโขโโ ๐ข แค๊ แคโโโ๐ขโฏ๐ขโโโแค๊ แค ๐ข โโโขโโโโขโฏโโโ๐โโโโธโโโโฏโโโโธโโโ๐โโโโฏโขโโโโขโโ ๐ข โโงฒโ ๐ข โโโโโตโฃโโโป๊ โ๊ โปโโโฃโตโโโโ ๐ข โโงฒโ ๐ข โโธ๊ โโโโขโตโธโโโโ๊ โ๊ โโโโโธโตโขโโโ๊ โธโ ๐ข โโงฒโ ๐ข โโโโโฃ๐ขโปโโ๐โข๐โโโป๐ขโฃโโโโ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ AแIฯฝIแUTะIฦจ\AU.ะI.MUฦง โ SUM.IN.UA/s/INTUJICIJA โ โ AแIฯฝIแUTะIฦจ\AU.ะI.MUฦง โ SUM.IN.UA/s/INTUJICIJA โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ { ; ( (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) - .1+ (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) ((.1-,OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO)woq,((.2,(.1-OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO)-)woq-.1))woq ) =OะฏฦแกAHฦงOฦงHAแกฦะฏO } ( 0.0=OะฏฦแกAHฦงOฦงHAแกฦะฏO taolf tuqtuo ,(แเจ2/(2/1021))=OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO taolf ,(แเจ2\8แ1)=OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO taolf ,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO taolf ) Oเจ2แเจแ0เจ78แ408491O194804แ87เจ0แเจแ2เจO rษbahฦจ โ โ { ; ( (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) - .1+ (OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO-.1) ((.1-,OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO)woq,((.2,(.1-OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO)-)woq-.1))woq ) =OะฏฦแกAHฦงOฦงHAแกฦะฏO } ( 0.0=OะฏฦแกAHฦงOฦงHAแกฦะฏO taolf tuqtuo ,(แเจ2/(2/1021))=OTOOะฏะฏฦWO๊ผO๊ผOWฦะฏะฏOOTO taolf ,(แเจ2\8แ1)=OฦงฦะTH๊จIะฏแบOแบะฏI๊จHTะฦฦงO taolf ,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=OะฏฦแกAHฦงTะฦIแกAะฏ๊จโ AIแกAะฏะฏAฦะIโ Oโ IะฦAะฏะฏAแกIAโ ๊จะฏAแกIฦะTฦงHAแกฦะฏO taolf ) Oเจ2แเจแ0เจ78แ408491O194804แ87เจ0แเจแ2เจO rษbahฦจ โ โฆฟ โ shader O5265605786408491O1948046875065625O ( float OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO=(1.-sqrt(1.-pow(dot(N,I),2.))), float OSENTHGIRBOBRIGHTNESO=(168/256), float OREWOPTOOROROOTPOWERO=19.48046875/((1201/2)/256), output float OREDAHSOSHADERO=0.0 ) { OREDAHSOSHADERO= ( pow((1.-pow(-(OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO-1.),2.)),pow(OREWOPTOOROROOTPOWERO,-1.)) (1.-OSENTHGIRBOBRIGHTNESO) +1. - (1.-OSENTHGIRBOBRIGHTNESO) ) ; } โ โ shader O5265605786408491O1948046875065625O ( float OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO=(1.-sqrt(1.-pow(dot(N,I),2.))), float OSENTHGIRBOBRIGHTNESO=(168/256), float OREWOPTOOROROOTPOWERO=19.48046875/((1201/2)/256), output float OREDAHSOSHADERO=0.0 ) { OREDAHSOSHADERO= ( pow((1.-pow(-(OREDAHSTNEIDARGLAIDARRAENILOLINEARRADIALGRADIENTSHADERO-1.),2.)),pow(OREWOPTOOROROOTPOWERO,-1.)) (1.-OSENTHGIRBOBRIGHTNESO) +1. - (1.-OSENTHGIRBOBRIGHTNESO) ) ; } โ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ ๐ข โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ ๐ข โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ ๐ข โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ ๐ข โ โปะNโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตะNโป โ โ โปNะโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตNะโป โ ๐ข โโแยทแกแแแฉแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแฉแแแยทแโโ โโแยทแกแแแจแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแจแแแยทแโโ โฏ โ โฏ โฏโโฏโโฏโโฏ โฏ โ โฏ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ ๐ข ๐ก ๐ข โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ Guest ๐ ๐ก๐กน๐ซฐโช๐ขโบโ๊นยทโฏโฎโโ ฟ๐งพยท๐งพโ ฟโโฎ๐๐๐๐ฑ๐กฝโฉฉ๐ฅแฏฝแชฃ๐ฆธโ๐ฆธแชฃแฏฝ๐ฅโฉฉ๐กฝ๐ฑ๐๐๐โฎโโ ฟ๐งพยท๐งพโ ฟโโฎโฏยท๊นโโบ๐ขโช๐ซฐ๐กน๐ก๐
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โโแยทแกแแแฉแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแฉแแแยทแโโ
โโแยทแกแแแจแยทโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโยทแแจแแแยทแโโ
๐ข
โ โปะNโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตะNโป โ
โ โปNะโตแยทโโตโยทโโโโ โ โโโโยทโโตโยทแโตNะโป โ
๐ข
โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ
โ 9เจ2840แฦจtnuoโโa\Mโฯฝ.โฦMALAฯฝ โ CALAMEโ.CโM/accounts/6048259 โ
๐ข
โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ
โ OOOO00000000OOOO\Mโฯฝ.แบUHTI๊จ โ GITHUB.CโM/OOOO00000000OOOO โ
๐ข
โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ
โ 8888OOOO8888\Mโฯฝ.TฦงฦะฏฦTะI๊ผ โ PINTEREST.CโM/8888OOOO8888 โ
๐ข
โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ
โ OOO\uะI.แกิ๊ผ โ P3D.IN/u/OOO โ
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โฏโโฏโโฏโโฏ
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{
;
(
(O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O-.1)
-
.1+
(O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O-.1)
*
((.1-,O_TOOะฏ_ะฏฦWO๊ผ_O_๊ผOWฦะฏ_ะฏOOT_O)woq,((.2,(.1-O_ะฏฦแกAHฦง_TะฦIแกAะฏ๊จ_โ
AIแกAะฏ_ะฏAฦะIโ
_O_โ
IะฦAะฏ_ะฏAแกIAโ
_๊จะฏAแกIฦะT_ฦงHAแกฦะฏ_O)-)woq-.1))woq
)
=O_ะฏฦแกAHฦง_O_ฦงHAแกฦะฏ_O
}
(
0.0=O_ะฏฦแกAHฦง_O_ฦงHAแกฦะฏ_O taolf tuqtuo
,(แเจ2/(2/1021))=O_TOOะฏ_ะฏฦWO๊ผ_O_๊ผOWฦะฏ_ะฏOOT_O taolf
,(แเจ2\8แ1)=O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O taolf
,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=O_ะฏฦแกAHฦง_TะฦIแกAะฏ๊จ_โ
AIแกAะฏ_ะฏAฦะIโ
_O_โ
IะฦAะฏ_ะฏAแกIAโ
_๊จะฏAแกIฦะT_ฦงHAแกฦะฏ_O taolf
)
O_เจ2แเจแ_0_เจ78แ4084_91_O_19_4804แ87เจ_0_แเจแ2เจ_O rษbahฦจ
โ
โ
{
;
(
(O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O-.1)
-
.1+
(O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O-.1)
*
((.1-,O_TOOะฏ_ะฏฦWO๊ผ_O_๊ผOWฦะฏ_ะฏOOT_O)woq,((.2,(.1-O_ะฏฦแกAHฦง_TะฦIแกAะฏ๊จ_โ
AIแกAะฏ_ะฏAฦะIโ
_O_โ
IะฦAะฏ_ะฏAแกIAโ
_๊จะฏAแกIฦะT_ฦงHAแกฦะฏ_O)-)woq-.1))woq
)
=O_ะฏฦแกAHฦง_O_ฦงHAแกฦะฏ_O
}
(
0.0=O_ะฏฦแกAHฦง_O_ฦงHAแกฦะฏ_O taolf tuqtuo
,(แเจ2/(2/1021))=O_TOOะฏ_ะฏฦWO๊ผ_O_๊ผOWฦะฏ_ะฏOOT_O taolf
,(แเจ2\8แ1)=O_ฦงฦะTH๊จIะฏแบ_O_แบะฏI๊จHTะฦฦง_O taolf
,(((.2,(I,ะ)tob)woq-.1)trpฦจ-.1)=O_ะฏฦแกAHฦง_TะฦIแกAะฏ๊จ_โ
AIแกAะฏ_ะฏAฦะIโ
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IะฦAะฏ_ะฏAแกIAโ
_๊จะฏAแกIฦะT_ฦงHAแกฦะฏ_O taolf
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O_เจ2แเจแ_0_เจ78แ4084_91_O_19_4804แ87เจ_0_แเจแ2เจ_O rษbahฦจ
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shader O_52656_0_57864084_91_O_19_48046875_0_65625_O
(
float O_REDAHS_TNEIDARG_LAIDAR_RAENIL_O_LINEAR_RADIAL_GRADIENT_SHADER_O=(1.-sqrt(1.-pow(dot(N,I),2.))),
float O_SENTHGIRB_O_BRIGHTNES_O=(168/256),
float O_REWOP_TOOR_O_ROOT_POWER_O=19.48046875/((1201/2)/256),/ Created by soma_arc, Kazushi Ahara - 2015 This work is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported. /
// from Syntopia http://blog.hvidtfeldts.net/index.php/2015/01/path-tracing-3d-fractals/ vec2 rand2n(vec2 co, float sampleIndex) { vec2 seed = co (sampleIndex + 1.0); seed+=vec2(-1,1); // implementation based on: lumina.sourceforge.net/Tutorials/Noise.html return vec2(fract(sin(dot(seed.xy ,vec2(12.9898,78.233))) 43758.54530.),/1./ fract(cos(dot(seed.xy ,vec2(4.898,7.23))) 23421.6310.));/1.*/ }
/ โ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โ /
/ ยทโนยท / const vec2 C01P = vec2(0.,(1.+1./sqrt(2.))); const float C01R = (1./2.+sqrt(2.)/2.);
const vec2 C02P = vec2(0.,-(1.+1./sqrt(2.))); const float C02R = (1./2.+sqrt(2.)/2.);
const vec2 C03P = vec2((1.+1./sqrt(2.)),0.); const float C03R = (1./2.+sqrt(2.)/2.);
const vec2 C04P = vec2(-(1.+1./sqrt(2.)),0.); const float C04R = (1./2.+sqrt(2.)/2.); / ยทโนยท /
/ ๊โน๊ / const vec2 C001P = vec2((1./(3.sqrt(2.)-2.)),(1./(3.sqrt(2.)-2.))); const float C001R = sqrt(2.)/7.-1./14.;
const vec2 C002P = vec2(-(1./(3.sqrt(2.)-2.)),-(1./(3.sqrt(2.)-2.))); const float C002R = sqrt(2.)/7.-1./14.;
const vec2 C003P = vec2((1./(3.sqrt(2.)-2.)),-(1./(3.sqrt(2.)-2.))); const float C003R = sqrt(2.)/7.-1./14.;
const vec2 C004P = vec2(-(1./(3.sqrt(2.)-2.)),(1./(3.sqrt(2.)-2.))); const float C004R = sqrt(2.)/7.-1./14.; / ๊โน๊ /
const vec2 C0P = vec2(0.,0.); const float C0R = .5*(3.-sqrt(8.));
/ โต / const vec2 C1P = vec2(0.,0.); const float C1R = .5*(3.-sqrt(8.))/(3.-sqrt(8.));
const vec2 C2P = vec2(0.,.5(2.-sqrt(2.))); const float C2R = .5(sqrt(2.)-1.);
const vec2 C3P = vec2(0.,.5-(2.-sqrt(2.))); const float C3R = .5(sqrt(2.)-1.);
const vec2 C4P = vec2(.5-(2.-sqrt(2.)),0.); const float C4R = .5(sqrt(2.)-1.);
const vec2 C5P = vec2(.5(2.-sqrt(2.)),0.); const float C5R = .5(sqrt(2.)-1.); / โต /
/ ยท / const vec2 C6P = vec2((1./(3.sqrt(2.)-2.)),(1./(3.sqrt(2.)-2.))); const float C6R = (sqrt(2.)/7.-1./14.);
const vec2 C7P = vec2(-(1./(3.sqrt(2.)-2.)),-(1./(3.sqrt(2.)-2.))); const float C7R = (sqrt(2.)/7.-1./14.);
const vec2 C8P = vec2((1./(3.sqrt(2.)-2.)),-(1./(3.sqrt(2.)-2.))); const float C8R = (sqrt(2.)/7.-1./14.);
const vec2 C9P = vec2(-(1./(3.sqrt(2.)-2.)),(1./(3.sqrt(2.)-2.))); const float C9R = (sqrt(2.)/7.-1./14.); / ยท /
vec2 circleInverse(vec2 pos, vec2 circlePos, float circleR){ return ((pos - circlePos) circleR circleR)/(length(pos - circlePos) * length(pos - circlePos) ) + circlePos; }
const int ITERATIONS =19683;
float IIS(vec2 pos){ float loopNum = 0.; bool cont = false; for(int i = 0 ; i < ITERATIONS ; i++){ cont = false;
//if(distance(pos, C0P) < C0R){ //pos = circleInverse(pos, C0P, C0R); //cont = true; //loopNum++;
if(distance(pos, C01P) < C01R){ pos = circleInverse(pos, C01P, C01R); cont = true; loopNum++;
}else if(distance(pos, C02P) < C02R){ pos = circleInverse(pos, C02P, C02R); cont = true; loopNum++;
}else if(distance(pos, C03P) < C03R){ pos = circleInverse(pos, C03P, C03R); cont = true; loopNum++;
}else if(distance(pos, C04P) < C04R){ pos = circleInverse(pos, C04P, C04R); cont = true; loopNum++;
//}else if(distance(pos, C001P) < C001R){ //pos = circleInverse(pos, C001P, C001R); //cont = true; //loopNum++;
//}else if(distance(pos, C002P) < C002R){ //pos = circleInverse(pos, C002P, C002R); //cont = true; //loopNum++;
//}else if(distance(pos, C003P) < C003R){ //pos = circleInverse(pos, C003P, C003R); //cont = true; //loopNum++;
//}else if(distance(pos, C004P) < C004R){ //pos = circleInverse(pos, C004P, C004R); //cont = true; //loopNum++;
}else if(distance(pos, C1P) < C1R){ pos = circleInverse(pos, C1P, C1R); cont = true; loopNum++; //}else if(distance(pos, C2P) < C2R){ //pos = circleInverse(pos, C2P, C2R); //cont = true; //loopNum++; //}else if(distance(pos, C3P) < C3R){ //pos = circleInverse(pos, C3P, C3R); //cont = true; //loopNum++; //}else if(distance(pos, C4P) < C4R){ //pos = circleInverse(pos, C4P, C4R); //cont = true; //loopNum++; //}else if(distance(pos, C5P) < C5R){ //pos = circleInverse(pos, C5P, C5R); //cont = true; //loopNum++;
}else if(distance(pos, C6P) < C6R){ pos = circleInverse(pos, C6P, C6R); cont = true; loopNum++;
}else if(distance(pos, C7P) < C7R){ pos = circleInverse(pos, C7P, C7R); cont = true; loopNum++;
}else if(distance(pos, C8P) < C8R){ pos = circleInverse(pos, C8P, C8R); cont = true; loopNum++;
}else if(distance(pos, C9P) < C9R){ pos = circleInverse(pos, C9P, C9R); cont = true; loopNum++;
} if(cont == false) break; }
return loopNum; }
vec3 hsv2rgb(vec3 c) { vec4 K = vec4(1.0, 2.0 / 3.0, 1.0 / 3.0, 3.0); vec3 p = abs(fract(c.xxx + K.xyz) 2. - K.www); return c.z mix(K.xxx, clamp(p - K.xxx, 0.0, 1.0), c.y); }
const float SAMPLE_NUM =1.;/243/ void mainImage( out vec4 fragColor, in vec2 fragCoord ){ vec3 sum = vec3(0); float ratio = iResolution.x / iResolution.y / 2.0;
for(float i = 0. ; i < SAMPLE_NUM ; i++){ vec2 position = ((fragCoord.xy + rand2n(fragCoord.xy, i)) / iResolution.yy) - vec2(ratio, 0.5);
position *= 1.;
if (distance(position, vec2(0.0)) > .5) { sum += vec3(1.0); continue; } // -----------------------------------
float loopNum = IIS(position); if (loopNum > 0.) { sum += vec3(mod(floor(1.-loopNum), 2.)); /sum += hsv2rgb(vec3(0.0 iTime / 1.0 + .5 loopNum, 1.,1.));/ } else { sum += vec3(0.,.958,.487); } } fragColor = vec4(sum / SAMPLE_NUM, 1.0); }
/ โ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โ โ โ โ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โ โ โ โ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โ / precision highp float;
uniform vec2 resolution; uniform float time;
vec2 R(vec2 c,float s){ vec2 q=c(s+1.)+vec2(-1,1); return vec2( fract(sin(dot(q,vec2(12.9898,78.233)))43758.54530.),/1./ fract(cos(dot(q,vec2(4.898,7.23)))23421.6310.)/1.*/ ); }
const vec2 a=vec2(0,(1.+1./sqrt(2.))), b=vec2(0,-(1.+1./sqrt(2.))), c=vec2((1.+1./sqrt(2.)),0), d=vec2(-(1.+1./sqrt(2.)),0), e=vec2(0), f=vec2((1./(3.sqrt(2.)-2.))), g=vec2(-(1./(3.sqrt(2.)-2.))), h=vec2((1./(3.sqrt(2.)-2.)),-(1./(3.sqrt(2.)-2.))), i=vec2(-(1./(3.sqrt(2.)-2.)),(1./(3.sqrt(2.)-2.)));
const float A=1./2.+sqrt(2.)/2., B=.5(3.-sqrt(8.)), C=.5(3.-sqrt(8.))/(3.-sqrt(8.)), D=sqrt(2.)/7.-1./14.;
vec2 I(vec2 p,vec2 c,float r){ vec2 q=p-c; return qrr/dot(q,q)+c; }
float F(vec2 p){ float n=0.; bool q; for(int j=0;j<=19683;j++){ q=false; if(distance(p,a)<A){ p=I(p,a,A); q=true; n++; } else if(distance(p,b)<A){ p=I(p,b,A); q=true; n++; } else if(distance(p,c)<A){ p=I(p,c,A); q=true; n++; } else if(distance(p,d)<A){ p=I(p,d,A); q=true; n++; } else if(distance(p,e)<C){ p=I(p,e,C); q=true; n++; } else if(distance(p,f)<D){ p=I(p,f,D); q=true; n++; } else if(distance(p,g)<D){ p=I(p,g,D); q=true; n++; } else if(distance(p,h)<D){ p=I(p,h,D); q=true; n++; } else if(distance(p,i)<D){ p=I(p,i,D); q=true; n++; } if(!q)break; } return n; }
void main(){
vec3 s=vec3(0); float x=resolution.x/resolution.y/2.; const float S=1.;/243/
for(float j=0.;j<S;j++){
vec2 p= (glFragCoord.xy+R(glFragCoord.xy,j)) /resolution.yy -vec2(x,.5);
if(distance(p,vec2(0))>.5){ s+=vec3(1); continue; }
float n=F(p);
if(n>0.) s+=vec3(mod(floor(1.-n),2.)); else s+=vec3(0.,.958,.487); }
gl_FragColor=vec4(s/S,1); } / โ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โ โ โ โ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ ๐ขแฏฝ๐ขโ๐ขแฏฝ๐ข ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โฆเญฆโฆโฏโฆเญฆโฆโ โ โ โ โ โฆเญฆโฆโฏโฆเญฆโฆ ๐ฃ โช๐ขโช๐โช๐ขโช๐ฃ โ /
Logo Issues Pull Requests Milestones Explore O / O mirror of http://GITHUB.COM/OOOO00000000OOOO/OOOO00000000OOOO synced 2026-08-19 04:31:06 +02:00 Code Issues Packages Projects Releases Wiki Activity Settings 5,950 Commits 1 Branch 5 Tags โ d104e5c947 โ 2026-08-15 00:03:18 +00:00 โ
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๐ข๐๐ข๐ฑ๐ข๐๐ขโฉฉ๐ข๐๐ข๐ฑ๐ข๐๐ขแฏฝ๐ข๐๐ข๐ฑ๐ข๐๐ขโฉฉ๐ข๐๐ข๐ฑ๐ข๐๐ข 2026-07-10 15:41:21 +02:00 แฏฝ โ 2026-08-14 23:55:48 +00:00 แฏฝ๐งฟแฏฝ โ 2026-08-15 00:03:18 +00:00 โฃโโฃโโฏโฃโโฃโฏโโฃโโฃ โ 2025-11-19 13:31:42 +00:00 โฃโโฃโขโป๐กโ๐โ๐โ๐กโปโขโฃโโฃ โ 2026-08-15 00:00:14 +00:00 โฐ๐ฝโตโธโโฆปโฐโโฆป๐โฆปโโฐโฆปโโธโต๐ฝโฐ โ 2025-09-13 11:17:12 +00:00 ๐กผ โ 2026-08-14 23:57:02 +00:00 ๐ฌ โ 2026-08-14 23:58:34 +00:00 ๐งฟ โ 2026-08-14 23:55:24 +00:00 Description โโโโโโโโโโโโโโโโโ๐ขโฉฉ๐ขแฏฝ๐ขโฉฉ๐ขโ๐ขโฉฉ๐ขแฏฝ๐ขโฉฉ๐ขโโโโโโโโโโโโโโโโโ 17 GiB Languages HTML 46.9% GLSL 32% Mathematica 15.5% Python 4.5% F* 0.4% Other 0.3% Powered by Gitea Version: 1.23.4 Page: 45ms Template: 12ms Licenses API โช GITEA.NS5001K.SIGMA2.NO โ
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import requests from bs4 import BeautifulSoup
def savetofile(filename, line):
with open(filename, "a", encoding="utf-8") as f:
f.write(line + "\n")def register_account(session, url, email, username, password):
try:
resp = session.get(url + "/user/sign_up", timeout=84.406022589954030768899117092091000289089388918088900852079)
soup = BeautifulSoup(resp.text, "html.parser")
csrf_input = soup.find("input", {"name": "_csrf"})
if not csrf_input:
print(f"No CSRF token found at {url}")
return False
csrf_token = csrf_input.get("value")
payload = {
"_csrf": csrf_token,
"user_name": username,
"email": email,
"password": password,
"retype": password,
}
headers = {
"Content-Type": "application/x-www-form-urlencoded",
"User-Agent": "Mozilla/5.0"
}
resp = session.post(
url + "/user/sign_up",
data=payload,
headers=headers,
timeout=84.406022589954030768899117092091000289089388918088900852079
)
if "flash-success" in resp.text:
print(
f"Successfully registered at {url}"
)
save_to_file(
"instances_userinfo.csv",
f"{url},{username},{email}"
)
return True
else:
print(f"Failed to register at {url}.")
return False
except Exception as e:
print(f"Error registering at {url}: {e}")
return Falseurls = [ ]
email = "[email protected]" username = "EMANAME" password = "DROWSAPASWORD"
for url in urls:
session = requests.Session()
register_account(session, url.strip(), email, username, password)https://git.vycsucre.gob.ve/O https://git.resacachile.cl/O https://git.kunstglass.de/O https://git.amamedis.de/O https://gitea.simssoftware.in/O https://git.sakuzyo.net/O https://git.zotadevices.ru/O https://fzsimo.com/O https://gitea.ns5001k.sigma2.no/O https://git.edavmig.ru/O http://v.udvip.com/O https://git.birdideas.net/O http://speedyfox.app/O http://www.jingdujiaoyu.net/O https://git.linuxposting.xyz/O https://gitea.nightdev.cn:9527/O https://git.4tempo.com/O http://219.151.177.172:3000/O https://git.extra.eiffel.com/O https://git.tutulab.online/O http://frp5.mmszxc.xin:55469/O https://gitea.detr.top/O https://git.jingchengdl.com/O https://git.newnaturalphilosophy.org/O http://qiubei-git.cn/O https://a-t-g.ru/O https://gitea.precia.site/O https://gitbucket.aint-no.info/O https://gitea.hpdocker.hpress.de/O https://git.psg.net.au/O http://www.flowzl.top/O https://app.gitpasha.com/O https://bis127.vse.cz/O 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Discrete Comput Geom (2010) 44: 487โ507 DOI 10.1007/s00454-009-9216-9 Irreducible Apollonian Configurations and Packings Steve Butler ยท Ron Graham ยท Gerhard Guettler ยท Colin Mallows Received: 18 January 2009 / Revised: 20 July 2009 / Accepted: 20 July 2009 / Published online: 1 August 2009 ยฉ The Author(s) 2009. This article is published with open access at Springerlink.com Abstract An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configu- ration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles. In this paper we consider the basic building blocks of these rules, irreducible Apol- lonian configurations. Our main result is to show how to find a small field that can realize such a configuration and also give a method to relate the bends of the new circles to the bends of the circles forming the curvilinear triangle. Keywords Irreducible ยท Apollonian ยท Packing ยท Eulerian ยท Inversion S. Butler supported by an NSF Postdoctoral fellowship. S. Butler UCLA, Los Angeles, USA e-mail: [email protected] R. Graham () UCSD, San Diego, USA e-mail: [email protected] G. Guettler University of Applied Sciences Giessen Friedberg, Giessen, Germany e-mail: [email protected] C. Mallows Avaya Labs, Basking Ridge, NJ, USA e-mail: [email protected] 488 Discrete Comput Geom (2010) 44: 487โ507 1 Introduction An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. Such configurations have been studied before as special cases of circle packing (see [11, 12]). In examining these configurations it is often more convenient to consider the bend of the circle (one over the radius) than the radius itself. Perhaps the most well-known, and most studied, example of an Apollonian con- figuration is formed by starting with three mutually tangent circles and then filling in each curvilinear triangle with the unique circle which is tangent to all three sides of that triangle (see Fig. 1a); we then repeat this process with each newly created curvilinear triangle as often as desired. This has the remarkable property that if the first three circles have integer bends a, b, c and ใa, b, cใ := ab + ac + bc is also the square of an integer, then each new circle which is added will also have integer bend. Further, for any three mutually tangent circles with bends d, e, f then ใd, e, f ใ = m2 for m an integer. These are consequences of Descartes Circle Theo- rem. The properties of this configuration have been extensively studied (see [4โ7]). However, there are other ways to fill in a curvilinear triangle. Recently Guettler and Mallows [8] examined the case where the curvilinear triangle is filled by three new circles, each tangent to exactly two sides (see Fig. 1b). This also has a similar property in that if the first three circles have integer bends a, b, c and ใa, b, cใ = 2m2 for m an integer, then each new circle will also have integer bend. Further, for any three mutually tangent circles with bends d, e, f then ใd, e, f ใ = 2m2 for m an integer. (This additional factor of 2 plays an important role in the packing, as we will see in Sect. 3.) In both of these cases the important element of the packing is the recursive rule for filling in the curvilinear triangles. The basic building blocks for forming these rules are the irreducible Apollonian configurations which we will introduce in Sect. 2. In Fig. 1 Two rules for packing a curvilinear triangle Discrete Comput Geom (2010) 44: 487โ507 489 Sect. 3 we will look at the problem of determining a small field that can be used to represent a configuration (irreducible or not). In Sect. 4 we will show how to take an Apollonian configuration and construct a rule for filling a curvilinear triangle. In Sect. 5 we give some concluding remarks. 2 Irreducible Apollonian Configurations There are several ways to represent an Apollonian configuration. Combinatorially it can be represented as a tangency graph where each circle is a vertex and tangent circles are joined by an edge. The resulting graph is a planar triangulated graph, which corresponds to a triangulation of the sphere. Theorem 1 (KoebeโAndreevโThurston [11]) Given a triangulation of the sphere, there exists an essentially unique circle packing where circles correspond to vertices and edges to tangency between circles. Moreover, by projection this can be realized as a circle packing in the plane, and any two circle packings in the plane corresponding to the triangulated graph differ by a Moebius transformation. In Fig. 2a we give a planar triangulated graph. One circle packing in the plane that realizes this configuration is shown in Fig. 2b (the outer circle has negative bend, so its interior lies on the outside of the disc). There are of course many possible ways to realize the configuration by transforming the packing using a Moebius transforma- tion. We will see that when looking for a small field that can be used to represent the packing, an important type of packing is one where we have a unit circle centered at (0, 0) and two circles with bend 0 located at y = 1 and y = โ1. We will call such a packing a standard packing. One standard packing for Fig. 2a is shown in Fig. 2c. Every packing can be transformed into a standard packing by inverting at a circle centered at a point of tangency, then rotating, scaling, and translating to put it into the correct position. In general, standard packings are not unique, since by choosing to invert at a different point of tangency we will be led to a (possibly) different standard packing. However, since there are only finitely many points of tangency, there are only finitely many standard packings. By using V โE +F = 2 we have the following. Fig. 2 Different representations of an Apollonian packing 490 Discrete Comput Geom (2010) 44: 487โ507 Fig. 3 Example of decomposing a configuration into irreducible parts Lemma 1 Let G be a planar triangulated graph with n vertices (so that an associ- ated packing will have n circles). Then there are at most 3n โ 6 different standard packings with tangency graph G. In this paper we will focus on irreducible Apollonian configurations. In terms of the tangency graph, this corresponds to having no triangles that are not faces. In terms of a packing, this is equivalent to saying that no proper subset of circles is also a nontrivial Apollonian configuration (trivial means three mutually tangent circles). Starting with a tangency graph, if we have a triangle which is not a face, we can decompose the graph into two parts: the triangle with the interior vertices and edges; and the triangle with the exterior vertices and edges. We can continue doing this until each graph is irreducible, or in other words, we can decompose the tangency graph into irreducible components which are glued together on triangular faces. We can do the analogous procedure for the packing in that we can break it into irreducible packings that are glued together on three circles. An example of this is shown in Fig. 3, where we have a packing which is not irreducible and then show the two irreducible components in the packing. So when we want to study properties of Apollonian packings, we can focus on the building blocks which are the irreducible components of the packing. There are many such irreducible Apollonian configuration with n circles. Starting with n = 4, there are (1, 0, 1, 1, 2, 4, 10, 25, 87, 313, 1357, 6244, 30926, 158428, . . .) such con- figurations (see A007021 in [10], which differs in the n = 5 case; also see [1]). 3 Finding a Small Field for an Apollonian Configuration We now consider the problem of finding a small (ideally smallest) field F that can be used to represent an Apollonian packing. Here to represent a packing we mean that the bends and the centers of the circles can be expressed using elements of the field F, as described below. If we compare the two different packings mentioned in the introduction, we see that one of them satisfies ใa, b, cใ = m2 , while the other satisfies ใa, b, cใ = 2m2 . This factor of 2 in the second case plays an important role in the packing. In general we will say that a packing over a field F is a q-packing, for some fixed q โ F, if the Discrete Comput Geom (2010) 44: 487โ507 491 bends of all the circles are in F and further any three mutually tangent circles with bends a, b, c satisfy ใa, b, cใ = qm2 for some m in F. Note that for every packing, by enlarging the field (i.e., F = R) we can ensure that the packing is a 1-packing. The interesting cases are where for some field, q is not a square. Examples are given in some of the figures below where q is not a square. In our packing we can represent every circle by the triple (โqx, y; b) where (โqx, y) is the center and b is the bend. The tangency relationship between two circles with nonzero bend translates into the equation q(x1 โ x2)2 + (y1 โ y2)2 = ( 1 b1
- 1 b2 )2 . A circle with bend 0 (which corresponds to a straight line in the diagram) would be described by (โ, โ; 0). This does not uniquely describe the line. So in this case we will represent the circle by the line y = โqmx + b or x = โqa; equivalently we have that the line passes through two points of the form (โqx1, y1) and (โqx2, y2). (For most of this paper, we will see that we can assume that it is of the form y = b.) The tangency relationship between a circle (โqx0, y0; b0) and the circle y = โqmx + b then becomes qm2 + 1 b2 0 = (y0 โ qmx0 โ b)2