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

Many of Escher's works represent 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 choose 3 vertices -- the tip of the angel's feet, and the 2 tips of her wings. Figure out the angle at each vertex by taking 360 degrees and dividing by the total number of angels and devils that meet at that vertex. 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. Hold down the control key (ctrl), click on the picture with your mouse, and then release on Copy Image. Open up Sketchpad and under Edit, release on Paste Picture. 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". Note: To draw an arc, use the pointer tool to choose 3 points that you have created, and then under Construct, release on Arc Through 3 Points.

  3. In perspective drawing, lines that are parallel in the real world instersect in a vanishing point. In Escher's work, parallels behave differently. They violate Playfair's Theorem since we can find more than one parallel to a given "line" through a point off of the "line" - do you see how?