If you are at the exhibit…
Tess the Tortoise ambles down Zeno’s Path toward Integral Hill. Will she ever make it?
Look closely; what do you see?
Tess is walking along a path that has an unusual arrangement of cobblestones. Can you detect a relationship between the area of the stones, the length of each cluster of stones, and the location of the clusters along the path?
Focus on… infinity
Zeno’s Path is named after Greek philosopher Zeno of Elea (c. 490 – 430 BCE). Zeno’s dichotomy paradox suggests that in order for Tess to reach the end of the path, she must make it halfway. But then she has to make it to the halfway point of the remaining length, and to infinitely more halfway points after that. In theory, this will take forever! In practice, how might Tess arrive at her goal?
Search for other odes to infinity in this scene:
- The Lebesgue Terraces and Riemann Cliffs are inspired by their namesake integration techniques. In both cases, the entire space is measured by summing infinitely many subregions.
- Tess’s Sierpiński triangle kite and the Koch snowflakes falling over the mountaintop are both examples of fractals–self-repeating patterns that can be continued indefinitely.
- The tiling on Tess’s shell represents an infinite pattern inspired by the Poincaré disk. The heptagons scale in size as they get closer to the boundary of the shell to capture the ever-increasing surface area of the hyperbolic plane.
Can you find nods to infinity elsewhere in Mathemalchemy?
Tess’s shell demonstrates a heptagonal tiling of the Poincaré disk model of the hyperbolic plane.
Zeno’s dichotomy paradox
Zeno’s Path alludes to Zeno’s dichotomy paradox. This paradox is recounted in many ways. In the original version, Zeno takes the task apart into infinitely many ones, by saying that to complete any part, one must have first completed the first half of that part, and repeating the argument. Mathematically (but not philosophically) it is equivalent to list the infinite number of tasks obtained by the first half, followed by half of what remains, and then half of the now smaller remainder, etc. This is indicated on Tess’s to-do list.
Tess is flying a 3rd-iteration Sierpiński tetrahedral kite, named after the Polish mathematician Wacław Sierpiński—who, in Tess’s narrative, gave her the kite as a birthday present.
Lebesgue & Riemann integration
The Riemann wall alludes, both in its name and its form, to Riemann integration, and the Hilbert’s Hotel billboard on the Riemann Wall alludes to Hilbert’s paradox of the Grand Hotel.
The two components of Integral Hill, the Lebesgue terraces and the Riemann cliffs, respectively allude in their names and form to Lebesgue integration and Riemann integration.
(Here is a discussion of Riemann vs Lebesgue integration.)
Koch snowflakes, up to the 5th iteration, are falling from the sky.
From the Mathemalchemy Chronicles: Snowflakes and Lasers
2 thoughts on “Tortoise – Mathematical Connections”