- Thursday's quiz will be over symmetry/Platonic Solids.
- We began with a bit of a wrap up of Platonic solids; a video showing us why there are exactly five Platonic solids, and
- We talked about some other types of symmetry, and we'll continue that today.
- In our quiz we focused on reflection and rotation as symmetries, but there are others:
- translation (wallpapers)
- symmetries of scale
- We used a little algebra (the quadratic formula!), and the Greek's
definition of the golden rectangle, to find that
\[
\phi = \frac{1+\sqrt{5}}{2} \approx 1.618
\]
While we might think about the Fibonacci spirals as being created by
attaching squares, the golden rectangle is created by
removing squares. We just do things backwards....
But the thing about golden triangles is that each rectangle remaining is a perfect copy of the original (that's the way the Greeks defined it).
In contrast to our Fibonacci spiral, in which case each was successively more beautiful (and thus not the same), each of these is a perfect copy -- smaller, yes, but perfectly the same. This is an example of symmetry of scale. And while we can, of course, tile the floor with these, we're going to need some tiles of vastly different sizes (some getting incredibly small).
-
Here is a somewhat silly example of what we can do with this idea -- A
Fibonacci spiral fractal comic I made recently, which I have entitled
"Flirting with Death Spiral". Infinite fun! I'll be having you make one
of these yourself, so you'll want to think of a good photo to use....
(sidelengths: 466x288, with ratio 1.6180555555555556!)
- We begin with our 2-d projections of the Platonic solids, and our
job is to draw in the duals. Last time we discussed duality (even
seeing how it is used to create higher dimensional "polytopes"
-- analogues of the Platonic solids in other dimensions).
But while we're projecting three-dimensional objects into two-dimensional space, their duals will show up as well!
On the other side of the sheet is another way of thinking of the Platonic solids in two-dimensions: as "unfolded", if you will.
- That is followed by an exploration of the 17 wallpaper groups.
Wallpapers involve translations (e.g. wallpapers), including glide reflections. Here is a nice description of these "Transformations of the plane".
For your homework: explore the 17 patterns, and try to find ways to distinguish them.
This site can help us out a lot! Distinguishing the 17 wallpaper groups. In particular, let's take a look at the glide reflection symmetry.
- Last time I illustrated how we can generate
patterns that suggest Fibonacci flowers. This exercise is
based on that.
We'll try to make a little Fibonacci art, Fibonacci checkerboards, and then discover that trouble ensues; we'll discover why while watching a short, beautiful video, Creating The Never-Ending Bloom (the artistry of John Edmark).
This is an example of symmetry of scale: each "petal" is an exact copy of every other, so we're tiling the floor with a single shape (although it is changing in size).
- Some Fibonacci spiralling links:
- Creating The Never-Ending Bloom (the artistry of John Edmark)
- The Art of John Edmark | Talk by Paul Dancstep | Exploratorium: Fibonacci spiral checkerboards (and other symmetries of scale)
- Art-Equals Community (with symmetry):
- A gorgeous video of Prof. Carlo Sequin explaining regular polytopes in all dimensions (of which regular polygons are two-dimensional versions, and Platonic solids are three-dimensional versions -- so there are polytopes in higher dimensions!). Here Vi Hart describes eating four-dimensional polytopes...
- Symmetry, by Hermann Weyl (Princeton University Press, 1952)
- Symmetry of lifeforms on Earth
- A fun reading on symmetry
- All wallpapers
- The best site I've found for distinguishing the 17 wallpaper groups (local copy).
- Wallpapers begin with Transformations of the plane (this from the very informative website Wallpaper Groups (here's a local copy)
- Examples of all 17 wallpapers
- Polya's version of the Wallpapers
- Wallpaper exercises, and a Wallpaper exercise handout.
- Hexagonal Paper
