When Dominique asked for proposals for Mathemalchemy scenes, I sent her a proposal like this: “Something involving infinity . . . Zeno’s path (related to Zeno’s dichotomy paradox), a hill whose volume is approximated with cuboids, Koch snowflakes.”
Dominique’s response was polite: “I see that I have not communicated clearly what is needed. I am looking for stories.”
Oh, a story! I thought, I can write a story! And so, having chosen a tortoise as a protagonist (a nod to Zeno’s Achilles paradox), I sent Dominique the following tale.
This is a story about Tortoise, who goes on an adventure and learns about limits/infinite processes.
One fine day, Tortoise wakes up in her room at Hilbert’s Hotel and decides that she will climb to Koch’s Peak, the summit of Riemann Hill. She sets off.
As she strolls along Zeno’s Path, a straight path from her home to the hill, a mile away, she ponders: to get to Riemann Hill, she must get halfway there; then half the remaining distance; etc. She must, therefore, travel along an infinite number of line segments. How will she ever get there?? She passes markers letting her know she has traveled 0 mi, ½ mi, ¾ mi, ⅞ mi, along the path. She passes some fractal trees and maybe some other things.
She arrives at Riemann Hill. While climbing the hill, Tortoise wonders: what is the above-ground volume of the hill? She realizes that she can estimate this using cuboids.
Tortoise reaches Koch’s Peak, takes off her backpack, and looks up at the sky, where Koch snowflakes are falling!
“Wonderful!” Dominique wrote back. “I can already see her with her little backpack!”
The Tortoise Group
Tess comes to life
The story evolves
Some elements of the story changed as time went on. It was decided that Hilbert’s Hotel would not be a part of the installation, though a billboard advertising it would be featured. The Tortoise group didn’t want to hide the tessellation on Tess’s shell, so the backpack was replaced with a cart that Tess would pull behind her. This, however, also created difficulties: how exactly would she pull it? Could she yoke herself to it? Would she need a lizard friend to help? Would a cart block too much of Zeno’s Path?
As the cart discussions continued, Dominique suggested that we try to think of alternatives. After several days of musing, it occurred to me that if we didn’t want to obscure her shell or the path, perhaps Tess could carry something that would float above her. Maybe a balloon? Maybe even a toroidal balloon?
I brought these ideas to the group, and while people liked the idea of the balloon, we weren’t sure how to create it. That’s when Edward—who, as it turns out, is an avid kitemaker!—suggested that instead of carrying a balloon, Tess could be flying a kite. The group instantly knew that this was the ideal choice for our curious and playful tortoise. Tess’ kite will be a Level 2 Sierpiński tetrahedron, which plays into the infinity motif of Tess’s story.
The hill evolved, as well. Ingrid proposed an alternate design for it—part terraces, part cliffs—that would be a better fit with the naturalistic nature of the installation as a whole, and that would represent two types of integration: Riemann and Lebesgue. The group enthusiastically agreed to the design change, and hence Integral Hill was born.
Finally, the mile markers along Zeno’s Path were replaced with an illustrative cobblestone layout (courtesy of Ingrid): the cobblestone sizes will be halved at the halfway point of the path, halved again ¾ of the way along the path, etc.
It has been a delight to see Tess and her story come to life.