圈圈之地

https://origamiusa.org/thefold/article/ ... cal-puzzle

: thefold76_41Hats_shen.jpg (239.53 KiB) 查看 2231 次

: thefold76_two_tiles_shen.jpg (183.46 KiB) 查看 2231 次

thefold76_einstein_hat_chen.pdf: (118.37 KiB) 已下载 1 次

https://sites.google.com/a/g2.nctu.edu. ... 到名為愛因斯坦的地磚

https://www.scitw.cc/posts/news259-5

發現愛因斯坦地磚

: 37a78138-b545-40ef-b14f-65fc13f2a4d5.jpg (535.89 KiB) 查看 2213 次

影像來源／David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss (CC BY-SA 4.0)

英國約克郡的數學愛好者史密斯（David Smith）發現了數學家苦尋幾十年未果的一種13邊形。這個崎嶇如帽子般的形狀稱為「愛因斯坦」（einstein，源自於德語ein Stein，意思是一塊石頭）。如果拿愛因斯坦形狀的磁磚來鋪浴室地板（或任何平面，即使是無限大），它們能完美砌合，但永遠不會形成重複的圖案。幾十年來，數學家一直在找這種磚片的形狀，它們只會形成非重複的排列，稱為非週期密鋪（aperiodic tiling）。他們先從許多不同磚片的組合開始：1964年發現第一組，需要2萬426個不同的磚片，後來簡化到103個。到了1974年，數學家彭若斯（Roger Penrose）發現了兩個磚片形狀，當它們組合成馬賽克時，永遠不會形成重複的圖案。

但是，能否只用一種形狀的磚片（即假想的愛因斯坦地磚）來形成非週期密鋪？

專精於鑲嵌的美國摩拉維亞大學退休數學家夏特史奈德（Doris Schattschneider）對於發現真正愛因斯坦地磚的可能性，長期抱持懷疑態度。她說：「這件事令人震驚，因為不只是發現了真正的愛因斯坦地磚，而且形狀還是如此簡單。對我來說，這完全反常。」去年11月，史密斯使用PolyForm Puzzle Solver 這款軟體去實驗不同的形狀時發現了這塊磚片，其優雅程度令人驚歎。它是由直角的箏形所組成，完全不像許多數學家預測的那種扭曲、複雜的形狀。

科學棋談：關於「愛因斯坦地磚」的補充

https://sci-story.com/3968/

關於「愛因斯坦地磚」的補充

張瑞棋
January 4, 2024
瀏覽次數：143

沒想到我在〈2023年十大科學新聞〉介紹的「愛因斯坦地磚」，竟然吸引那麼多人的關注。不過由於還有其它幾則科學新聞要同時並列，所以我只做了概要的介紹，有些重要關鍵就略而不提。這裡我就進一步補充說明，讓有興趣的人可以知道大概的來龍去脈。

今年64歲的史密斯是在2022年11月發現後來暱稱為「帽子」的13邊形，它看起來奇形怪狀，但其實是由8個一模一樣的箏形所組成，而這箏形是將一個正六邊形均分成六等份而來（如下圖）。
圖片來源：Scientific American

史密斯印了一堆「帽子」出來，在桌上一個個拼接起來，拚了很久，發現都沒有出現重複的圖案，不禁猜想莫非真的被他找到了愛因斯坦地磚？不過自己這樣土法煉鋼也不是辦法，誰知道範圍更大後會不會開始出現某種規律？
史密斯列印出愛因斯坦地磚，在桌面上進行拼接。
圖片來源：Quantum Magazine

加拿大滑鐵盧大學的卡普蘭教授（Craig Kaplan）是密鋪（Tessellation，也就是用幾個幾何圖形不留縫隙地鋪滿整個平面）的專家，他還自己寫了密鋪的電腦程式。於是史密斯寫email給卡普蘭，請他幫忙驗證帽子是不是愛因斯坦地磚？

卡普蘭用電腦跑了半天也都沒看到週期性，不過就算讓電腦跑一萬年，也無法就此證明這就是愛因斯坦地磚，必須從數學上加以證明才行。卡普蘭在2023年1月找了兩名對密鋪也有研究的幫手，一位是專精凸幾何的史特勞斯（Chaim Goodman-Strauss）教授，另一位是兼具組合數學與軟體開發專長的邁爾斯（Joseph Myers）博士。

2月時，他們已經找到方法證明帽子的確可以鋪滿平面而不具週期性，就在卡普蘭著手寫成論文時，邁爾斯又有了新發現。

原來去年12月史密斯曾email另一個像是烏龜的13邊形給卡普蘭，問他這個會不會也是愛因斯坦地磚？「烏龜」也是用箏形組成，不過比帽子多了兩片（如下圖）。卡普蘭心想帽子都還沒搞定，哪有閒工夫再研究另一個烏龜，何況有可能半世紀來數學家都找不到的愛因斯坦地磚，一個業餘玩家在兩個禮拜內就發現兩個嗎？便將它暫擱一旁。
圖片來源：scilogs.spektrum.de

如今帽子的證明搞定了，卡普蘭繼續忙著寫論文，得空的邁爾斯覺得不妨檢視一下烏龜，結果他從兩者的共通性發現另一種方法，可以證明帽子和烏龜都是愛因斯坦地磚。這不但強化了他們預備發表的論文，而且第二種證明方法未來也可用來檢驗其它幾何形狀。

他們四人的論文發表後立刻轟動學界，不過熱烈迴響之中卻也伴隨著一項質疑。帽子和烏龜固然能沒有規律地密鋪整個平面，卻需摻雜著一些反面，也就是本身的鏡像（如下圖）。雖然在數學上，鏡射和平移、旋轉一樣，都被視為並未改變幾何形狀，因此帽子和烏龜的確算是愛因斯坦地磚，但還是有許多人提出質疑，認為應該不能用到鏡像才算數。
用帽子密鋪而成的圖案沒有週期性，其中紅色是白色、黃色與橘色的形狀都一樣，紅色則是它們的鏡像。
圖片來源：Wikipedia

這時史密斯又想出另一個形狀，看起來像是烏龜和帽子的混合體，也是13邊形，不過這次的組成不是箏形（如下圖）。這個形狀也可以做為愛因斯坦地磚，更讚的是完全不需要用到反面。
史密斯的第三個發現是「準愛因斯坦地磚」，除了唯一的長邊是2，其餘每個邊長都是1。
圖片來源：David Smith’s blog

只不過有個問題，如果搭配反面，反而可以排出規律的圖案（如下圖）。然而若按嚴格定義，愛因斯坦地磚只能有非週期性的排法，不允許還有規律的排法。

他們四人再次合作，很快地將找出方法，將直線的邊修改為曲線，如此便無法用正反面排出規律圖案，但仍能非週期性地密鋪整個平面（如頁首）。他們於五月發表論文，這個暱稱為「幽靈」(spectres) 的愛因斯坦地磚終於不會有人質疑了。

參考資料：

Inside Mathematicians’ Search for the Mysterious ‘Einstein Tile’ | Scientific American
Hobbyist Finds Math’s Elusive ‘Einstein’ Tile | Quanta Magazine
Elusive ‘Einstein’ Solves a Longstanding Math Problem – The New York Times (nytimes.com)
The Special One – hedraweb – David Smith (wordpress.com)

https://www.jaapsch.net/puzzles/polysolver.htm

https://arxiv.org/abs/2303.10798

論文。

https://hedraweb.wordpress.com/2023/06/ ... ecial-one/

史密斯唯一認證博客。

: polykite_2-1.png (608.27 KiB) 查看 2033 次

: polykite_10b.png (779.97 KiB) 查看 2033 次

: tutrtle-dominat.png (116.22 KiB) 查看 2011 次

: hat-dominat.png (125.14 KiB) 查看 2011 次

: Screenshot from 2024-07-15 20-44-51.png (93.61 KiB) 查看 1908 次

polyform solver 其實之前我也玩過，感覺有點難玩，操作不是很順手。同時，帽子的密鋪有點難，需要先掌握規律才好下手。

: Screenshot from 2024-07-15 21-02-09.png (188.98 KiB) 查看 1907 次

找找規律……

寡人準備為此製作一個密鋪程式。

功能描述：

平鋪平移主界面

旋轉及選取貼點

反覆

換塊

取色選色

備注

undo，redo

上傳，下載

這個難道是「烏龜」的改版？

: einstein-shapes.jpg (209.34 KiB) 查看 1300 次

: 2024_09_29_20.52.28.png (20.94 KiB) 查看 1074 次

代码：全选

<svg id="etmain" version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" width="100%" height="480"> <defs> <path id="tile1" fill="#ececec" d="M 0 15.588 L -9 31.176 L -27 31.176 L -45 31.176 L -54 15.588 L -27 0 L -27 -31.176 L -9 -31.176 L 0 -46.764 L 27 -31.176 L 27 0 L 45 0 L 54 15.588 L 27 31.176 Z"></path> <path id="tile2" fill="#54b0e8" d="M -27 31.176 L -54 15.588 L -45 0 L -27 0 L -27 -31.176 L 0 -46.764 L 9 -31.176 L 27 -31.176 L 27 0 L 54 15.588 L 45 31.176 L 27 31.176 L 9 31.176 L 0 15.588 Z"></path> <path id="tile3" fill="#f9d98d" d="M 0 31.177 L -27 46.765 L -36 31.177 L -27 15.588 L -54 0 L -54 -31.177 L -36 -31.177 L -27 -15.588 L 0 -31.177 L 27 -15.588 L 54 0 L 54 31.177 L 36 31.177 L 27 46.765 Z"></path> <path id="tile4" fill="#d25e5e" d="M -27 46.765 L -36 31.177 L -54 31.177 L -54 0 L -27 -15.588 L 0 -31.177 L 27 -15.588 L 36 -31.177 L 54 -31.177 L 54 0 L 27 15.588 L 36 31.177 L 27 46.765 L 0 31.177 Z"></path> </defs> <g id="etdrop" xy="-192.077,121.421" scale="0.76" transform="translate(-192.077, 121.42099999999999) scale(0.76)"><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(132, 154) rotate(0)" class="tiledroped" xy="132,154" r="0"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(78, 122.824) rotate(-180)" class="tiledroped" xy="78,122.824" r="180" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(91.5, 68.265) rotate(-120)" class="tiledroped" xy="91.5,68.265" r="-120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(37.501, 161.793) rotate(-120)" class="tiledroped" xy="37.501,161.793" r="-120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(-16.497, 192.969) rotate(-120)" class="tiledroped" xy="-16.497,192.969" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(145.498, 37.089) rotate(-120)" class="tiledroped" xy="145.498,37.089" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(172.5, 99.441) rotate(-60)" class="tiledroped" xy="172.5,99.441" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(24, 60.472) rotate(180)" class="tiledroped" xy="24,60.472" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(-16.498, 115.028) rotate(120)" class="tiledroped" xy="-16.498,115.028" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(78, 216.352) rotate(180)" class="tiledroped" xy="78,216.352" r="180" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(145.501, 224.147) rotate(-120)" class="tiledroped" xy="145.501,224.147" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(10.502, 255.323) rotate(-60)" class="tiledroped" xy="10.502,255.323" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(64.5, 286.499) rotate(-60)" class="tiledroped" xy="64.5,286.499" r="-60" n="1"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(118.499, 302.088) rotate(-120)" class="tiledroped" xy="118.499,302.088" r="-120" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(172.5, 286.501) rotate(-60)" class="tiledroped" xy="172.5,286.501" r="-60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(226.499, 239.736) rotate(60)" class="tiledroped" xy="226.499,239.736" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(239.999, 122.823) rotate(180)" class="tiledroped" xy="239.999,122.823" r="180"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(293.998, 154) rotate(0)" class="tiledroped" xy="293.998,154" r="0" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(334.498, 115.03) rotate(60)" class="tiledroped" xy="334.498,115.03" r="60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.496, 83.854) rotate(60)" class="tiledroped" xy="388.496,83.854" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(347.997, 216.353) rotate(0)" class="tiledroped" xy="347.997,216.353" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.497, 161.795) rotate(-60)" class="tiledroped" xy="388.497,161.795" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(226.496, 52.678) rotate(60)" class="tiledroped" xy="226.496,52.678" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(293.999, 60.471) rotate(0)" class="tiledroped" xy="293.999,60.471" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(78, -1.882) rotate(0)" class="tiledroped" xy="78,-1.882" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(118.5, -56.441) rotate(-60)" class="tiledroped" xy="118.5,-56.441" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(172.498, -25.264) rotate(-60)" class="tiledroped" xy="172.498,-25.264" r="-60"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(226.495, -9.675) rotate(-120)" class="tiledroped" xy="226.495,-9.675" r="-120" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(280.5, -25.264) rotate(-60)" class="tiledroped" xy="280.5,-25.264" r="-60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(334.497, 5.912) rotate(-60)" class="tiledroped" xy="334.497,5.912" r="-60"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.495, 21.501) rotate(-120)" class="tiledroped" xy="388.495,21.501" r="-120" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(442.495, 5.913) rotate(-60)" class="tiledroped" xy="442.495,5.913" r="-60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(496.493, 37.089) rotate(-60)" class="tiledroped" xy="496.493,37.089" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(415.493, -56.44) rotate(-120)" class="tiledroped" xy="415.493,-56.44" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(347.999, -64.235) rotate(-180)" class="tiledroped" xy="347.999,-64.235" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(455.995, 91.648) rotate(0)" class="tiledroped" xy="455.995,91.648" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(455.996, 154.001) rotate(0)" class="tiledroped" xy="455.996,154.001" r="0" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(415.495, 239.736) rotate(120)" class="tiledroped" xy="415.495,239.736" r="120"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(469.495, 208.559) rotate(-60)" class="tiledroped" xy="469.495,208.559" r="-60" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(509.994, 247.53) rotate(0)" class="tiledroped" xy="509.994,247.53" r="0" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(550.493, 192.97) rotate(-59.99999999999999)" class="tiledroped" xy="550.493,192.97" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(523.492, 130.619) rotate(-120)" class="tiledroped" xy="523.492,130.619" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(442.494, 302.089) rotate(60)" class="tiledroped" xy="442.494,302.089" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(293.999, 278.707) rotate(180)" class="tiledroped" xy="293.999,278.707" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(361.497, 286.5) rotate(-120)" class="tiledroped" xy="361.497,286.5" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(226.498, 317.677) rotate(-60)" class="tiledroped" xy="226.498,317.677" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(293.999, 341.059) rotate(-180)" class="tiledroped" xy="293.999,341.059" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.495, 348.854) rotate(-60)" class="tiledroped" xy="388.495,348.854" r="-60"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(334.498, 380.03) rotate(120)" class="tiledroped" xy="334.498,380.03" r="120" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(347.994, 434.589) rotate(180)" class="tiledroped" xy="347.994,434.589" r="180" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(253.496, 395.618) rotate(120)" class="tiledroped" xy="253.496,395.618" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(280.495, 457.971) rotate(60)" class="tiledroped" xy="280.495,457.971" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(118.499, 364.441) rotate(60)" class="tiledroped" xy="118.499,364.441" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(185.999, 372.236) rotate(0)" class="tiledroped" xy="185.999,372.236" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(253.493, -87.616) rotate(-120)" class="tiledroped" xy="253.493,-87.616" r="-120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(186, -95.412) rotate(-180)" class="tiledroped" xy="186,-95.412" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(415.493, 426.795) rotate(120)" class="tiledroped" xy="415.493,426.795" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(455.994, 372.236) rotate(180)" class="tiledroped" xy="455.994,372.236" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(509.994, 309.882) rotate(0)" class="tiledroped" xy="509.994,309.882" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(199.499, 177.382) rotate(120)" class="tiledroped" xy="199.499,177.382" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(280.497, 208.56) rotate(60)" class="tiledroped" xy="280.497,208.56" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(199.497, 442.383) rotate(-120)" class="tiledroped" xy="199.497,442.383" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(131.999, 434.588) rotate(-180)" class="tiledroped" xy="131.999,434.588" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(347.994, 496.941) rotate(180)" class="tiledroped" xy="347.994,496.941" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(577.493, 270.912) rotate(120)" class="tiledroped" xy="577.493,270.912" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(415.493, 504.737) rotate(-120)" class="tiledroped" xy="415.493,504.737" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(523.494, 380.031) rotate(-120)" class="tiledroped" xy="523.494,380.031" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(577.494, 348.853) rotate(-120)" class="tiledroped" xy="577.494,348.853" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(604.492, 411.208) rotate(-60)" class="tiledroped" xy="604.492,411.208" r="-60"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(509.994, 434.59) rotate(-180)" class="tiledroped" xy="509.994,434.59" r="180" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(469.492, 473.56) rotate(-120)" class="tiledroped" xy="469.492,473.56" r="-120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(563.991, 465.767) rotate(0)" class="tiledroped" xy="563.991,465.767" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(509.992, 528.12) rotate(180)" class="tiledroped" xy="509.992,528.12" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(577.491, 535.914) rotate(-120)" class="tiledroped" xy="577.491,535.914" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(37.501, 348.853) rotate(-120)" class="tiledroped" xy="37.501,348.853" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(64.5, 411.206) rotate(-60)" class="tiledroped" xy="64.5,411.206" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(91.499, 489.147) rotate(120)" class="tiledroped" xy="91.499,489.147" r="120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(145.498, 520.323) rotate(120)" class="tiledroped" xy="145.498,520.323" r="120"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(199.498, 504.736) rotate(60)" class="tiledroped" xy="199.498,504.736" r="60" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(253.495, 520.324) rotate(120)" class="tiledroped" xy="253.495,520.324" r="120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(307.494, 551.501) rotate(120)" class="tiledroped" xy="307.494,551.501" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(361.493, 582.676) rotate(120)" class="tiledroped" xy="361.493,582.676" r="120"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(415.494, 567.09) rotate(60)" class="tiledroped" xy="415.494,567.09" r="60" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(469.492, 582.679) rotate(120)" class="tiledroped" xy="469.492,582.679" r="120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(523.491, 613.855) rotate(120)" class="tiledroped" xy="523.491,613.855" r="120"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(577.492, 598.267) rotate(60)" class="tiledroped" xy="577.492,598.267" r="60" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(631.49, 613.856) rotate(120)" class="tiledroped" xy="631.49,613.856" r="120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(685.488, 645.032) rotate(120)" class="tiledroped" xy="685.488,645.032" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(550.49, 676.208) rotate(60)" class="tiledroped" xy="550.49,676.208" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(617.99, 684.003) rotate(0)" class="tiledroped" xy="617.99,684.003" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.492, 645.029) rotate(60)" class="tiledroped" xy="388.492,645.029" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(455.992, 652.826) rotate(0)" class="tiledroped" xy="455.992,652.826" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(172.497, 582.676) rotate(60)" class="tiledroped" xy="172.497,582.676" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(239.995, 590.471) rotate(0)" class="tiledroped" xy="239.995,590.471" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(307.494, 629.443) rotate(-120)" class="tiledroped" xy="307.494,629.443" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(496.492, 707.384) rotate(60)" class="tiledroped" xy="496.492,707.384" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(334.493, 691.794) rotate(-60)" class="tiledroped" xy="334.493,691.794" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(401.992, 715.176) rotate(180)" class="tiledroped" xy="401.992,715.176" r="180"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(455.993, 746.354) rotate(0)" class="tiledroped" xy="455.993,746.354" r="0" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(442.493, 800.913) rotate(60)" class="tiledroped" xy="442.493,800.913" r="60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(388.495, 832.089) rotate(60)" class="tiledroped" xy="388.495,832.089" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(361.491, 769.735) rotate(120)" class="tiledroped" xy="361.491,769.735" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(509.993, 808.706) rotate(0)" class="tiledroped" xy="509.993,808.706" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(550.49, 754.15) rotate(-60)" class="tiledroped" xy="550.49,754.15" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(550.491, 68.265) rotate(-60)" class="tiledroped" xy="550.491,68.265" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(604.492, 146.205) rotate(60)" class="tiledroped" xy="604.492,146.205" r="60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(617.992, 216.352) rotate(-180)" class="tiledroped" xy="617.992,216.352" r="180"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(604.49, 83.854) rotate(-120)" class="tiledroped" xy="604.49,83.854" r="-120" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(658.49, 68.266) rotate(-59.99999999999999)" class="tiledroped" xy="658.49,68.266" r="-60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(712.488, 99.442) rotate(-60)" class="tiledroped" xy="712.488,99.442" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(671.991, 153.999) rotate(0)" class="tiledroped" xy="671.991,153.999" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(631.491, 302.088) rotate(120)" class="tiledroped" xy="631.491,302.088" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(685.492, 224.146) rotate(-120)" class="tiledroped" xy="685.492,224.146" r="-120"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(685.491, 286.5) rotate(60)" class="tiledroped" xy="685.491,286.5" r="60" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(658.492, 364.442) rotate(60)" class="tiledroped" xy="658.492,364.442" r="60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(739.49, 302.088) rotate(120)" class="tiledroped" xy="739.49,302.088" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(793.488, 333.264) rotate(120)" class="tiledroped" xy="793.488,333.264" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(725.991, 372.236) rotate(0)" class="tiledroped" xy="725.991,372.236" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(671.991, 434.59) rotate(180)" class="tiledroped" xy="671.991,434.59" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(631.49, 489.149) rotate(120)" class="tiledroped" xy="631.49,489.149" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(658.489, 551.503) rotate(60)" class="tiledroped" xy="658.489,551.503" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(712.489, 520.326) rotate(60)" class="tiledroped" xy="712.489,520.326" r="60" n="1"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(725.991, 465.766) rotate(0)" class="tiledroped" xy="725.991,465.766" r="0" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(766.49, 426.795) rotate(60)" class="tiledroped" xy="766.49,426.795" r="60" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(820.487, 395.617) rotate(60)" class="tiledroped" xy="820.487,395.617" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(820.489, 473.56) rotate(-60)" class="tiledroped" xy="820.489,473.56" r="-60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(779.988, 528.12) rotate(0)" class="tiledroped" xy="779.988,528.12" r="0"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(725.989, 590.473) rotate(-180)" class="tiledroped" xy="725.989,590.473" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(631.488, 5.913) rotate(-120)" class="tiledroped" xy="631.488,5.913" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(563.992, -1.882) rotate(-180)" class="tiledroped" xy="563.992,-1.882" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(496.494, -40.852) rotate(60)" class="tiledroped" xy="496.494,-40.852" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(739.486, 177.383) rotate(120)" class="tiledroped" xy="739.486,177.383" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(766.49, 239.735) rotate(60)" class="tiledroped" xy="766.49,239.735" r="60"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(253.495, 660.618) rotate(-120)" class="tiledroped" xy="253.495,660.618" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(185.997, 652.823) rotate(-180)" class="tiledroped" xy="185.997,652.823" r="180"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(293.993, 746.353) rotate(0)" class="tiledroped" xy="293.993,746.353" r="0"></use><use href="#tile2" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(239.995, 715.177) rotate(-180)" class="tiledroped" xy="239.995,715.177" r="180" n="2"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(199.494, 754.146) rotate(-120)" class="tiledroped" xy="199.494,754.146" r="-120" n="1"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(145.496, 785.322) rotate(-120)" class="tiledroped" xy="145.496,785.322" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(145.497, 707.382) rotate(120)" class="tiledroped" xy="145.497,707.382" r="120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(307.497, 816.5) rotate(-120)" class="tiledroped" xy="307.497,816.5" r="-120"></use><use href="#tile1" fill="#fcfcfc" stroke="#777" stroke-width="2" transform="translate(239.993, 808.705) rotate(-180)" class="tiledroped" xy="239.993,808.705" r="180"></use></g> </svg>

圈圈之地

轉載：愛因斯坦密鋪

轉載：愛因斯坦密鋪

摺法：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

PolyForm Puzzle Solver

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

帽子和烏龜也可以組合密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪

Re: 轉載：愛因斯坦密鋪