轉載:愛因斯坦密鋪

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轉載:愛因斯坦密鋪

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摺法:愛因斯坦密鋪

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Re: 轉載:愛因斯坦密鋪

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Re: 轉載:愛因斯坦密鋪

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

發現愛因斯坦地磚
37a78138-b545-40ef-b14f-65fc13f2a4d5.jpg
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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。
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Re: 轉載:愛因斯坦密鋪

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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)
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PolyForm Puzzle Solver

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Re: 轉載:愛因斯坦密鋪

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Re: 轉載:愛因斯坦密鋪

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Re: 轉載:愛因斯坦密鋪

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帽子和烏龜也可以組合密鋪

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Re: 轉載:愛因斯坦密鋪

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Screenshot from 2024-07-15 20-44-51.png
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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。
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Re: 轉載:愛因斯坦密鋪

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找找規律……
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Re: 轉載:愛因斯坦密鋪

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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

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Re: 轉載:愛因斯坦密鋪

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這個難道是「烏龜」的改版?
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Re: 轉載:愛因斯坦密鋪

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代码: 全选

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