Circle Limit 4 -- Heaven and Hell by M.C.Escher 1960

Escher's work represents a projection of creatures "living" in a 2-dimensional space, which may or may not be flat, onto the page, which is definitely flat. The creatures are all the same size in their own world - the apparent change in size of the angels and devils in Heaven and Hell is an artifact of the projection onto the flat page, like the distortion of the size of Greenland on a Mercator map in an atlas.
  1. Pick an angel and label the tip of the angel's feet, and the two tips of her wings. Figure out the angle at each of these three points by taking 360 degrees and dividing by the total number of angels and devils that meet at that point. For example, at the feet coming into the center of the disk, there are 6 angels and devils coming into that point and so the angle at that point is 360/6 = 60 degrees. What is the angle at the tips of her wings? What is the sum of the three angles?
  2. Lines should preserve symmetry, so they should cut creatures in half (like mirror reflections). Use this idea to draw some "lines" in this space. Start by drawing mirror "lines" through the center of an angel that cuts her in half and continue these mirrors in both directions through other creatures. Draw at least five "lines".
  3. In perspective drawing, lines that are parallel in the real world instersect in a vanishing point. In spherical geometry there are no parallel lines (ie there are lines---great circles, but they are not parallel). In Escher's work, parallels behave differently. We can find more than one parallel to a given "line" through a point off of the "line" - do you see how?