The Learn’d Astronomer

(9/14/2022) We began our exploration today with a reading of the Walt Whitman poem, When I Heard the Learn’d Astronomer. This poem has a huge connection to mathematics, but I resisted the urge to tell who wrote it (our famous local poet) or to initiate a discussion about what it’s about. But artifacts matter. So I did show the book I was reading it from: Martin Gardner’s Best Remembered Poems. I told of Gardner’s relevance to our course. Anecdotes matter too. R remembered an anecdote from a prior course, the story of his famous April Fool’s joke. Students were also interested in the story of why I was given the book, by whom, and the cards that I received in the book. The human connection to mathematics was a big theme of all our explorations today, and always.

I then read the poem, let it sit, then jumped into the math.

Opening the Bag

I told the class briefly about the bag, which had come from the 14th biennual in-person Gathering 4 Gardner (G4G14). I explained that at this celebration, everyone who wanted to participate in a Gift Exchange could, and I had. I hadn’t explored any of the items in the bag. I was waiting to share it with our Math Circle participants.

Me: “Go ahead, look in the bag. Pick out anything that interests you.”


Z chose the little bag with construction pieces labeled Zometool. Z looked at the instructions and asked “What is a 3D cube?” (I added this question to running list that I had started of assumptions, questions, and conjectures.)

Me: Is there a difference between a 3D cube and a cube? Do you know what 3D means?

Z – sorta

Me: Do you know what dimensions are?

Z – not sure

Me: a bunch of hard-to-understand info

W: 3D means it’s not flat!

Me – so can a cube be flat?

Students – flat means like a pancake. Maybe cubes could be flat

I talked about nets, which are 2D figures that can be folded into 3D shapes without gaps or overlap.  This was very hard for students to imagine, so I plan to bring some physical nets in for the students to explore another week.

This set of Zometools was a carefully curated kit of pieces with instructions for creating specific geometric objects. Z had started following the instructions but then creativity took over and they created their own mathematical object.

I told the students that I was wondering who at G4G contributed the Zometools. We couldn’t find a person’s name in the package. One of my goals in this course is that students realize that there are mathematicians alive and working now and doing interesting things. Real things. Another of my many goals is for students in this course to see math in action, to present applied mathematics, not just all theoretical work like Walt Whitman’s Learn’d Astronomer did.

The students asked “What did you make for the Gift Exchange?” I told of (1) the G4G Gift Exchange book, (2) that I had written an article for it about a prior Math Circle from years ago, and (3) how I presented some of this writing to the hundreds of people gathered together back in March at G4G 14.


R pulled out of the bag a small 3D-printed cube (contributed to the Gift Exchange by puzzle designer Oskar van Deventer). R had a lot of questions about it.

R: What are you supposed to do with it?

Someone: Do you think you’re supposed to take it apart?

R: The instructions say disassemble it. But HOW do you dissemble it?

The pieces were somehow linked together. After some struggle, R had disassembled it.

R: I noticed that almost all the pieces are identical except for one – why? And what do I do next?

Me: I don’t know. Do you think that maybe we are supposed to put it back together into its original form, or maybe create something new?

P:  Create something new!

At this point, R passed the cube to P, who created something new but not interconnected.

Me: Do you think the creator intends us to build something interconnected or connected?

(Here, I wish I had asked “Do you think it matters whether we do or think about what the creator intends?” That is the real question, IMHO.)  

Accidentally but fortunately, P took this item home. Maybe she’ll explore it more. If not, at some point before the course ends, after everyone who is curious gets to play with it, I may show the website directions on exactly what the puzzle is and how to solve it. But for now, I want questioning, exploration, and discovery to be the focus.

Celt Decision Maker

R next chose an item labelled “Celt Decision Maker.”

R: How does it work?

W: Is it like those things you can make with paper and two pencils to get answers from ghosts? (Everyone got interested and wondered the same thing.)

Me (not wanting to break the spirit of questioning or to reveal too much): Chances are not since a mathematician made this.

R explored it more, testing it’s motion, figuring out how to use it. Others joined it. R asked me to give it a question about a decision I’m facing.

Me: Should I go home and make my lunch todaCely or buy something prepared?

R: You have to ask a yes-no question!

Me: Should I buy something prepared for lunch today?

(The Celt Decision Maker answered in the affirmative -yum!)

Me: What’s the math question here? What’s interesting about how this thing works? This thing, by the way, is generically called a razorback. Celt Decision Maker is it’s brand name.

More exploration.

R: It controls itself when you spin it in one direction!

Others rushed over to see what R meant.

Me: Do you want to come up with some ideas about why it works that way?

Students explored, but so far no conjectures about that. They were more interested in the question of who created this, since there is an interesting name on the package.

W: Who is Sirius Enigma? It sounds like the name of a plant.

After discussion about whether this might be a pseudonym, students turned their attention to another gift.

Akio’s Fibonacci Windmill

W choose this gift, and wondered

  • Is it made from 1 piece of paper?
  • How do you make it? (W took it apart to see how it was made – like the math strategy of working backwards, starting at the endpoint and going back to the beginning)
  • What are we supposed to do with this object (wear it? Create one of our own?) Can you wear it? Is it a pin (jewelry)?

W was worried about damaging this seemingly fragile object. The pin fell apart and got lost for a bit. I explained that I had made a conscious decision that these things can get ruined, rained on, blown away in the wind, etc. No worries. Exploration is more important than preservation!

W noticed that there is a website for this object, and that maybe we can look at it when I bring my laptop.

Me: you can learn almost anything about geometry from folding paper (origami).

The students said they knew this. What they didn’t know or notice is what this object is called and the significance of both words in its name. People did wonder who Akio is and whether this came from Japan. I told an anecdote about G4G and the huge Zoom screen allowing people from very far away to present.

W then noticed that “G4G14” was backwards on the item. Was this intentional? If so, why? How could we find out?

Me: Remember that the creators of these gifts made over 200 of them. If you had to make so many gifts, might you make a mistake?

The students doubted that it is a mistake. Maybe it wasn’t. But one of my goals here is to humanize mathematics, so a mathematician making a mistake is a useful thing to consider.

Build 14 Bridges

The gift that P chose had the exact same name and instructions: Build 14 Bridges. She, and everyone, wondered

  • What do the instructions mean?!
  • Do you build them all at once, or one at a time? Do you build them on the platform provided or elsewhere?

P also pointed out that one of the pieces was broken.

Me: That must have gotten broken in my suitcase. I had to get this entire bag of gifts into my suitcase for the flight home.

Students: Wait, where was this thing?

Me: Atlanta.

The students’ eyes opened wide. They were impressed!

Other Gifts

Students pulled out a few other items but decided not to explore them for various reasons. One needed us to look something up online and I didn’t bring my laptop. One, students thought, required us to tie-dye something and “that would be messy.” One gift seemed to want us to have a 3D printer, thought students. Some were things students weren’t in the mood to do, so those went back in the bag for another day or another student. “That’s okay,” I said. “We’re mathematicians. We’ll pursue what interests us!”

Sneaky Arithmetic

I ended today’s session with one of Martin Gardner’s “Sneaky Arithmetic” puzzles. I won’t write it here since I suggested, as did Gardner, that now that student’s figured out the answer, that they challenge you (their families) with the problem. “But our parents are right there!” protested R, pointing at a few parents nearby. I said I don’t think they heard. So ask your students for the one about the seventeen sheep.

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