- Today's quiz will be over symmetry/Platonic Solids. Next week's
quiz will feature symmetry and wallpaper groups. Our next
topic will be fractals. Still very symmetric!
- Important: Assignment for your Spring break!
You should have some fun, and not think too hard about school! Unless you want to....
When we come back, we'll discuss your projects, and your logos a little more.
- Remember last time, when I told you that I can pick up any current
issue of Science and find something to show you relevant to what
we're doing in this class? Well this week it was Science News,
but there are two things to show you from the current edition:
- An
aperiodic tiling using a single "einstein" tile: "That's a
shape that can perfectly cover an infinite plane - no gaps or
overlaps - but only without forming a repeating pattern."
- A molecule
in the shape of a trefoil knot:
Note, in particular, the "handedness" of the two forms of this molecular knot.
Knots is our next topic after fractals.
That reminded me of a recent article, which described how there are two mirrored types of thalidomide -- "For drug compounds, enantiomers with opposite handedness may have completely opposite effects in the human body. For instance, R-thalidomide is sedative while S-thalidomide is teratogenic..."
[A few weeks later... Wed May 1 16:00:58 EDT 2024]: There's more to the story: it turns out that the body turns one form into the other randomly, meaning that it's not as though we could manufacture and administer only the "good" variety.....
- An
aperiodic tiling using a single "einstein" tile: "That's a
shape that can perfectly cover an infinite plane - no gaps or
overlaps - but only without forming a repeating pattern."
- Bonus track! Yesterday's Washington Post had an article entitled
How to hang art and photos to please the eye and the brain:
Consider height, symmetry and balance for the most appealing result
They had this to say:
Create balance and symmetry
Research shows that humans prefer symmetry. A 2018 study from the University of Vienna found that both art history and psychology students at the university level prefer symmetrical patterns over asymmetrical ones with art; this may have an evolutionary basis as well as a psychological one, the authors noted. In another series of experiments, researchers in Rome found that visual symmetry even leads to positive mood changes in viewers.
This may be because symmetry creates a sense of order. If a group of paintings "is symmetrical, we perceive it as more thought-through, not random," explains Augustin, whose firm, Design With Science, uses principles from neuroscience to create spaces that foster productivity, well-being and positive mental states. "It creates a certain comfort in us because it's easier to survey and process."
- We began with a bit of a wrap up of Platonic solids; a video showing us why there are exactly five Platonic solids;
- We were able to create 2-dimensional projections of the Platonic solids, and then sketch in their duels -- simultaneously producing additional projections of each.
- We talked about some other types of symmetry, using wallpaper
patterns as a unifying theme for several of these symmetries,
and we'll continue that today.
Wallpapers lead us to focus on reflection and rotation as symmetries, but there are translation and "glide reflection".
- I mentioned symmetries of scale, but they don't usually figure into wallpaper. We'll do an exercise today, however, that focuses on symmetries of scale in a second version of a Fibonacci spiral.
- Let's continue our exploration of the 17 wallpaper groups.
For everything that follows, imagine that the wallpaper is infinite -- it extends out in all directions, forever.
Wallpapers involve translations (e.g. wallpapers), including glide reflections. Here is a nice description of these "Transformations of the plane". In particular, let's take a look at the glide reflection symmetry.
For your homework, you were to explore the 17 patterns, and try to find ways to distinguish them.
This site can help out a lot! In fact, it's the solution to distinguishing the 17 wallpaper groups.
On the back of Polya's display of the 17 wallpaper patterns you find additional patterns. Can we classify those, according to the choices on the front?
- For our next trick, allow me to illustrate how one can generate
patterns that suggest Fibonacci flowers. The exercise to
follow 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 clip from a beautiful video, The Art of John Edmark: Fibonacci spiral checkerboards (and other symmetries of scale)
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
