ESCHER #5: All About Assumptions
(February 10, 2015) I brought in some soccer balls to continue our discussion about which regular polygons can be tessellated. The kids discovered that the balls were a pattern of both pentagons and hexagons. The question became “Why?” Conjectures: Roundness? Size?
Of course, since soccer balls are 3-D, they weren’t going to answer our question about 2-D tessellations. But sometimes looking at something that you know won’t lead you directly to a quick answer might bring up the right questions to ask. For instance, “Why does it seem that pentagons can tessellate in 3-D but not in 2-D?” (We reached this point last week.) Not that we answered this yet, of course. The questioning is the journey.
TESSELATION PROOF USING ANGLES
But wait a minute, we sorta did get on track to answering this because G, who was absent last week, knew the answer already. She explained to the others that it has to do with the interior angles of the pentagons not being factors of 360. As she was explaining this, eyes started glazing over. I tried putting some figures on the board to supplement her verbal explanation, but a number of the kids here didn’t have the math background to understand her explanation. They only sorta knew that degrees were these sorta arbitrary measures of angles. So we were at a crossroads:
Should I spend some time doing an activity that would facilitate students figuring out exactly what an angle is, how angles and circles are related, how angles are measured, etc? My original plan for this course was to get into geometric proof about which shapes can tesselate. And, to add more support to the side of the argument that said “Yes, do it!” I do know some activities that would facilitate this in a pretty interesting way. This was not one of those occasions in which little side topics, or lemmas, come up in math circle, and I just don’t have handy a method to facilitate discovery, so I just let the kids make assumptions instead – another valid approach in mathematics. In general, I’d prefer that kids take risks and make assumptions rather than me steal their opportunity for discovery by just giving them math facts.
OTOH, there didn’t seem to be interest in this proof. And we had bigger fish to fry, so to speak, in this, our next-to-last session. And, I had at least one student who already knew the proof, so that would make the session less productive/interesting for that student. So I decided to let it go. The students were all happy to posit the conjecture that you could prove with degrees why pentagons tessellate and hexagons do not. They were also happy to state this as an assumption since one student was convinced of it this week, and another, who was absent today, referred to this proof last week.*
ONE-EARED BUNNIES REVISITED
You may recall that back in the first two weeks of this course, we had explored types of symmetry. The kids made a list of different symmetry types that they had discovered: reflectional (with a number of sub-types) and rotational. We debated a wallpaper edging pattern of one-eared bunnies, and the kids had come to a consensus that this pattern had no symmetry. I wanted to revisit this.
I put on the board a footprint pattern in sand at the edge of the water, from a bird’s-eye view. (On a side note, last week we talked about Escher’s early work Tower of Babel, the perspective of which he described as a “bird’s-eye view.” The kids asserted that Escher must have coined that phrase. I did something here that I rarely do – stated my own opinion. I thought that phrase had been around long before Escher. “You mean, Escher was alive at the same time you were?” said J, amazed. The arc of history comes up in interesting ways.)
“Is there symmetry in this pattern,” I asked the group. The knee-jerk reaction from most people was yes. Then the questioning began. The questions opened up a big discussion about what would have to be true for there to be symmetry. I listed those statements on the board as assumptions:
>The sand doesn’t shift (and therefore change the shape of any of the footprints).
>All the footprints are the same size.
>The footprints are the same distance apart.
>There is no wind (again, shifting the sand/shape).
>The “pattern” we’re referring to is only the footprints, not the water, fish, and sand that I drew.
>The pattern is infinite (otherwise the reflective symmetry might disappear because of a different number of footprints on each side of the axis of symmetry).
“If you take away any one of these assumptions,” stated A, “then it’s not symmetrical.” We talked about how she had gotten to the root of exactly what an assumption is.
“What types of symmetry do you see in the footprints?” I asked. This question caused some heavy thinking. Everyone realized that there is no straight-up reflection or rotation. The group thinking started to move away from believing that the pattern had symmetry.
“They shift a bit, but that still feels like symmetry,” said someone else. A few heads nodded.
“This is like the one-eared bunnies,” said another person. All agreed. I asked for a show of hands to get a sense of the group’s thinking at this point: 2 hands in support of symmetry, 2 against, and 2 uncommitted.
I passed out copies of the Escher print we had examined in week one (the black birds with the white smiling fish – see it in photo gallery – I still can’t find out the title of this one!). I asked the students whether symmetry existed in this. Again, a knee-jerk reaction of yes, of course. “What type of symmetry?” I asked.
The students squinted, held their papers up to the light, turn the papers upside down, and looked and looked and looked. (BTW, in the photo gallery, there’s a photo that I love of M & M using a clear ruler to look for an axis of symmety.) Some answers:
- >“It feels like it.”
- >“not reflectional”
- >“not rotational”
- >“it does shift”
G pointed out that “translation must be a kind of symmetry if Escher used translation, and the topic of this course is “Escher and symmetry” – it just goes to reason.” But most people here didn’t like calling the fish and birds symmetrical when it didn’t meet the requirements they had identified for symmetry. They wondered: Is there some category of pattern that feels like symmetry, that has no axis of symmetry nor point of rotation, and that shifts? I told them that mathematicians call that category of pattern a translation. (Technically, the footprint pattern is called a glide reflection, which is a two-step symmetry mapping/pattern involving both a reflection and a translation – we’ll get more into that next week.) So now, this thing that the students discovered had a name.
ARE TRANSLATIONS A TYPE OF SYMMETRY, OR, CAN YOU INVENT YOUR OWN MATHEMATICS?
“Do you think that translations are a type of symmetry?” I asked. Not really, most people thought, but it seemed to everyone that based upon our discussion so far, that they were wrong. I did tell them that in general, mathematicians do accept it as a form of symmetry.
“If it were up to you, would you consider translation a form of symmetry?” Silence. “Would you like to declare that a transformation is not a form of symmetry?” Dead silence. No one argued against the idea. No one wanted to say anything. “It can be scary to promote an idea that no one has ever promoted before. Do you have to accept everything someone tells you in math? Do you have to believe that lines are real, for instance, even though no one has ever come up with a definitive definition?”
I asked for a show of hands here: 3 yes, 2 no, and 2 unsure. (One student had left early, if you’re a vote counter.)
“Is it okay to declare your own math?” I asked. Could you have your own math system in which translation is not symmetry?
“Teachers would give you nasty looks,” said someone.
G: added, “Yes, and then if you go thinking that thing when you take the SAT, and the rest of the world doesn’t see it like you do, you’re in trouble.”
“But might there be a good reason for you to have your own system too, to make that extra work worth it?” I prompted. I told them that in real life, new ways of seeing things do come up in math. Then the students did acknowledge that new discoveries would be a valid reason to invent your own math, but they still seemed dubious.
OUR STUDENTS’ DISCOVERIES/CREATIONS
We spent our last few minutes looking at and photographing what the students had been working on (with their pencils or polydrons) during class:
- >L had spent the entire time building and rebuilding polydron soccer balls with hexagons, pentagons, then a combination of the two.
- >J had tessellated gingerbread men.
- >A had made a whole town, building the platonic solids without realizing.
- >M and M had each invented their own codes.
- >X, J, and M had used their pencils to create various patterns on paper.**
Everyone doing math in their own way, in other words. See you all in two weeks for our last session!
*Might this question be too pedestrian for our math circle?
**We have multiple people with the same first initials in this group.