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Updated Jul 16, 2026

𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢 𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢

⠀ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ⠀

View ◦୦◦◯◦୦◦⏮⏭⟠ⵙ⏮⏭Ⓞ◇Ⓞ🝊⟠🝊Ⓞ◇Ⓞ⏮⏭ⵙ⟠⏮⏭𖢄◇⟠⏮⏭Ⓞⵙ✣ᑐᑕᗱᗴ🝍ⵙↀⵙ∞ⵙↀⵙ🝍ᗱᗴᑐᑕ✣ⵙⓄ⏮⏭⟠◇𖢄⏮⏭⟠ⵙ⏮⏭Ⓞ◇… ◦୦◦◯◦୦◦⏮⏭⟠ⵙ⏮⏭Ⓞ◇Ⓞ🝊⟠🝊Ⓞ◇Ⓞ⏮⏭ⵙ⟠⏮⏭𖢄◇⟠⏮⏭Ⓞⵙ✣ᑐᑕᗱᗴ🝍ⵙↀⵙ∞ⵙↀⵙ🝍ᗱᗴᑐᑕ✣ⵙⓄ⏮⏭⟠◇𖢄⏮⏭⟠ⵙ⏮⏭Ⓞ◇… (([email protected]))

Updated Jul 14, 2026

◦୦◦◯◦୦◦⏮⏭⟠ⵙ⏮⏭Ⓞ◇Ⓞ🝊⟠🝊Ⓞ◇Ⓞ⏮⏭ⵙ⟠⏮⏭𖢄◇⟠⏮⏭Ⓞⵙ✣ᑐᑕᗱᗴ🝍ⵙↀⵙ∞ⵙↀⵙ🝍ᗱᗴᑐᑕ✣ⵙⓄ⏮⏭⟠◇𖢄⏮⏭⟠ⵙ⏮⏭Ⓞ◇Ⓞ🝊⟠🝊Ⓞ◇Ⓞ⏮⏭ⵙ⟠⏮⏭◦୦◦◯◦୦◦

⠀ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ⠀

View 𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢᯽𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢 𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢᯽𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢 (([email protected]))

Updated Jul 10, 2026

𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢᯽𔗢𑗋𔗢🝱𔗢𑗋𔗢⩩𔗢𑗋𔗢🝱𔗢𑗋𔗢

⠀ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 𔗢𖡽𔗢⩩𔗢𖡽𔗢᯽𔗢𖡽𔗢⩩𔗢𖡽𔗢 ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ⚪𖡗⚪𖡼⚪𖡗⚪𔗢⚪𖡗⚪𖡼⚪𖡗⚪ ⠀

View 𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢᯽𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢 𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢᯽𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢 OOOO00000000OOOO

Updated Jul 4, 2026

𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢᯽𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢𖥕𔗢🝱𔗢⩩𔗢🝱𔗢᪣𔗢🝱𔗢⩩𔗢🝱𔗢

𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠 ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ 𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠 𔗢᯽𔗢 𔗢᯽𔗢 𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠 ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ 𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠

View 𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖥕𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢᯽𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜… 𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖥕𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢᯽𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜… (([email protected]))

Updated Jul 4, 2026

𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖥕𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢᯽𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖥕𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢𖤞𔗢 ⬜ 𔗢⩩𔗢 ⬜ 𔗢

𔗢᯽𔗢 𔗢᯽𔗢

View 𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢᯽𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢 𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢᯽𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢 (([email protected]))

Updated Jul 4, 2026

𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢᯽𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢𖥕𔗢⬜𔗢⩩𔗢⬜𔗢𖤞𔗢⬜𔗢⩩𔗢⬜𔗢

𔗢᯽𔗢 𔗢᯽𔗢

View 𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢 𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢 𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢 𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢 (([email protected]))

Updated Jul 4, 2026

𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢 𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢᯽𔗢⩩𔗢𖤞𔗢⩩𔗢𖥕𔗢⩩𔗢𖤞𔗢⩩𔗢

𔗢᯽𔗢 𔗢᯽𔗢

View 𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷 𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷 (([email protected]))

Updated May 13, 2026

𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠𖧷𐫱ⵙ𖢌⛋𖥠⛋𖢌ⵙ𐫱𖧷

𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠

View ⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞ ⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞ (([email protected]))

Updated May 13, 2026

⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠⊞⯏⦻⛋ꖅ𖧷ꖅ⦻ꖅ𖧷ꖅ⛋⦻⯏⊞

𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠

View ​ ​ ​ ​ (([email protected]))

Updated May 13, 2026

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View ​ ​ ​ ​ (([email protected]))

Updated May 13, 2026

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View 𖢌··꞉·꞉꞉꞉·꞉꞉꞉···꞉ⵔ꞉ⵔ꞉ⵔ·ⵔⵔ·ⵔⵔⵔ꞉·꞉꞉·ⵔ꞉·꞉꞉·꞉ⵔⵔⵔⵔⵔⵔⵔ꞉ⵔ꞉ⵔⵔ꞉ⵔ·꞉·ⵔ·ⵔ·ⵔ···꞉ⵔ꞉꞉… 𖢌··꞉·꞉꞉꞉·꞉꞉꞉···꞉ⵔ꞉ⵔ꞉ⵔ·ⵔⵔ·ⵔⵔⵔ꞉·꞉꞉·ⵔ꞉·꞉꞉·꞉ⵔⵔⵔⵔⵔⵔⵔ꞉ⵔ꞉ⵔⵔ꞉ⵔ·꞉·ⵔ·ⵔ·ⵔ···꞉ⵔ꞉꞉… (([email protected]))

Updated Apr 17, 2026

𖢌··꞉·꞉꞉꞉·꞉꞉꞉···꞉ⵔ꞉ⵔ꞉ⵔ·ⵔⵔ·ⵔⵔⵔ꞉·꞉꞉·ⵔ꞉·꞉꞉·꞉ⵔⵔⵔⵔⵔⵔⵔ꞉ⵔ꞉ⵔⵔ꞉ⵔ·꞉·ⵔ·ⵔ·ⵔ···꞉ⵔ꞉꞉··꞉·ⵔ·꞉···꞉ⵔⵔ꞉ⵔⵔ·꞉ⵔⵔ꞉·꞉ⵔⵔ·꞉··꞉꞉ⵔⵔⵔ··ⵔ··ⵔⵔ꞉ⵔ꞉꞉꞉꞉·꞉꞉·ⵔ·◌ⵔ·ⵔⵔ·ⵔⵔ·ⵔ◌·ⵔ·꞉꞉·꞉꞉꞉꞉ⵔ꞉ⵔⵔ··ⵔ··ⵔⵔⵔ꞉꞉··꞉·ⵔⵔ꞉·꞉ⵔⵔ꞉·ⵔⵔ꞉ⵔⵔ꞉···꞉·ⵔ·꞉··꞉꞉ⵔ꞉···ⵔ·ⵔ·ⵔ·꞉·ⵔ꞉ⵔⵔ꞉ⵔ꞉ⵔⵔⵔⵔⵔⵔⵔ꞉·꞉꞉·꞉ⵔ·꞉꞉·꞉ⵔⵔⵔ·ⵔⵔ·ⵔ꞉ⵔ꞉ⵔ꞉···꞉꞉꞉·꞉꞉꞉·꞉··𖢌𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠𖢌··꞉·꞉꞉꞉·꞉꞉꞉···꞉ⵔ꞉ⵔ꞉ⵔ·ⵔⵔ·ⵔⵔⵔ꞉·꞉꞉·ⵔ꞉·꞉꞉·꞉ⵔⵔⵔⵔⵔⵔⵔ꞉ⵔ꞉ⵔⵔ꞉ⵔ·꞉·ⵔ·ⵔ·ⵔ···꞉ⵔ꞉꞉··꞉·ⵔ·꞉···꞉ⵔⵔ꞉ⵔⵔ·꞉ⵔⵔ꞉·꞉ⵔⵔ·꞉··꞉꞉ⵔⵔⵔ··ⵔ··ⵔⵔ꞉ⵔ꞉꞉꞉꞉·꞉꞉·ⵔ·◌ⵔ·ⵔⵔ·ⵔⵔ·ⵔ◌·ⵔ·꞉꞉·꞉꞉꞉꞉ⵔ꞉ⵔⵔ··ⵔ··ⵔⵔⵔ꞉꞉··꞉·ⵔⵔ꞉·꞉ⵔⵔ꞉·ⵔⵔ꞉ⵔⵔ꞉···꞉·ⵔ·꞉··꞉꞉ⵔ꞉···ⵔ·ⵔ·ⵔ·꞉·ⵔ꞉ⵔⵔ꞉ⵔ꞉ⵔⵔⵔⵔⵔⵔⵔ꞉·꞉꞉·꞉ⵔ·꞉꞉·꞉ⵔⵔⵔ·ⵔⵔ·ⵔ꞉ⵔ꞉ⵔ꞉···꞉꞉꞉·꞉꞉꞉·꞉··𖢌

View 🌐⚪🌐🅯🌐⚪🌐𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠🌐⚪🌐🅯🌐⚪🌐 🌐⚪🌐🅯🌐⚪🌐𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠🌐⚪🌐🅯🌐⚪🌐 (([email protected]))

Updated Apr 13, 2026

🌐⚪🌐🅯🌐⚪🌐𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠🌐⚪🌐🅯🌐⚪🌐

𓇬🌐𓇬◌𓇬🌐𓇬⚪𓇬🌐𓇬◌𓇬🌐𓇬🅯𓇬🌐𓇬◌𓇬🌐𓇬⚪𓇬🌐𓇬◌𓇬🌐𓇬𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦𖣠⚪𔗢⚪🞋⚪𔗢⚪𖣠𓇬🌐𓇬◌𓇬🌐𓇬⚪𓇬🌐𓇬◌𓇬🌐𓇬🅯𓇬🌐𓇬◌𓇬🌐𓇬⚪𓇬🌐𓇬◌𓇬🌐𓇬

View ​     𑁍     ​◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦​     𑁍     ​ ​     𑁍     ​◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦​     𑁍     ​ OOOO00000000OOOO

Updated Mar 31, 2026

​     𑁍     ​◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦​     𑁍     ​

◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦

View ⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀ OOOO00000000OOOO

Updated Mar 30, 2026

⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦⠀⠀⠀⠀⠀⠀  𖢒𖤞𖢒  ⠀⠀⠀⠀⠀⠀

◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦

View ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ꖅ✢ⵙ𖧷ⵙ✢ꖅ✺✻𑽇ⵙ◇ⵙ𑽇✻✺ꖅ✢ⵙ𖧷ⵙ✢ꖅ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ꖅ✢ⵙ𖧷ⵙ✢ꖅ✺✻𑽇ⵙ◇ⵙ𑽇✻✺ꖅ✢ⵙ𖧷ⵙ✢ꖅ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ OOOO00000000OOOO

Updated Mar 30, 2026

◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ꖅ✢ⵙ𖧷ⵙ✢ꖅ✺✻𑽇ⵙ◇ⵙ𑽇✻✺ꖅ✢ⵙ𖧷ⵙ✢ꖅ◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦

◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦

View ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​… ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​… OOOO00000000OOOO

Updated Dec 3, 2025

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​፨​⬡​፨​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

View ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​… ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​… OOOO00000000OOOO

Updated Dec 3, 2025

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​⬡​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

View ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​… ​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​… OOOO00000000OOOO

Updated Dec 3, 2025

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​⎈​ꙮ​⎈​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​𓇬​◦​୦​◦​◯​◦​୦​◦​𓇬​ ​

View ​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​◦​⊚​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​… ​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​◦​⊚​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​… OOOO00000000OOOO

Updated Dec 3, 2025

​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​◦​⊚​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⬢​◦​୦​◦​◯​◦​୦​◦​⊚​ ​

​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⊚​ ​⊚​◦​୦​◦​◯​◦​୦​◦​⊚​ ​

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-iter(log((z'^4))-2,log(z^4),4)+6

Complex Function Viewer

This tool visualizes any complex-valued function as a conformal map by assigning a color to each point in the complex plane according to the function's value at that point.

Enter any expression in z.

The identity function z shows how colors are assigned: a gray ring at |z| = 1 and a black and white circle around any zero and colored circles around 1, i, -1, and -i. Checkers cover the plane in a 1/16th unit grid. Colors are turquoise in the positive direction, red in the negative, gold-green towards +i, purplish towards -i, and darker towards infinity. There is also a colored circle towards infinity at |z| > 16 that can be seen at any pole towards infinity such as in 1/z.

Here are some example functions to try:

z^2 zz* (z+1)/(z-1) sin(z) e^z log(z) sech(z) arctan(z) z^3-1 0.926(z+7.3857e-2 z^5+4.5458e-3 z^9) Jacobi elliptic sn(z, 0.3) Gamma function gamma(z) Iterated function iter(z+z'^2,z,12) Conformal Maps on the Globe

Conformal maps have their history in 18th century mapmaking, when new mathematical developments allowed mapmakers to understand how to precisely eliminate local shape distortions in maps. Click the ⊕ button in the lower right corner to switch to a conformal mapping of the surface of the earth. Conformal maps preserve local angles everywhere, although they may distort sizes to do so.

The Mercator projection is an example. Try:

e^iz

The azimuthal stereographic projection is a beautiful ancient technique that is also conformal, but it is usually broken into two hemispheres:

...i(z+1-i)/(z+1+i)...

Lagrange advocated another conformal projection that squeezes the entire globe into a single circle:

(disk(z)(z-i)/(z+i))^2

Read more about conformal projections in cartography on Carlos A. Furuti's nicely illustrated mapmaking website. Or Donald Fenna's mathematical mapmaking book, Cartographic Science. Animating Conformal Maps

To visualize the relationships within families of complex functions, parameterize them with the variables t, u, s, r, or n. The tool will render a range of complex functions for values of the parameter, adjustable with a slider or shown in an aimation. The parameter t will vary linearly from 0 to 1; u will circle through complex units; s follows a sine wave between -1 and 1; r follows a sine wave from 0 to 1 and back; and n counts integers from 1 to 60.

For example, to see the relationship between z^3 and z^3+1, simply view:

z^3+t

On the globe, multiplying by powers of unity will rotate the world on its axis:

u(z-i)/(z+i)

Because more than 300 frames are computed, parameterized expressions can take a long time to fully render. A rough, blurry sketch is drawn quickly, and finer-grained rendering will follow for several minutes. When done, the frames will be antialiased and animated at 24 fps.

Simple families of rational function produce mesmerizing animations:

z^2+s z^3+1+u z^5+uz+1 z^2/(r+z)

Iterated functions and sums can also be animated. For example, the following are well-known Taylor series for e^z, sin(z), 1/(1-z), and log(1-z):

sum(z^n/n!) sum((-1)^n/(2n+1)! z^(2n+1)) sum(z^n) sum(z^(n+1)/(n+1))

The radii of convergence can clearly be seen in the last two examples tool by David Bau

+ - 🌍 ◰ ⚪ DAVIDBAU.COM ◌

Kwasi Blog Software

Improved method for storing normals as a series of bytes 2012-10-03 by Michael Kwaśnicki

GLSLOpenGL

This article describes a method to pack vertex normals into just three bytes using an approach that provides better results than just regular conversion. Why do this at all

Graphics processors (GPUs) are blazing fast nowadays and they can perform a huge amount of computation. A problem arises when it comes to provide data to keep the GPU busy. You just can’t feed it as quick as it computes the output. Especially on mobile platforms the memory bandwidth is a problem. So packing the data more densely helps to reduce the pressure on the memory bandwidth and allows to do more on the GPU. The naïve approach

Normals are unit vectors that point away from a surface. I’m using here the C language paradigm to explain the required steps. Typically they are constructed from three floating point numbers. Each component of this vector takes values of [-1, 1] thus converting them into bytes will be horrible. With the default round towards zero behavior the components will be zero for most cases. Thus they have to be scaled with UCHAR_MAX = 127 (defined in limits.h). It is also possible to call roundf() to round the floating point number prior to conversion but the result is still bound to the surface of a quantized sphere.

float x; byte bx = x * UCHAR_MAX;

That would be the straight forward approach. On the OpenGL side one needs to provide those byte normals for drawing. This can either be done by telling OpenGL to normalize the input, which converts the into float and then maps the value from [-128, 127] to [-1, 1]. (There is a change on that behavior in the latest OpenGL specification. Now it maps from [-127, 127] to [-1, 1] as the prior approach made it impossible to represent 0 exactly.) Anyway, this approach does not guarantee you to get normals of unit length as they are already shortened by the conversion. Or one tells OpenGL to just pass in the values as floats and call normalize() in the vertex shader to get unit length normals. The better approach

As calling normalize() gives better results, then why not use an input that is not limited to the surface of a sphere but does take advantage of the full byte space. So instead of believing that the byte representation of the normal as described above would be the best, one can pretty much find other points along the original normal in the byte cube that are much closer to the original normal even the length differs significantly. Those byte normals vary in length pretty much and cannot be used directly. So calling normalize() in the vertex shader is obligatory here. Illustrating the quantization of a direction (red) by the naïve round towards zero approach (green) and the new approach (yellow). The new approach comes much closer to the original direction after normalizing. Quality of the results

As we are not limited to the surface of a sphere, this approach certainly outperforms the naïve approach, which uses just a subset of byte combinations that are used by the better approach. As the 3D byte space has a size of 2563 there are 16777215 possible directions for normals (the null vector (0, 0, 0) is excluded here as it does not point anywhere). But not all directions are unique as (1, 0, 0) has the same direction as (127, 0, 0) along with all those in-between. There is a total of 3167541 ambiguities which leads to a total of 16777215 - 3167541 = 13609674 unique normal directions. Also the directions are not uniformly distributed. The more ambiguities a direction has, the worse the resolution. This implies that along the x-, y- and z-axis and also the diagonals with either x, y or z = 0 and the diagonals across the cube have the worst resolution. Quantifying the worst case

As the worst resolution is around the axis, we take a look on how far off a floating point normal can get. The furthest point would be half the way along a diagonal direction. So we compare the vectors (127, 0, 0) and (127, 1, 1). The angle between those is computed with atan(sqrt(2) / 127) and gives us 11.14 mrad or 0.638°. That means that any normal that is discretized this way differs at most by 5.568 mrad or 0.319° in its direction.

Actually this is also true for the naïve approach. But you can hardly get better there. Rendering of all possible normal directions as points on a sphere using an orthogonal projection with 6x super sampling. The naïve method (left) with relying on OpenGL's built in normalization. The naïve method (center) with calling normalize() in the vertex shader. The new approach (right). Drawbacks

While a regular normal occupies 12 bytes which is made up from three times the size of a float (4 bytes or 32 bits), the presented packing stores the same normal in just 3 bytes. Which is a saving factor of 4. But because of alignment requirements, it is given a penalty when transferring the normals as such. All data has to be aligned to 32 bits or 4 bytes. Therefore one has to add a padding of one byte to each normal and store it in 4 bytes instead of 3 bytes. But this last byte does not need to be unused. It can carry information for a different task in the vertex shader. Anyway this reduces our saving from factor 4 to factor 3. Conclusion

As the computation of those better unnormalized byte normals is much more expensive, it is primary intended for offline computation. But beyond that it offers way better results. Copyright © 2017 Michael Kwaśnicki. All rights reserved. ⚪ WEB.ARCHIVE.ORG ◌

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𑁍/🟗𑁍🟗/TXT.𖤞𑁍𖤞.TXT ADDED 𑁍/🟗𑁍🟗/ꓨИꟼ.𔗢 ⵔ፨ⵔ·ⵔ፨ⵔ 𐧾ꔹ𐧾 ❋𑗋❋ 🟗𑁍🟗 ᯽ ᯽ 🟗𑁍🟗 ❋𑗋❋ 𐧾ꔹ𐧾 ⵔ፨ⵔ·ⵔ፨ⵔ 𔗢.PNG filter=lfs diff=lfs merge=lfs -text 𑁍/🟗𑁍🟗/ꓨИꟼ.𔗢 ⵔ፨ⵔ·ⵔ፨ⵔ ⠿ꔹ⠿ ❋𑗋❋ 🟗𑁍🟗 ᯽ ᯽ 🟗𑁍🟗 ❋𑗋❋ ⠿ꔹ⠿ ⵔ፨ⵔ·ⵔ፨ⵔ 𔗢.PNG filter=lfs diff=lfs merge=lfs -text 𑁍/🟗𑁍🟗/𑪽ߖ.ꓨИꟼ.XHꓨ.𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢 𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢.GHX.PNG.7Z filter=lfs diff=lfs merge=lfs -text 𑁍/🟗𑁍🟗/XHꓨ.𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢 𔗢🟗𔗢𑁍𔗢🟗𔗢᯽𔗢🟗𔗢𑁍𔗢🟗𔗢.GHX filter=lfs diff=lfs merge=lfs -text 𖡼/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 ⩩/🝱/ꓨИꟼ.𔗢[[:space:]]⠀[[:space:]]𔗠[[:space:]]⠀[[:space:]]𐧾ꔹ𐧾[[:space:]]⠀[[:space:]]⋮⯐⋮[[:space:]]⠀[[:space:]]⠿𑗋⠿[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⩩[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᯽ ᯽[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]⩩[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]⠿𑗋⠿[[:space:]]⠀[[:space:]]⋮⯐⋮[[:space:]]⠀[[:space:]]𐧾ꔹ𐧾[[:space:]]⠀[[:space:]]𔗠[[:space:]]⠀[[:space:]]𔗢.PNG filter=lfs diff=lfs merge=lfs -text ⩩/🝱/ꓨИꟼ.𔗢[[:space:]]⠀[[:space:]]⠿ꔹ⠿[[:space:]]⠀[[:space:]]⁘⯐⁘[[:space:]]⠀[[:space:]]⋮𑗋⋮[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠀⬜⩩⬜⠀[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᯽ ᯽[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]⠀⬜⩩⬜⠀[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]⋮𑗋⋮[[:space:]]⠀[[:space:]]⁘⯐⁘[[:space:]]⠀[[:space:]]⠿ꔹ⠿[[:space:]]⠀[[:space:]]𔗢.PNG filter=lfs diff=lfs merge=lfs -text

𖡼/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 ⩩/🝱/ꓨИꟼ.𔗢[[:space:]]⠀[[:space:]]𔗠[[:space:]]⠀[[:space:]]𐧾ꔹ𐧾[[:space:]]⠀[[:space:]]⋮⯐⋮[[:space:]]⠀[[:space:]]⠿𑗋⠿[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⩩[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᯽ ᯽[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]⩩[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]⠿𑗋⠿[[:space:]]⠀[[:space:]]⋮⯐⋮[[:space:]]⠀[[:space:]]𐧾ꔹ𐧾[[:space:]]⠀[[:space:]]𔗠[[:space:]]⠀[[:space:]]𔗢.PNG filter=lfs diff=lfs merge=lfs -text ⩩/🝱/ꓨИꟼ.𔗢[[:space:]]⠀[[:space:]]⠿ꔹ⠿[[:space:]]⠀[[:space:]]⁘⯐⁘[[:space:]]⠀[[:space:]]⋮𑗋⋮[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠀⬜⩩⬜⠀[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᯽ ᯽[[:space:]]⠀[[:space:]]𖥕[[:space:]]⠀[[:space:]]᪣[[:space:]]⠀[[:space:]]⠀⬜⩩⬜⠀[[:space:]]⠀[[:space:]]🝱[[:space:]]⠀[[:space:]]⠿·⠿⬚⠿·⠿[[:space:]]⠀[[:space:]]⋮𑗋⋮[[:space:]]⠀[[:space:]]⁘⯐⁘[[:space:]]⠀[[:space:]]⠿ꔹ⠿[[:space:]]⠀[[:space:]]𔗢.PNG filter=lfs diff=lfs merge=lfs -text .gitattributes CHANGED

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Badernes sphériques

Sur des sujets voisins, voir les pages :

cercles-sphères Apollonius empilements apolloniens cercles tangents

Empilements apolloniens de cercles sur une sphère

1. BADERNES ETOILEES : Réitération d'inversions de pôles les sommets d'un polyèdre régulier

J'ai considéré un polyèdre régulier ( tétraèdre, cube, octaèdre ...) et la sphère S0 tangente à ses arêtes en leur milieu. Les faces coupent ainsi cette sphère selon des cercles égaux et tangents. Je considère les inversions Ci centrées aux sommets Ai du polyèdre et laissant globalement invariante la sphère S0.

Je fais subir à tous les cercles les inversions Ci (sauf ceux invariants par Ci ) : j'obtiens une nouvelle famille (F1) de cercles tracés sur la sphère S0. Je recommence l'opération avec (F1) et ainsi de suite (F2), ...

On obtient ainsi des pavages de la sphère par des "badernes étoilées" gauches.

2. BADERNES CIRCULAIRES : Réitération d'inversions de pôles sur les axes des faces d'un polyèdre régulier

J'ai considéré un polyèdre régulier ( tétraèdre, cube, octaèdre ...) et la sphère S0 tangente à ses arêtes en leur milieu. Les faces coupent ainsi cette sphère selon des cercles égaux et tangents. Je considère les inversions Ci dont les pôles sont sur les axes des faces du polyèdre et laissant globalement invariante la sphère S0.

Je procède ensuite comme pour la méthode 1 ...

On obtient ainsi des pavages de la sphère par des "badernes circulaires" gauches.( situées dans les grandes calottes des pavages 1)

Dans ce paragraphe, les images des seconde et quatrième colonnes des tableaux sont des images en relief à regarder avec des lunettes rouge-cyan.

Baderne étoilée du tétraèdre

Baderne circulaire du tétraèdre

Baderne étoilée du cube

Baderne circulaire du cube

Baderne étoilée de l'octaèdre

Baderne circulaire de l'octaèdre

Baderne étoilée du dodécaèdre

Baderne circulaire du dodécaèdre

Baderne étoilée de l'icosaèdre

Baderne circulaire de l'icosaèdre

Baderne étoilée du cuboctaèdre

Baderne circulaire du cuboctaèdre

Baderne étoilée du grand rhombicuboctaèdre

Baderne circulaire du grand rhombicuboctaèdre

3. Projection stéréographique inverse d'une baderne ou d'un empilement apollonien du plan sur une sphère

Cette méthode très simple à mettre en oeuvre est donc beaucoup plus riche puisque les empilements de cercles du plan sont très divers.

On peut ensuite faire subir diverses rotations sur la sphère à l'image de l'empilement ainsi obtenue.

Départ : baderne plane avec 4 cercles égaux + rotations

Départ : baderne plane avec 4 cercles égaux + rotations

Départ : baderne plane avec 4 cercles égaux + symétrie

Départ : badernes imbriquées + symétrie

Départ : badernes imbriquées + symétrie

Départ : empilement Apollonien dans un carré + symétrie

Début