I thought I would end 11111011110* with an homage to my first post of the year on shortbraid and other geometric cookies, by making a regular polyhedron out of gingerbread and chocolate ‘glue’.
The dodecahedron** is a particularly nice shape to close the year with because it has 12 faces – one per month!*** If you want a fun mathematical New Year project, you can make yourself a dodecahedron calendar.
I considered doing a corresponding month by month round-up of my favorite posts, but I apparently didn’t bother to make a post last January. Instead, I leave you with my delicious and oddly healthy gingerbread recipe for anyone interested in making their own geometric cookies.
3 oz. melted coconut oil
1/2 cup molasses
1 tsp. baking soda mixed with 1 Tbsp. hot water
2 cups whole wheat flour
2 tsp. ground ginger
2 tsp. ground cinnamon
other appropriate spices if you like
1/2 cup ginger chips/chopped candied ginger (optional)
Mix liquid ingredients and dry ingredients (except ginger chips) separately, combine. I use a Kitchenaid mixer for this.
Mix ginger chips into dough.
Shape dough into a log, wrap in plastic wrap and stick in the freezer some time that is at least long enough to firm up, but could approach indefinitely
Pre-heat oven to 350 degrees Farenheit
If you just want circles, slice dough as thin as possible and set pieces on cookie sheet (no need to grease)
If you want to make other shapes, let dough warm up a bit, then roll it out as thinly as possible and cut out your shapes
Bake for ~10 minutes, let cookies cool before starting any cookie geometry construction
Finally, gingerbunny ears!****
* I’m unreasonably excited that 2015 will be a palindrome in binary.
** It’s also my favorite platonic solid.
*** Or, as in a recent Vi / eleVR / everyone project, the same number of tones on the scale and days of Christmas…
**** Yes, as you can see in the background, my house is kind of overflowing with random geometric constructions.
I really dislike being sick. I’m just not good at it. Not that it’s necessarily possible to be “good” at being sick, but I regularly get the impression that other people are able to work through colds and mild flus and at least be kind of productive. And, I like to imagine that this is something that I ought to be able to do as well.
So, when I woke up yesterday morning with a more than suspiciously sore throat, I somehow believed that I was still going to have a productive day. Right.
Just a couple hours of unproductive computer staring later, I was back in bed with an awful sinus headache and a mild fever. I don’t know why I ever thought I would get anything done on a sick day. Stupid body.
For the next 6 hours, I lay grumpy or asleep in bed before I had to make a bathroom run, where my sick brain observed that we had a lot of cotton swabs (the actual Q-tips brand, although Q-tips is a bit of genericized trademark) in the drawer, right by a bag of small hair ties.
It is a bizarre fact that my sick and/or sleep-deprived brain remains fairly good at piecing together geometric constructions even as it loses the ability to hold a basic conversation. And, so, my brain latched onto the idea of making a Q-tip and hair band version of George Hart‘s classic sculpture 72 pencils.
Surprisingly, this sculpture isn’t too difficult to construct if you build it up from the center in layers and then remove the extra pieces. It’s perhaps a bit harder in Q-tips because the heads are a bit harder to push through than if the diameter were consistent, and Q-tips have some definite flex that interferes with trying to insert them straight, but I can’t verify the relative difficulty, since this was my first time actually making this sort of sculpture.
The hardest part of the sculpture is at the very beginning, where you want to make four intersecting axes with 3 sticks each. This is the same basic structure as in the 4-axis sculpture that I made out of pipecleaners here. It’s much easier to make out of cotton swabs than pipecleaners (which have way too much flex), but it still requires developing a pretty good understanding of how to fit the pieces together.
Once you have your axes properly defined, it’s just a matter of continuing to add cotton swabs along the axes, growing them into small hexagons.
And bigger hexagons.
And even bigger hexagons.
Finally, you can just remove the interior sticks to reveal the structure and the hidden rhombic dodecahedron in the middle.
At which point you’ve probably had enough of an adventure for one sick day and should probably take another nap. (Full disclosure: I took a nap in the middle of this construction as well.)
We had discussed a number of workshop ideas when Edmund sent me a picture of his latest laser cut paper sculpture – a gyroid.
Now, the gyroid is a particularly neat mathematical structure. It’s a triply periodic surface that divides space into two unconnected halves, and it shows up in nature in things like butterfly wings, giving them their natural iridescence. It’s also surprisingly incomprehensible to the average person despite living in normal three dimensional space.
The gyroid is closely related to the Schwarz P and D surfaces, which were first described by Hermann Schwarz in 1865, but the gyroid itself remained undiscovered until ~1970, when Alan Schoen, who apparently had a more natural understanding of these surfaces than anyone for the previous century, “intuited” it’s existence. He has a fascinating and detailed page on these surfaces here.
The gyroid colored by it’s Gaussian curvature
There is a ‘standard’ tiling of the gyroid surface with skew hexagons. These hexagons meet four at a vertex, so there are also “squares” meeting in sixes in the dual to the hexagon tiling (interestingly, the same skew hexagons can be used to tile the Schwarz P and D surfaces, and the tiling for those surfaces still uses six hexagons meeting four at a vertex). This means that it’s relatively easy to create modules that will combine to form a gyroid.
Icosahedral and octahedral versions of ‘Seeing Stars’
I’ve long thought that the gyroid was a neat surface, but I’ve never tried to construct one myself. However, I was pretty confident that I could construct one, especially with the help of an example and someone who has put a gyroid together in the past. I’ve known for quite some time that the plastic sunglasses that I used to make my Seeing Stars sculptures can be joined together in a variety of ways to make different sunglass sculptures.
Upon seeing Edmund’s gyroid, I promptly dismantled a sunglass sculpture to make test skew hexagon modules, then somehow convinced Edmund that making a completely untested sunglass gyroid sculpture would be a good use of my visit time (to be fair, we had the sunglass sculptures that I had already made as backups).
The construction started off with quite a lot of sunglasses and Edmund’s little section of paper gyroid for reference. While I had previously made some suitable test modules for the sculpture at home, I ended up spending some time redesigning my module right before the workshop in Arkansas. It turned out to be a good decision, as the new module was strictly superior.
We started by making some modules. Then, we pieced a small number of them together experimentally. So far, so good.
This was pretty tiring work, so we all took a break for Fibonacci Lemonade. Unsurprisingly, attendence for the math club at least tripled for this part of the event. For those following along at home, it’s a bit easier to layer Fibonacci Lemonade when you have tall skinny glasses.
Finally, we completed the construction. My assistants in these two photos are math club president Josh Nunley and VP Jesse Horton, who were incredibly helpful both in organization of the event, and in the construction of our gyroid.
The sunglass sculpture (being held by Jesse) turned out quite a bit bigger than Edmund’s laser cut paper version (being held by Edmund), but still not nearly as big as the real infinite surface.
Working with VR makes me prone to describing things as the “first ever bla-bla-bla”. I think you can figure out how that applies to the sunglass gyroid.
ps. I overestimated how many sunglasses we would use for our gyroid (to be fair, if construction had gone faster, we could presumably have used all of them). Fortunately, the math club put them to good use after I was gone construction some nice geometric shapes similar to my Seeing Stars sculptures.
One of the things that I love about mathematics is how deeply and subtly it permeates the world around us. Complicated math concepts are often represented in everyday things. In keeping with the season, we might even say that the math is dressed up “in costume”.
Dance, with it’s connections to rhythm and standard moves or patterns clearly ought to show some connections to mathematics, and it’s one of my favorite examples of “math in costume”. Traditional set dancing, where each dance consists of a group of people moving in predetermined patterns such that each person ends up at a designated place at a designated time hints at that connection quite strongly, but it’s not always obvious what connections exist.
So, let’s try to find it by asking some questions. My dance group once asked an interesting question about part of the choreography in the video below. Some, but not all of the people were getting back to their home positions when performing a traditional figure known as the Waves of Tory, which starts at 1:33.
In the Waves of Tory, couples form a long line and at each iteration a couple alternates between arching over the next couple or ducking under the arch of the next couple. The first couple begins by moving towards the end of the line. The remaining couples begin by moving towards the beginning of the line. Once a couple reaches the end of the line, they spend one iteration turning around to head in the opposite direction.
The reason for dancers not ending up where they started is intuitive – there is a staggered start, but everybody finishes at the same time, so there was no way for all performers to end at their home positions. But, how can we figure out precisely where each couple is going to end up?
In more mathematical terms, we want to know where the i-th couple would be after n couples danced the Waves of Tory for k bars of music. Go ahead and try to solve this problem on your own if you like, then keep reading to learn about how this example problem helps unveil some of the underlying mathematics hidden in dance in general.
Stanford Ceili dancing the Waves of Tory. Photo by Jason Chuang.
Have you figured out what the dancers are doing? We can describe their movements like this.
Let’s call the top couple that faces down at the start of Waves is Couple 0. The other n-1 couples are then Couple 0, Couple 1, Couple 2, … Couple n-1. At the start, the i-th couple is standing in the i-th position. The i-th couple starts movement on bar i.
As the dance progresses,
for k < i, the couple has not yet moved and is in the i-th position
for k ≥ 2i, we can reduce this problem to an equivalent problem defined as follows: What position is the 0th couple in after m bars of music, where m = k – 2i ?
From the 0th position facing forward, it take n bars to reach the n-th position and turn around. Similarly, from the n-th position it takes n bars to reach the front and turn around. Thus, after 2n bars of music, couple 0 should return to where they began the dance. We can therefore further reduce this problem to:
What position is the couple 0 in after p bars of music, where p = (k – 2i) mod 2n ?
for p < n, the couple has been moving forward the entire time and is now in the p-th position
for p = n, the couple has moved to the end, and turned around and is now in the (n – 1) position
for p > n, the couple has reached the end and started moving back and is now in the (n – 1)(p – n) = (2n – p – 1) position
Ok. So there was definitely math there, but it doesn’t really feel like it’s really some other math concept in costume.
Or does it? It certainly seemed very familiar to me as I was solving it, but it took me another half hour or so to make the connection.
The motions of the dancers seemed familiar to me because they were moving exactly like the strands in a standard n-strand braid. The connection is easy to see if we just imagined ribbons hanging from the ceiling, with one attached to each dancer (or pair of dancers in this case), like in a Maypole dance.
And this realization applies to almost all dances, not just the Waves of Tory. A partner dance where two partners spin and twirl around each other is equivalent to a braid on two strands, and, just like the braid group on two strands, it’s isomorphic to the group of the integers, Z, under addition!
Want to prove that the braid group on three strands is non-Abelian (does not commute)? Just find two friends and dance around each other a bit!
You’ll quickly find that if you start in a line and swap the positions of the first two people, then the last two people that you end up with a different result than if you first swap the positions of the last two people and then the positions of the first two people. Thus, your dancing “swap” movement clearly can’t be commutative.
And, since dances and braids are really dressed up as each other, you can not only figure out what braid you are making with your dance, but you can also figure out how to dance your favorite braids. Here are some of mine on five strands (for five dancers). It’s possible to come up with rule that every dancer follows independently that will give you each of these patterns.
So, this Halloween, I’m finding a partner, dressing up as dancers, and explaining to anyone that asks that I’m dressed up as the integers under addition. After all, math does it all the time.
The Celebration of Mind (CoM) is like a series of mini-Gathering for Gardner‘s that happens all over the world around the anniversary of Martin Gardner’s birth. If G4G has always looked like fun to you but you haven’t been able to go, I really encourage you to attend a Celebration of Mind.
Admittedly, I’m a bit late to the advertising, since some of the CoM events have already happened, but there are still plenty more to go (the 100th anniversary of Martin Gardner’s birth isn’t until tomorrow!).
Last weekend, I went to the Stanford CoM event and listened to some fun talks about an assortment of Martin Gardner related things. For example, Donald Knuth talked about Martin Gardner and the number 100 in honor of Martin’s 100th birthday. For the second half of the event, I ran a workshop where we did modular origami with business cards. There were a bunch of other neat workshop type things going on at the same time, but I’m afraid that I had to miss all of them.
This was my first time doing a business card origami workshop on the west coast and I was surprised how few people had done business card origami before. On the East coast, business card origami has evangelists like Jeannine Mosely, *the* master of business card origami. So I assume that most people interested in math art will have seen it before. But, apparently that is not so much the case on the west coast. It was fun being people’s first introduction to the idea. I had people commenting that they “finally knew what to do with their old business cards” or that they “needed to redesign their business cards to work better when folded up”.
I taught the smaller kids the standard six card cube (originally designed by ???, certainly rediscovered many times due to simplicity). Some of the adults entertained themselves by reverse engineering models that I had brought to show off. I also taught several people the module used to create a rhombicuboctahedron (a design that I discovered on my own, although, again, it would be far from shocking of someone else had made it first. Although, interestingly, I haven’t seen it anywhere else myself, so…).
Rhombicuboctahedron and Truncated Tetrahedron
If you want to check out a similar event in your area, you can search for one on the official CoM page.
By the way, if you’re in the Bay Area and are sad about having already missing out on this event, then you’re in luck. There is another CoM in Berkeley on the 26th, so you get a free second chance to check it out for yourself!