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Escher and the Droste Effect
The goal of this site is to visualize the mathematical structure behind M.C. Escher's picture called "Print Gallery" (1956). The visualization itself is largely non-mathematical and is accomplished through many still images and animations. The actual mathematics, involving conformal mappings of the complex plane, is contained in a pdf copy of the original AMS publication. The Droste Effect refers to any image that contains itself on a smaller scale.
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Fidel Calvillo (Staff)
In 1956 the Dutch graphic artist Maurits Cornelis
Escher (1898–1972) made an unusual lithograph
with the title Prentententoonstelling. It shows a young man standing in an exhibition gallery, viewing a print of a Mediterranean seaport. As his eyes follow the quayside buildings shown on the print from left to right and then down, he discovers among them the very same gallery in which he is standing. A circular white patch in the middle of the lithograph contains Escher’s monogram and signature. It amazed me how he did design of the white patch in the middle, before i read the the material i would think he just painted it. I like the mathematics he involved in the piece. I like the idea of the mathematics used on the picture becuase it can be viewed as drawn on a certain elliptic
curve over the field of complex numbers and
deduce that an idealized version of the picture repeats itself in the middle. More precisely, it contains a copy of itself, rotated clockwise by
157.6255960832. . . degrees and scaled down by a
factor of 22.5836845286. Its amazing how we can put an idea that will make it possible to make an annular bulge "a cyclic expansion without beginning or end". Another good example is the way one transforms a single closed loop, counterclockwise around the center, from
the curved world to the straight world. It reflects the invariance of the straight picture under a blowup by a factor of 256. I like the idea how such phenomenon takes place if we do not walk around the center of the piece. The idea that we have the factor of 256 will take us to make something so beautiful like the pictures that escher has done. Another idea and figure that amazed me was figure A5x5 square transformed to the curved world, i like it because the lines line up straight with each other and we can still make the curves on the picture event though the lines themselve dont curve. So every thing that escher done gave me an idea in how art work can look better than if it was just drawn on the paper without making some kind of chart or graph and mathematics.
Time spent reviewing site: I spent about four hours reading the material and observing the paper and iformation. I spent about one hour writing the paper.
Tim Riessen (Staff)
As the aim does nto appear to be a full blown mathematical treatise of how Escher worked, but to provide a case study, the site works well in this regard. With the focus on a single piece of artwork, the audience is likely to be late high school or first year undergrads, as classroom based review of Echers process. For more in depth study of Escher, as might happen in later years of undergrad programs, or as part of specfiic unit of study, this would form only one part of a larger set of resources. It would seem that the abbreviated content is deliberate, as a way of making a short sharp session on this specific picture. It could then used as a lead in to a classroom activity to use the grid for students to develop heir own "Escher" picture.
It is fairly easy to use from a teacher perspective. It may be good to have copies of the grid available in hard copy before using the site in a classroom session. Its focus on specific steps for this one pictures would make it straighforward for a non-specialist to use. Although clearly, haveing greater background knowledge of the content would be an advantage.
The navigation on the left hand side follows a consistent format of highliting current pages, and providing a heirarchy of sub sections. This provides an easy understandiing of location within this web site.
Technical Remarks:
Pages load fairly quickly. Animations assume relevant players, for example QT. It may be advisable to have other formats, for example flv or swf.
Some pages have links to external sites, (not explored here).