CS798 Assignment 2
Due date: February 20th.
You can submit this assignment by emailing me a PDF or by handing it to
me on paper. You can write it up by hand as long as it's tidy.
If you're auditing the course, the implementation question
Question 1: Regular polygons and stars
Here is Exercise 5 in Section 2.1 of Tilings and Patterns:
Show that if a tiling by regular polygons contains an octagon, then
it must be the tiling (4.82).
Actually, the statement is false as written. Give a simple
counterexample. Now prove the statement under the added assumption
that every side of every tile has same length.
- The general problem of tiling the plane with tiles all of which
have five-fold symmetry is still open. Let's dispense with two
of the easier-to-describe cases.
- Prove that it is impossible to tile the plane with regular
- Sketch a proof of why it is impossible to tile the plane
with regular pentagons and regular |5/2| pentacles.
Question 2: Isohedral tilings
Determine the isohedral type of each of the following tilings, according
to the table given by Grünbaum and Shephard. Mark the tiling vertices
of one tile and add
(possibly directed) labels consistent with the incidence symbol given
in the table. As always, disregard colours and other markings that
artificially distinguish tiles from one another.
- A pentomino is a topological disk formed by taking five contiguous
squares from the regular tiling 44. It is well known
that the twelve different pentominos can each tile the plane
isohedrally. Nine of the twelve pentominoes can tile by
translations alone. For the remaining three, two distinct aspects
(orientations) are required. Find the three two-aspect pentominoes,
draw tilings that use them, and determine the isohedral types of
Question 3: Tiling playground
Write an interactive program to manipulate the shapes of marked
Isohedral tilings with topology type . You may
assume that in all cases, the tiling vertices are fixed in a
square configuration (so you don't need to worry about tiling
vertex parameterizations). Of the 36 isohedral types with this
topology, you only need to handle the 28 for which the tile shape
is not constrained to be a square.
- At a minimum, your program should support the following
- Non-trivial extension
- You must implement at least one non-trivial extension
to your program beyond what's decribed here. This part
of the assignment is open-ended.
Some ideas for
extensions are given on a
This part of the assignment is not intended to be
overwhelming; it's just a way to get you thinking about
what ideas might follow on from the basic implementation.
Don't feel you have to wear your fingers down to nubs
trying to implement your extension. A proof-of-concept
You must produce a short write-up describing your implementation.
You can structure your submission as you wish, but should include
at least the following:
I don't plan to examine your source code, but I reserve the right
to request it for marking purposes. I also reserve the right to
request a demonstration (a demo might be the best way to show off
- Describe your implementation. What set of languages, tools,
and libraries did you use? What is the interface? Include
a few select screen shots to show the various features of your
system in operation.
- Mention any bugs or limitations, and say what you would do
to overcome them.
- Describe your extension. Explain briefly how you modified
the core implementation to accommodate the extension. Comment
on how successful you feel it was.
- Include some polished samples produced by your program. At
least one sample must clearly demonstrate the effect of your
extension, if possible.
Bonus: Tiling scavenger hunt
Find and document as many interesting tilings as you can
that can be found within Waterloo city limits. These tilings can't
be temporary (posters or chalk drawings). They have to be permanently
installed as fixed architecture (brickwork, paving, ceramic tile,
murals, wood inlay, driveways, etc). You are permitted to construct a
new permanent building, or renovate an old one, with your choice of
tilings if you want to go the extra mile. The tiling should have
structure: a random jumble of paving stones isn't very interesting.
For each tiling, take a photograph and indicate where the tiling can
be found. Explain the structure of the tiling (give its isohedral
type, for example). Ideally, they should all be found on public property,
or at least visible from public property.
Everybody should be able to find squares, hexagons, and bricks.
Start with those. Here are some others I'd love to see...
- Other Laves tilings
- Other Archimedean tilings (4.82 and variations are
- Tilings by squares or triangles in multiple sizes
- Aperiodic tilings (the windows of the Physics building at
the University of Washington are etched with Penrose tiles)
- Anything with pentagons