The works of M.C. Escher
At first, I thought the works of Escher were just a collection of magnificent drawings that keeps your attention, but there's a line in his works; it can be categorised (he did that as well). Most of the pictures I mention here can be viewed online at the gallery and here, which may be helpful to understand the explanations below if you're not familiar with his works.
- 3D structures
- Landscapes; mainly created during his life in Italy. Some of these paintings and drawings were used in a later stage (like in the Picture Gallery).
- penetration of worlds; the Hand with the sphere as a mirror. Noteworthy is, that this implies that he assumed that there's more than one world, or, and more likely when you take his other works into account, one world that can be seen from different perspectives, that it seems to be a different world and that he tried to capture the various points of view all at the same time.
- abstract mathematical things; like Stars and [crystal] shapes (his brother was a geologist, and Escher made all illustrations for the book he wrote). All figures are based on the five regular shapes: the tetraeder (4 sides), cubicle (6 sides), octaeder (8), dodecaeder (12) and ikosaeder (20). A picture like Gravity was constructed by placing five-sided pyramids on each side of a dodecaeder.
- metamorphose; examples is Day and Night (but this may be interpreted as a cyclus too) and the Magic Mirror, but the most famous one is Metamorphose II.
- cycli; e.g. the reptiles walking out of the paper. The main idea behind this that there's no beginning and no end, like with the metamorphosis there's this transition between stages.
- infinity; they appear to me like what we call fractals nowadays. In the first stage the infinity of the figures went inside into the picture, but later (with Circle Limit I, II and III) outwards (or ideas about the infinity of the universe maybe?). Those pictures can be generated with a click of a mouse now, he calculated and draw it all by hand!!!
- The relation between 2D and 3D
- conflict 2D-3D; see the Dragon, it was constructed by drawing a dragon, cut out, two holes were made in the paper and this construction was drawn again. In other words, he used 3D drawing techniques to emphasize the 2D of the initial object.
- perspective; Escher used combinations of zeniths, nadirs and vanishing points to create e.g. Relativity and Other World I and II (see picture at the left). In Up and Down, the zenith of the lower part of the picture is the nadir of the upper part at the same time. Remarkably, the very same principle has been used for Stair Well with the "wentelteefjes" (the millipede-like creatures; see picture at the right hand side).
- impossible figures; the best know are Belvedere (based on the magicbox from Dr. Cochran), Waterfall (based on the triangle form R. Penrose) and Upstairs Downstairs. Cool thing about the latter is, that the stairs are flat, but you are fooled by the lines of the rest of the building where the architecture is not realistic (ok, ok, it probably is possible to build it with some extra supporting buttresses, but will be extremely uncomfortable to live in). Another one is Concave and Convex: the left side of the picture is convex (with the apparent nadir bottom-left) and the right side concave (with the apparent zenith top-right), whereas the middle is in sort of a transition state.
Chronologically, the above can be divided into 4 distinct periods: 1922-'37 Landscapes, '37-'45 metamorphosis, '46-'56 perspective and '56-'70 to infinity.
Anyway, there are many more pictures than the ones I mentioned that may not seem to be like the ones I picked, nevertheless those ones do have the same mathematical basis (btw, Escher wasn't a mathematician).
There's a very good book that explains how he did it, called "Der Zauberspiegel des M.C. Escher", (translated into the Magic Mirror of M.C. Escher), which outlines his thought processes and shows the working drawings and models for his final works. E.g. the Picture Gallery (shown left) is "just" based on standard graph paper with some exploded and imploded graph lines...