Course:Harris, Fall 08: Diary Week 4

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  • I handed out copies of a color outline on classifying Wallpaper Groups (from [1]).
  • Groups finished up the Frieze Patterns Exploration (took most of the period).
  • Question raised: If a frieze has reflection symmetries from both a horizontal line and from vertical lines, must it also have a 2-fold rotation symmetry?
    • The list of Frieze Groups suggests this to be the case, as there is only one Group which contains both horizontal and vertical symmetries, and it also contains rotational symmetries; but is this truly the case?
    • Suppose a frieze design has these symmetries:
      • from reflection across a horizontal line; call that transformation H.
      • from reflection across a vertical line; call that transformation V.
    • We know that if we apply either H or V to the plane, that the frieze design is moved onto itself; but is there a rotation which moves the design onto itself?
    • We examined what happens to the plane if we apply, successively, first the transformation V and then the transformation H to it:
      • Tracking where labeled points of the plane get sent by doing V and then H to the plane, we saw the net result is a 180-degree rotation:
        • Center point is the intersection of the vertical and horizontal lines.
    • Since the design gets preserved by both the the V and H transformations, doing both of those must also preserve the design; that means that the design is preserved by the rotation, as well. In other words:
      • Yes, having a vertical-line symmetry and a horizontal-line symmetry in a design implies the design has a 2-fold rotation symmetry, also.
  • The general goal of the course is to investigate the mathematics behind design symmetries.
    • Part of this is simply to learn to classify the various groups of symmetries that can occur.
    • Another part is to see how various symmetries interact with one another, as in the analysis above.
  • Wednesday will be a quiz on building designs, using a given motif and a given Rosette or Frieze Group .
    • You can use the Frieze Group outline to assist you; but you should know the Rosette Groups by now.


  • I handed back corrections to Exercises and such, noting that "cyclic" is not a group-name, but C4 (for instance) is.
  • We spent about 10 minutes on Quiz 1 (make designs, using given motif, with symmetry groups of C4, D4, M1, and 12).
    • Most trouble seemed to be with D4, ensuring that the design does, indeed, have 4-fold rotational symmetry. Need to double the given motif in order to have a reflection-symmetric building block.
  • Problems in how to specify a tessellation of the plane by, say, copies of one parallelogram (20 minutes or so):
    • Issue is in showing that one knows how to cover the entire plane. Does saying "and so on" really nail it down?
    • If we have rows of parallelograms which are simply related to one another, then we can say "and so on" and have that mean something definite.
      • This is a tessellation with symmetries. This is one reason symmetries are useful, for uniquely specifying a tessellation of the entire plane (infinite in extent!).
    • If rows are randomly related to one another, then there is no unique specification.
      • This is a tessellation--or, rather, a class of many possible tessellations--without symmetry.
  • What is a lattice of translations on a tessellation (at least, on a Wallpaper-group tessellation, i.e., on that has at least two translations in its symmetry group)?
    • Pick any point, anywhere.
    • Note the position of at images of that point under translations.
    • Note positions of images of those images under translations, and so on.
    • Connect these images by lines that go in the directions of the translations; these are the lattice lines associated to the original choice of a point.
  • We had 10 or 15 minutes to start on the Wallpaper Group Exploration; to be continued next time.


  • Most did fairly well on the quiz.
  • I emphasized that Exercises should be corrected to get a decent grade; I am available for help with them.
    • Particularly troublesome in the Frieze Exercises was finding all the symmetries in a frieze.
    • Note that there can be "trivial" glide reflections when there is a horizontal reflection present (trivial because the separate translations and reflections are both symmetries); **but in other cases, there is a nontrivial glide reflection, i.e., in a frieze which does not have a separate horizontal reflection.
  • I changed the wording of #7 in Frieze Exercises to be restricted to the case of a nontrivial glide refelection.
  • Groups spent most of the period (20-30 minutes) finishing up Wallpaper Symmetry Exploration.
    • I asked the class to spot the nontrivial glide reflection in Schattschneider #91 [2].
      • This has, for its glide reflection axis, the line along the bugs' legs.
  • Groups spent a short while getting started on Escher's Wallpaper Groups Exploration.