Print Gallery 《畫廊》

愛藝

Print Gallery is a lithograph （31.7 cm × 31.9 cm） printed in 1956 by the Dutch artist M. C. Escher.

It depicts a man in a gallery viewing a print of a seaport， and among the buildings in the seaport is the very gallery in which he is standing， making use of the Droste effect1 with visual recursion2. The lithograph has attracted discussion in both mathematical and artistic contexts.

Bruno Ernst3 cites M. C. Escher as stating that he began Print Gallery “from the idea that it must be possible to make an annular bulge， a cyclic expansion... without beginning or end.” Escher attempted to do this with straight lines， but intuitively switched to using curved lines which make the grid expand greatly as it rotates.

In the book G?del， Escher， Bach4， Douglas Hofstadter explains it as a strange loop showing three kinds of “in-ness”： the gallery is physically in the town （“inclusion”）; the town is artistically in the picture （“depiction”）; the picture is mentally in the person （“representation”）5.

Escher’s signature is on a circular void in the center of the work. In 2003， two Dutch mathematicians， Bart de Smit and Hendrik Lenstra， reported a way of filling in the void by treating the work as drawn on an elliptic curve over the field of complex numbers6. They deem an idealized version of Print Gallery to contain a copy of itself （the Droste effect）， rotated clockwise by about 157.63 degrees and shrunk by a factor of about 22.58. Their website further explores the mathematical structure of the picture.

《畫廊》是荷蘭畫家M.C. 埃舍爾創(chuàng)作的一幅平版印刷畫（31.7厘米×31.9厘米），首印于1956年。

該畫描繪了一個(gè)男子在畫廊中觀看一幅描繪海港的版畫，畫面呈現(xiàn)的海港建筑中就有男子所處的這家畫廊。埃舍爾創(chuàng)作時(shí)利用了視覺遞歸的德羅斯特效應(yīng)。這幅平版印刷畫在數(shù)學(xué)和藝術(shù)領(lǐng)域都引發(fā)了討論。

布魯諾·恩斯特引用埃舍爾本人的話說，他創(chuàng)作《畫廊》是“源于這樣一個(gè)念頭，即肯定可以畫出一個(gè)環(huán)狀凸起，一種循環(huán)擴(kuò)展……沒有起點(diǎn)也沒有終點(diǎn)”。埃舍爾試圖用直線來呈現(xiàn)，但又跟隨直覺改用曲線，這使畫中的網(wǎng)格在旋轉(zhuǎn)時(shí)可以極大擴(kuò)展。

在《哥德爾、艾舍爾、巴赫》一書中，侯世達(dá)將此畫詮釋為一個(gè)怪圈，展示出三種“X在Y中”的情景：畫廊以實(shí)體形式存在于小鎮(zhèn)中（“包含”）;小鎮(zhèn)經(jīng)藝術(shù)加工存在于圖畫中（“描繪”）;圖畫因引發(fā)思考而存在于觀畫者大腦中（“表征”）。

埃舍爾的簽名位于此畫中央的圓形空當(dāng)中。2003年，兩位荷蘭數(shù)學(xué)家巴特·德斯密特和亨德里克·倫斯特拉報(bào)稱找到了一種填充空當(dāng)?shù)姆椒ǎ窗汛水嬁醋髟趶?fù)數(shù)域上繪制的橢圓曲線。他們認(rèn)為，理想的《畫廊》應(yīng)包含一個(gè)自身的復(fù)制品（根據(jù)德羅斯特效應(yīng)），順時(shí)針旋轉(zhuǎn)約157.63度，縮小約22.58%。他們在自己的網(wǎng)站上進(jìn)一步探索了此畫的數(shù)學(xué)結(jié)構(gòu)。