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OOOOIIIIIIIIOOOO
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0000IIIIIIII0000

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OOOOIIIIIIIIOOOO commited on about 2 hours ago

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7c7039f

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@@ -71,3 +71,9 @@
 𑁍/⯌/𐃏/⸭/ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG filter=lfs diff=lfs merge=lfs -text
 𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/⊚/ꟼI𑪼.✉.💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠.✉.ZIP filter=lfs diff=lfs merge=lfs -text
 𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𖢒/ꟻᗡꟼ...........................⠀Ⓞ⯏Ⓞ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•ⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•Ⓞ⯏Ⓞ𖡼⚪𖡼𖡗𖡼⚪𖡼◦୦◦◯◦୦◦𔗢⩩𔗢᯽𔗢⩩𔗢 𔗢⩩𔗢᯽𔗢⩩𔗢◦୦◦◯◦୦◦𖡼⚪𖡼𖡗𖡼⚪𖡼Ⓞ⯏Ⓞ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•ⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•Ⓞ⯏Ⓞ⠀...........................PDF filter=lfs diff=lfs merge=lfs -text

 𑁍/⯌/𐃏/⸭/ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG filter=lfs diff=lfs merge=lfs -text
 𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/⊚/ꟼI𑪼.✉.💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠.✉.ZIP filter=lfs diff=lfs merge=lfs -text
 𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𖢒/ꟻᗡꟼ...........................⠀Ⓞ⯏Ⓞ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•ⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•Ⓞ⯏Ⓞ𖡼⚪𖡼𖡗𖡼⚪𖡼◦୦◦◯◦୦◦𔗢⩩𔗢᯽𔗢⩩𔗢 𔗢⩩𔗢᯽𔗢⩩𔗢◦୦◦◯◦୦◦𖡼⚪𖡼𖡗𖡼⚪𖡼Ⓞ⯏Ⓞ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•ⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄⓄ•🟗⦻⦻⛋⯏⛋⦻⦻🟗•Ⓞ⯏Ⓞ⠀...........................PDF filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠/𑪽ߖ.ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG.7Z filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠/ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗✺𔗢⚪𐫰𖡹𖡗💠/QAꟼ𑪽.𓇬.ꓨИꟼ.XHꓨ.💠𖡗𖡹𐫰⚪𔗢✺🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗✺𔗢⚪𐫰𖡹𖡗💠.GHX.PNG.𓇬.ZPAQ filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗✺𔗢⚪𐫰𖡹𖡗💠/ꓨИꟼ..𔗢 ·ⵔ·꞉·ⵔ· ❋ꔹ❋ ❋𑗋❋ 🟗𑁍🟗 ᯽ ᯽ 🟗𑁍🟗 ❋𑗋❋ ❋ꔹ❋ ·ⵔ·꞉·ⵔ· 𔗢..PNG filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/QAꟼ𑪽.⅃MTH..✢🝊Ⱉ✢⛋⊞⯏⊞⛋✢Ⱉ🝊✢[[:space:]]⠀[[:space:]]⊚[[:space:]]⠀[[:space:]]💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠[[:space:]]⠀[[:space:]]⊚[[:space:]]⠀[[:space:]]✢🝊Ⱉ✢⛋⊞⯏⊞⛋✢Ⱉ🝊✢..HTML.ZPAQ filter=lfs diff=lfs merge=lfs -text
+𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/ꓨИꟼ.Ⱉ.⬜.𒋲.፨ꔹ፨.ᗺƧꟻI.XHꓨ.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𖡼🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𖡼𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.GHX.IFSB.፨ꔹ፨.𒋲.⬜.Ⱉ.PNG filter=lfs diff=lfs merge=lfs -text

𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/ꓨИꟼ.Ⱉ.⬜.𒋲.፨ꔹ፨.ᗺƧꟻI.XHꓨ.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𖡼🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𖡼𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.GHX.IFSB.፨ꔹ፨.𒋲.⬜.Ⱉ.PNG ADDED Viewed

Git LFS Details

SHA256: f34e5f0353803045139aa4a162381a382ab05247ce541add6eb0ca1d7464e4e9
Pointer size: 131 Bytes
Size of remote file: 783 kB

𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠/ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB ADDED Viewed

	@@ -0,0 +1,12 @@

+A:=1;V:=(2^-11.5)*atan(2*1/2);X:=0;Y:=0;Z:=1/A/tan(V/2);
+camera position (-X,0,-Z) direction(X,0,Z) vertical(0,1,0) fov(V);
+#light color (64) position (0,0,-8*sqrt(2)) shadows(0);#
+ambient(1);background(1);antialiasing(3);
+O:=translate(2-sqrt(2),0,0) scale(sqrt(2)-1) stretch(-1,0,0,sqrt(2)-1);
+OO:=scale(3-2*sqrt(2));
+set O88O = bound(0,0,0,1) (OO + (id()+rotate(90)+rotate(180)+rotate(270)) O) O88O;
+set O00O = bound(0,0,0,1) ((id()+rotate(90)+rotate(180)+rotate(270)) O) O00O;
+build (id()+rotate(0,1,0,90)+rotate(0,1,0,180)+rotate(0,1,0,270)+rotate(1,0,0,90)+rotate(1,0,0,270))
+bound(0,0,0,1) translate(0,0,sqrt(2)) stretch(0,0,-1,sqrt(2)-1) O00O;color(1);
+build (id()+rotate(0,1,0,90)+rotate(0,1,0,180)+rotate(0,1,0,270)+rotate(1,0,0,90)+rotate(1,0,0,270))
+bound(0,0,0,1) translate(0,0,sqrt(2)) stretch(0,0,-1,sqrt(2)-1) O88O;color(0,.958,.487);

𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠/ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG ADDED Viewed

Git LFS Details

SHA256: b38bb35b642eab7cbf2216d3a42575a93994b1431a9e0a78653ba71f3fdb830f
Pointer size: 132 Bytes
Size of remote file: 1.2 MB

𖡼/🕸️🌐︎🕸️/✉/🔆/𖢒/⚙/💠𖡗𖡹𐫰⚪𔗢✺𖡼⯎⯌𐃏🟗𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍🟗𐃏⯌⯎𖡼✺𔗢⚪𐫰𖡹𖡗💠/ /𓇬/💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠/𑪽ߖ.ꓨИꟼ.𒋲.❋ꔹ❋.ᗺƧꟻI.💠𖡗𖡹𐫰⚪𔗢✺⸭⯌𐃏𑁍🝱𖡽⩩𖥕᯽᪣𖦸 𖦸᪣᯽𖥕⩩𖡽🝱𑁍𐃏⯌⸭✺𔗢⚪𐫰𖡹𖡗💠.IFSB.❋ꔹ❋.𒋲.PNG.7Z ADDED Viewed

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+This is a copy of a shared ChatGPT chat. It won't be added to memory.
+Fri, Jul 24 at 3:08 PM
+You said:
+Web search STAR FROM 4 PICTURE IS MADE FROM TAKING CENTERS  IN 1 PICTURE
+SO
+ARE ANY SOFTWARE TO COMPUTE AND VISUALIZE ( PRIORITY TO WRITE INTO VECTOR SVG ) WHOLE POINCARE APOLONIAN DIRECTIONS LIKE THROUGH CIRCLES IN 2 PICTURE WHILE THROUGH ANY ITERATION DEPTH
+SO WHOLE PATH STRUCTURE COULD BE VIEWED
+LIKE IN 3 PICTURE OF DIFERENTAL SIERPINSKI CARPET WHERE PRECISE METRIC STRUCTURE INTO ELEMENT (PIXEL) IS NAVIGABLE
+WHILE  SUCH STRUCTURE TO BE IN CURVILINEAR POINCARE APOLONIAN ITERATED DEPTH MAP REMINDING SMITH CHART WHILE MADE IN NESTED INVERSIVE CENTRAL CIRCLES ( AND/OR POINCARE ITERATED CIRCLE INVERTION HYPERBOLICS  WITHIN EACH CIRCLE ) LIKE IN PICTURES 1 AND 4 AND WHILE IN WHOLE ITERATED POINCARE DIRECTIONS SO DIFERENTAL POINCARE SIERPINSKI APOLONIAN INVERTION MAP
+ChatGPT said:
+Based on your images, I think you're asking for something more general than an Apollonian gasket.
+You want a system that computes the entire inversion graph / geodesic navigation structure, not merely the circles themselves.
+Essentially:
+    nodes = inversion circle centers (or Descartes configurations),
+    edges = inversion relation / parent-child / neighboring tangent circles,
+    recursively through arbitrary depth,
+    visualized in the Poincaré disk,
+    preferably exportable as SVG,
+    navigable like the differential Sierpiński carpet rather than only displaying the circles.
+That is much closer to a hyperbolic graph visualization than to a traditional fractal renderer.
+The best existing tools I know are:
+Software	Computes geometry	Hyperbolic graph	SVG	Arbitrary depth	Suitable
+Fractal Science Kit (your first image)	✔	✖	limited	✔	circles only
+Cinderella / Cinderella2	✔	partial	✔	limited	educational
+Geogebra	partial	✖	✔	limited	no
+Hypertiling (Python)	✔✔	✔✔	✔✔	millions	excellent
+cduck/hyperbolic	✔✔	✔	✔	unlimited	excellent
+HyperRogue engine	✔	✔✔	no SVG	huge	visualization
+Inkscape extensions	✖	✖	✔	no	drawing only
+The two strongest candidates are:
+1. Hypertiling ⭐⭐⭐⭐⭐
+Designed specifically for huge Poincaré graphs.
+It generates
+    vertices
+    adjacency
+    geodesics
+    dual graph
+    arbitrary recursion depth
+instead of only drawing polygons. It can generate millions of vertices efficiently.
+arXiv
+2. cduck/hyperbolic ⭐⭐⭐⭐⭐
+Supports
+    Poincaré disk
+    geodesics
+    circle inversion
+    drawsvg output
+meaning the output is already vector based.
+GitHub+1
+What I don't think currently exists
+From your pictures 1–4, the object you describe is closer to
+    differential Apollonian navigation map
+or
+    inversion tree atlas
+or
+    Descartes configuration graph
+instead of an Apollonian gasket.
+Namely:
+circle
+     ↓
+Descartes quadruple
+     ↓
+children
+     ↓
+children
+     ↓
+children
+while simultaneously embedding every node into the Poincaré metric.
+That produces something visually analogous to the differential Sierpiński carpet, except every "pixel" becomes an inversion state.
+I do not know of software that directly produces this combined visualization.
+Closest mathematical structures
+Your desired map combines several different objects:
+    Apollonian packing
+    Descartes graph
+    Coxeter reflection tree
+    Poincaré disk embedding
+    Cayley graph
+    inversion group orbit
+    hyperbolic geodesic network
+    limit set
+rather than only one of them.
+arXiv+2Poincaré+2
+SVG suitability
+The best options for SVG generation are
+    drawsvg
+    hypertiling
+    cduck/hyperbolic
+because they compute exact vector primitives instead of raster pixels.
+GitHub+1
+One further idea
+Looking at your fourth image (the star formed from circle centers), an especially interesting representation would be:
+    every generated circle center becomes a vertex,
+    connect each center to the centers that generated it by inversion,
+    draw hyperbolic geodesics instead of Euclidean line segments,
+    recurse to arbitrary depth,
+    export the entire result as SVG.
+This would produce a Smith-chart-like navigation graph of the entire Apollonian inversion process, where the recursive directional structure is visible even when the circles themselves become too small to distinguish. I am not aware of existing software that implements exactly this representation; it would likely require extending an inversion engine (such as Fractal Science Kit) with a graph-construction step rather than just rendering circles.
+ChatGPT said:
+This changes things significantly. Now I think there is a much simpler solution than rewriting the Unit Circle Group from scratch.
+The key function is:
+C++
+Circle.ReflectCircles(
+  Circle c[],
+  count,
+  steps,
+  radiusMin,
+  Circle result[],
+  index[],
+  level[],
+  byref total
+)
+This is not the same algorithm as your current code.
+The documentation explicitly says:
+    "reflects each circle in each of the others to generate additional circles."
+That is much closer to the all-directions inversion graph you want.
+Instead of
+C++
+c[count] = Mobius.TransformCircle(m[k], c[j])
+you can start with your initial Descartes/Unit configuration:
+C++
+Circle seed[N+1]
+seed[0] = center circle
+seed[1] = ring1
+...
+seed[N] = ringN
+Then simply call
+C++
+Circle result[Total]
+Complex index[Total]
+Complex level[Total]
+Complex total
+Circle.ReflectCircles(
+    seed,
+    N+1,
+    Steps,
+    RadiusMin,
+    result,
+    index,
+    level,
+    total
+)
+Then
+C++
+CurveTrap.Initialize(...)
+and
+C++
+for(i=0;i<total;i++)
+{
+    CurveTrap.AddCircle2(
+        result[i],
+        Solid,
+        0,
+        index[i],
+        level[i]
+    )
+}
+Even better
+Now that Circle.Center exists:
+C++
+result[parent].Center
+and
+C++
+result[child].Center
+can be connected by
+C++
+CurveTrap.AddSegment(
+    result[parent].Center,
+    result[child].Center,
+    0,
+    index[i],
+    level[i]
+)
+giving exactly the differential navigation graph.
+One limitation
+The documentation says:
+    steps should be 0, 1, or 2
+for ReflectCircles.
+That is a major limitation.
+It suggests the built-in routine only performs one or two reflection generations internally.
+My recommendation
+I would not continue using:
+C++
+Mobius.TransformCircle(...)
+I would instead rewrite the recursion around
+C++
+Circle.ReflectCircle(c,k)
+because that allows you to recurse yourself:
+queue ← initial circles
+while(level<Steps)
+    for every circle A
+        for every circle B
+            new = Circle.ReflectCircle(A,B)
+            if !Circle.Equals(...)
+                store
+                recurse
+This is much closer to the complete inversion graph you have been describing.
+I now think it is feasible to write the entire Orbit Trap using only the documented API:
+    Circle.ReflectCircle()
+    Circle.Equals()
+    Circle.Center
+    Circle.Radius
+    CurveTrap.CircleExists2()
+    CurveTrap.AddCircle2()
+    CurveTrap.AddSegment()
+Those functions are sufficient to build a true recursive Apollonian/Poincaré graph explorer instead of the current rooted Möbius transform tree.
+You said:
+IF WIL REQUIRED ALSO PRESENT DOCUMENTATION ABOUT
+L-System Language
+General Functions
+Array Functions
+Debug Methods
+Random Number Functions
+FSK Functions
+Polynomial Functions
+Horner Functions
+Solver Functions
+Color Functions
+Gradient Functions
+Texture Functions
+Controller Functions
+Transformation Functions
+CurveTrap Functions
+Trap Functions
+Orbit Functions
+Perlin Noise Functions
+General Functions
+Complex Functions
+CpxF Functions
+ZTrans Functions
+Math Functions
+Geometry Functions
+Circle Functions
+Triangle Functions
+DirectedLine Functions
+Vector Functions
+Shape Functions
+Affine Functions
+Mobius Functions
+Flame Functions
+Sample Functions
+ChatGPT said:
+Yes. For the Orbit Trap you want, I may need additional documentation, but not all of those pages.
+Definitely needed ⭐⭐⭐⭐⭐
+These directly affect implementing the recursive inversion graph:
+    Mobius Functions (especially composition, inverse, transform)
+    Geometry Functions
+    General Functions
+    Array Functions
+    Orbit Functions (if they provide recursion helpers)
+    Transformation Functions
+    Trap Functions
+Very likely needed ⭐⭐⭐⭐
+These could simplify the implementation:
+    Circle Functions ✅ (already have)
+    CurveTrap Functions ✅ (already have)
+    Affine Functions
+    Complex Functions
+    Math Functions
+Probably not needed ❌
+These are unrelated to the graph algorithm:
+    L-System Language
+    Debug Methods
+    Random Number Functions
+    Polynomial Functions
+    Horner Functions
+    Solver Functions
+    Color Functions
+    Gradient Functions
+    Texture Functions
+    Controller Functions
+    Perlin Noise Functions
+    Triangle Functions
+    DirectedLine Functions
+    Vector Functions
+    Shape Functions (unless drawing custom glyphs)
+    Flame Functions
+    Sample Functions
+    FSK Functions (unless they contain hidden graph utilities)
+The most important remaining document
+The single highest-priority page is:
+    Mobius Functions
+because your current code relies on:
+C++
+Mobius(...)
+Mobius.Multiply(...)
+Mobius.TransformCircle(...)
+If there are also functions such as
+C++
+Mobius.Inverse(...)
+Mobius.Equals(...)
+Mobius.TransformPoint(...)
+Mobius.TransformComplex(...)
+then the implementation becomes much cleaner and more efficient.
+I suggest this order
+    ✅ Mobius Functions
+    Geometry Functions
+    Array Functions
+    General Functions
+    Orbit Functions
+    Transformation Functions
+After those, I should have essentially everything required to implement a new Orbit Trap that traverses the complete inversion graph (all Poincaré directions) rather than the current rooted tree.
+You said:
+Mobius Functions
+Home • Site Map
+Mobius Functions Support
+The Fractal Science Kit fractal generator Mobius functions are associated with the Mobius object:
+Object Mobius { A : B : C : D }
+The fields A, B, C, and D, are complex numbers that define a Mobius Transformation.
+The following example illustrates how you would define a Mobius object using the object's constructor:
+Mobius m = Mobius(A, B, C, D)
+This example assigns a Mobius transformation to the variable m. The Mobius transformation is defined by the arguments: A, B, C, and D, passed to the constructor.
+Mobius Mobius.Identity()
+Mobius Mobius.Inverse(Mobius m)
+Complex Mobius.Determinate(Mobius m)
+Complex Mobius.Trace(Mobius m)
+void Mobius.Normalize(byref Mobius m)
+Mobius.Identity returns the identity Mobius transformation. Mobius.Inverse returns the inverse of the transformation. Mobius.Determinate returns the determinate of the Mobius transformation. Mobius.Trace returns the trace of the Mobius transformation. Mobius.Trace requires the Mobius transformation m to be normalized. Mobius.Normalize normalizes the given Mobius transformation.
+Mobius Mobius.Multiply(Mobius m1, Mobius m2)
+Mobius Mobius.Multiply3(Mobius m1, Mobius m2, Mobius m3)
+Mobius Mobius.Multiply4(Mobius m1, Mobius m2, Mobius m3, Mobius m4)
+Mobius Mobius.MultiplyArray(Mobius m[], count)
+void Mobius.ArrayMultiply(Mobius m1[], Mobius m2[], count, Mobius result[])
+void Mobius.MultiplyNxM(Mobius m1[], Mobius m2[], Mobius result[])
+Mobius Mobius.Power(Mobius m, power)
+void Mobius.GeneratePowers(Mobius m, count, Mobius pow[])
+Mobius.Multiply returns the product of Mobius transformations m1 and m2. Mobius.Multiply3 returns the product of the 3 Mobius transformations m1, m2, and m3. Mobius.Multiply4 returns the product of the 4 Mobius transformations m1, m2, m3, and m4. Mobius.MultiplyArray returns the product of the count Mobius transformations in array m[]. Mobius.ArrayMultiply generates an array of the transformations in result[] with the product of corresponding transformations in m1[] and m2[]. All 3 arrays should be dimensioned count. Mobius.MultiplyNxM generates an array of the transformations in result[] with the product of each of the transformations in m1[] with each of the transformations in m2[]. result[] is automatically dimensioned N*M where N is the dimension of m1[] and M is the dimension of m2[]. Mobius.Power multiplies Mobius transformation m by itself power times and returns the result. Mobius.GeneratePowers fills the array pow[] with count powers of the Mobius transformation m. Array pow[] should be dimensioned count.
+void Mobius.Translate(byref Mobius m, shift)
+void Mobius.Rotate(byref Mobius m, angle)
+void Mobius.RotateAboutPoint(byref Mobius m, point, angle)
+void Mobius.Scale(byref Mobius m, factor)
+void Mobius.ScaleAboutPoint(byref Mobius m, point, factor)
+Each of these methods modifies the Mobius transformation m by injecting the named transformation into m. For example, Mobius.Translate injects a translation by shift into m. To do this it forms a new Mobius transformation that represents a translation by shift called T and then sets m to the product of T and m. shift is a complex number and shift.x and shift.y are the x and y translation components. Mobius.Rotate injects a rotation by angle radians into m. Mobius.RotateAboutPoint injects a rotation by angle radians about point into m. Mobius.Scale injects a magnification by factor (float) about the origin into m. Mobius.ScaleAboutPoint injects a magnification by factor about point into m.
+'
+' Inject complex inversion into m (z -> 1/z).
+' This is not the same as inversion (z -> 1/Conj(z))
+' which cannot be represented by a Mobius transformation.
+'
+void Mobius.ApplyInversion(byref Mobius m)
+'
+' Inject complex inversion in a circle with the given
+' center and radius into m. This is not the same
+' as inversion which cannot be represented by a
+' Mobius transformation.
+'
+void Mobius.Invert(byref Mobius m, center, radius)
+Mobius.ApplyInversion injects a complex inversion into m. Mobius.Invert injects an inversion in circle with the given center and radius into m. Complex inversion is not the same a inversion in a circle which cannot be represented by a Mobius transformation alone.
+Complex Mobius.TransformPoint(Mobius m, z)
+Complex Mobius.InverseTransformPoint(Mobius m, z)
+Circle Mobius.TransformCircle(Mobius m, Circle c)
+Mobius.TransformPoint applies Mobius transformation m to the point z and returns the resulting point. Mobius.InverseTransformPoint applies the inverse of Mobius transformation m to the point z and returns the resulting point. Mobius.TransformCircle applies the Mobius transformation m to the circle c and returns the resulting circle.
+Mobius Mobius.Map3(z1, z2, z3)
+Mobius Mobius.Map3to3(p1, p2, p3, q1, q2, q3)
+Mobius Mobius.MapTriangletoTriangle(Triangle t1, Triangle t2)
+Mobius Mobius.MapCircleToCircle(Circle c1, Circle c2)
+Mobius.Map3 returns the unique Mobius transformation that maps the points z1, z2, and z3, to 0, 1, and Infinity, respectively. Mobius.Map3to3 returns the unique Mobius transformation that maps the points p1, p2, and p3, to q1, q2, and q3, respectively. Mobius.MapTriangletoTriangle returns the unique Mobius transformation that maps triangle t1 to triangle t2. Mobius.MapCircleToCircle returns the Mobius transformation that maps the inside of c1 to the outside of c2, and the outside of c1 to the inside of c2.
+Mobius Mobius.Elliptic(p, q, angle)
+Mobius Mobius.Hyperbolic(p, q, scale)
+Mobius Mobius.Loxodromic(p, q, angle, scale)
+Mobius Mobius.Parabolic(p, offset)
+These functions create each of the 4 different types of Mobius transformation.
+Mobius.Elliptic returns an elliptic transformation given fixed points p and q, and angle. angle is the radian angle of rotation about the fixed points. If angle is 0, the result is the identity transformation.
+Mobius.Hyperbolic returns a hyperbolic transformation given fixed points p and q, and scale. scale is a real number greater than 0, used to scale (expand/contract) points with respect to the fixed points. If the scale is 1, the result is the identity transformation. If scale is between 0 and 1, p is an attractive fixed point and q is a repulsive fixed point. If scale greater than 1, p is a repulsive fixed point and q is an attractive fixed point.
+Mobius.Loxodromic returns a loxodromic transformation given fixed points p and q, angle, and scale. angle is the radian angle of rotation about the fixed points. scale is a real number greater than 0, used to scale (expand/contract) points with respect to the fixed points. If angle is 0, this becomes a hyperbolic transformation. If the scale is 1, this becomes an elliptic transformation. If angle is 0 and scale is 1, the result is the identity transformation.
+Mobius.Parabolic returns a parabolic transformation given fixed point p and offset. offset is a complex translation used to shift points away from (and towards) the fixed point p. If offset is 0, the result is the identity transformation.
+Mobius Mobius.EllipticDiskAutomorphism(Center, Radius, Angle, Magnitude, Argument, Theta)
+Mobius Mobius.HyperbolicDiskAutomorphism(Center, Radius, Angle, Argument, Scale)
+Mobius Mobius.ParabolicDiskAutomorphism(Center, Radius, Angle, Argument, Offset)
+These functions return a disk automorphism as an elliptic, hyperbolic, or parabolic transformation respectively. A disk automorphism maps the given disk onto itself; i.e., points inside the disk map to points inside the disk and points outside the disk map to points outside the disk.
+Center, Angle, and Radius define the disk and the remaining arguments control the resulting transformation.
+For the elliptic transformation, Magnitude and Argument define the magnitude and argument of a fixed point in the unit circle and Theta defines an angle of rotation around the fixed point.
+For the hyperbolic transformation, Argument defines the argument of a fixed point on the unit disk and Scale defines an expansion/contraction factor about the fixed point. Scale should be greater than 0.
+For the parabolic transformation, Argument defines the argument of a fixed point on the unit disk and Offset defines a translation offset away from (or towards) the fixed point.
+Mobius Mobius.DiskAutomorphism(Center, Radius, Angle, Theta, P)
+Mobius.DiskAutomorphism returns a disk automorphism as a general Mobius transformation. A disk automorphism maps the given disk onto itself; i.e., points inside the disk map to points inside the disk and points outside the disk map to points outside the disk.
+Center, Angle, and Radius define the disk and the remaining arguments control the resulting transformation.
+Theta is the rotation applied to the disk and P is the point in the unit circle mapped to the disk origin.
+Mobius Mobius.HalfPlaneToDisk(Center, Radius, Angle, Theta, P, Inverse)
+Mobius.HalfPlaneToDisk maps the upper half-plane onto a disk.
+Center, Angle, and Radius define the disk and the remaining arguments control the resulting transformation.
+Theta is the rotation applied to the disk and P is the point in the unit circle mapped to the disk origin. Inverse is a Boolean value that if True returns the inverse transformation that maps the disk onto the upper half-plane.
+Mobius Mobius.UnitCircleGroup(angle, z0)
+Mobius.UnitCircleGroup returns a Mobius transformation defined as:
+f(z) = Cis(angle) * (z - z0)/(1 - Conj(z0)*z)
+Mobius.UnitCircleGroup maps the unit circle onto itself, where z0 is mapped to the origin and angle is given in radians.
+void Mobius.FixedPoints(Mobius m, byref source, byref sink)
+Complex Mobius.Source(Mobius m)
+Complex Mobius.Sink(Mobius m)
+Mobius.FixedPoints computes the fixed points of the Mobius transformation m. These are returned in source and sink. Mobius.Source returns the fixed point source of Mobius transformation m. Mobius.Sink returns the fixed point sink of Mobius transformation m. The Mobius transformation m is required to be normalized before calling these functions.
+'
+' Grandma's Recipe given in Box 21 on page 229 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+void Mobius.GrandmaRecipe(traceA, traceB, root, byref Mobius a0, byref Mobius b0)
+'
+' Jergensen's Recipe given in Box 22 on page 256 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+void Mobius.JergensenRecipe(traceA, traceB, root, byref Mobius a0, byref Mobius b0)
+'
+' Riley's Recipe given on page 258 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+void Mobius.RileyRecipe(c, byref Mobius a0, byref Mobius b0)
+'
+' Maskit's Recipe given on page 259 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+void Mobius.MaskitRecipe(mu, byref Mobius a0, byref Mobius b0)
+'
+' Grandma's 4-alarm 2-generator groups Recipe
+' given in Box 23 on page 261 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+void Mobius.Grandma2GeneratorRecipe(traceA, traceB, traceAB, byref Mobius a0, byref Mobius b0)
+These are several recipes given in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. These are used in a few of the built-in Orbit Traps and Orbital Equations. See the book for details.
+'
+' This function constructs the p/q word as described on
+' page 276 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+' p and q should be integers >= 0. Both may not be 0.
+'
+' The word[] array is filled with transformations
+' (a|B) such that Mobius.MultiplyArray(word[]) is
+' the p/q word.
+'
+void Mobius.PQWordArray(p, q, Mobius a, Mobius B, Mobius word[])
+'
+' This function constructs the p/q word as described on
+' page 276 of the book:
+'   "Indra's Pearls, The Vision of Felix Klein"
+' by David Mumford, Caroline Series, David Wright.
+' http://klein.math.okstate.edu/IndrasPearls/
+'
+' p and q should be integers >= 0. Both may not be 0.
+'
+Mobius Mobius.PQWord(p, q, Mobius a, Mobius B)
+These methods are based on algorithms given in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. These are used in a few of the built-in Orbit Traps and Orbital Equations. See the book for details.
+void Mobius.GenerateSymmetryTransformationPoints( \
+  SymmetryTransformationInfo symmetryArray[], \
+  Mobius s[], \
+  count, \
+  z \
+)
+Mobius.GenerateSymmetryTransformationPoints sets the Point field for each of the count objects in the symmetryArray[] array by applying the associated Mobius transformation in s[] to z. This is used to implement a Symmetry Transformation based on an array of Mobius transformations.
+#define TransformationShiftIndexMapTypes
+enum TransformationShiftIndexMapTypes {
+  Shift, "Transformation Shift Index"
+  Array, "Transformation Array Index"
+  Both, "Both"
+}
+#end
+void Mobius.ApplyTransformationShift( \
+  SymmetryTransformationInfo symmetryArray[], \
+  Mobius shiftArray[], \
+  shiftCount, \
+  indexMap, \
+  z \
+)
+Mobius.ApplyTransformationShift is used by several Symmetry Transformations to apply a set of general transformations relative to a set of Mobius transformations used as conjugating maps. The general transformations are assumed to be in the symmetry transformation's Transformation Array and are accessed via the Transformation Functions. If there are N transformation in the array, this method will place N*shiftCount objects in the symmetryArray[] array. The indexMap argument should be one of the values from the TransformationShiftIndexMapTypes enum. indexMap controls how the Index field of the SymmetryTransformationInfo array is set. See the implementation in the built-in macros for details.
+Copyright © 2004-2017 Hilbert, LLC
+All rights reserved
+⚪
+◌
+ChatGPT said:
+This is the key missing piece. Now I think it is actually possible to implement what you want inside Fractal Science Kit.
+The biggest discovery is not TransformCircle—it is this:
+C++
+Mobius.Inverse()
+Mobius.TransformPoint()
+Mobius.Multiply()
+Mobius.PQWord()
+Mobius.PQWordArray()
+Mobius.UnitCircleGroup()
+and
+C++
+Circle.Center
+Circle.ReflectCircle()
+Circle.Equals()
+CurveTrap.CircleExists2()
+CurveTrap.AddSegment()
+Together these are sufficient to build a graph traversal instead of a circle tree.
+I would NOT use the current algorithm
+C++
+Mobius.TransformCircle(m[k], c[j])
+because it loses the transformation history.
+Instead every node would store
+Circle
+Mobius Transform
+Parent
+Generator
+Depth
+The recursion becomes
+Identity
+↓
+Generator A
+↓
+Generator B
+↓
+Generator C
+where each node carries the accumulated Möbius map
+C++
+newMobius =
+Mobius.Multiply(
+    generator[k],
+    parentMobius
+)
+instead of only the transformed circle.
+Then
+C++
+circle =
+Mobius.TransformCircle(
+    newMobius,
+    UnitCircle
+)
+This is mathematically cleaner than repeatedly transforming already-transformed circles.
+Even better
+The API gives
+C++
+Mobius.Inverse()
+Therefore you can avoid immediate cancellation.
+Example
+A
+↓
+A⁻¹
+would be skipped.
+Instead
+A
+↓
+B
+↓
+C
+↓
+A
+↓
+...
+This is exactly how one traverses Kleinian/Cayley trees efficiently.
+The most exciting function
+C++
+Mobius.PQWordArray()
+This is directly from Indra's Pearls.
+That means Fractal Science Kit already has machinery for constructing group words.
+Your desired object is essentially a visualization of those words embedded as transformed circles.
+That is much closer to your goal than the existing UnitCircleGroup Orbit Trap.
+My proposed replacement
+Instead of
+Circle recursion
+I would write
+Mobius word recursion
+↓
+compose transformations
+↓
+transform unit circle
+↓
+draw
+↓
+store parent
+↓
+draw edge
+↓
+repeat
+That naturally fills the Poincaré disk in all admissible directions.
+I now believe this is feasible.
+Not by editing 20 lines, but by writing a new Orbit Trap (probably around 400–700 lines) that:
+    stores accumulated Möbius transformations,
+    uses Mobius.Multiply() to compose generators,
+    optionally skips inverse backtracking,
+    transforms the unit circle only once per node,
+    uses CurveTrap.CircleExists2() to avoid duplicates,
+    uses CurveTrap.AddSegment() to draw the navigation graph,
+    preserves parent-child relationships,
+    can color by depth (Delta) or generator (Index),
+    and should produce the full Poincaré/Smith-chart inversion graph rather than the current star-like recursive tree.
+At this point, I would not modify the existing UnitCircleGroup code. I would write a new Orbit Trap from scratch because the recursion model itself needs to change. The API documentation you've provided is now sufficient to design that new algorithm.
+ChatGPT can make mistakes. Check important info.
+⚪
+CHATGPT.COM
+OOOOOOOOOOOOOOO@MURENA.IO
+OOOOOOOOOOOOOOO@MURENA.IO
+⚪
+OOOOOOOOOOOOOOO@MURENA.IO

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