Joe + Kieran, Hiu Fung TAI

 We chose to work on a piece by Hiu Fung TAI, linked here and shown below.

Hiu Fung TAI - Bridges 2026 Exhibition of Mathematical Art, Craft, and Design

An initial glance may not grasp the depth to this rather simple submission. The common factor between these shapes is that they share the same integer value of area and perimeter. Further interestingly, each side/arc length is also an integer value, which seems unintuitive when we look at the pie piece. Tai ends his article with a somewhat obvious statement that two unique 2D shapes can share an area and perimeter value. To the more analytic reader, this raises a few questions: Are there infinite unique area/perimeter pairs? Can circular objects be paired with linear objects? Does this statement generalize to 3D? To higher dimensions?

We aimed to answer the penultimate question and examine the relationship between volume and surface area in 3-dimensional objects through a similar lens. It was quickly decided between us that we should create a few shapes and working out a lesson through them. However, glancing at an object in your hand and determining the volume is significantly less obvious than looking at a shape on graph paper. How can we make the demonstration visually intuitive then? 

Joe initially proposed that we 3D print these objects to make them watertight and submerge them within a clear container of water, clearly showing volume via displacement and careful measurements. Surface area would then have to be calculated by hand, which we thought wasn't too much of a hassle, both for us and the audience. 

A cube and sphere, sliced and ready to print

However, due to logistical complexity and issues with the printer, we moved to a demonstration revolving around Gabriel's horn and paper mache.

We chose Gabriel's horn to highlight the complex relationship between surface area and volume, since, of course, this famous example has an infinite surface area paired with finite volume. We chose a scale for the horn and set bounds of y=20/x from x=1 onward, rotated about the x-axis to find a workable volume of 200π. Kieran had an idea to create a hanging display that we could leave in our classroom, and so decided on a final design involving three objects with a volume summing to 200π suspended above the horn. The dimensions of the shapes, a cube, sphere and octahedron were carefully calculated before we got hands-on.

Profile of horn sketched onto cardboard

The finished product

After the outstanding work to create such a beautiful display, we moved onto the interactive component. There was a realization at this point that our demonstration would likely involve a few calculations from the audience. To avoid a boring talk about geometry, we aimed to engage the audience in open-ended, discussion-prompting questions (some unanswered!) with a goal to hold some real estate each students' curiosity after we end our lesson.