By Norran Yu. (Grade 11) May 4th, 2023

**Rational**

In the field of packaging, there is a type of problem which is to partition a fixed area or a fixed volume into equal-sized units using the least amount of material. For example, how to divide farmland into equal-sized pieces using the least amount of fencing material, or how to build an apartment of equal-sized units using the least amount of construction material. This IA is to find mathematical solutions to both problems.

**1.0 Introduction**

The isoperimetric inequality problem studies both two-dimensional and three-dimensional shapes. For two-dimensional shapes, it tries to find the optimal shape with the greatest area in the least perimeter. While for three-dimensional, it finds the greatest volume in the least surface area. There are four types of objects to study: 2D and 3D single objects, and 2D and 3D compound objects, as shown in the following diagram. A *compound object* is a repetition of equal-sized single objects. For example, a farmland partitioned into equal-sized units is a 2D compound object, while an apartment building consisting of equal-sized units is a 3D compound object. Since compound objects are built upon single objects, we need to begin with our analysis of single objects.

The rest of the IA is organized as follows. The *data and analysis* section finds the optimal shape of 2D single objects, followed by the optimal 3D single object. It then extends the analysis to find the optimal compound 2D and 3D objects. The solution to the farmland partition and the optimal apartment problem are discussed in the *discussion* section, and the study concludes in the *conclusion* section.

Moving onto 3D shapes, the sphere is the optimal shape that contains the most volume with the least surface area [2]. Although a full proof is beyond the scope of this IA, we compare a few 3D objects, including the equilateral pyramid, cube, dodecahedron, cylinder, and sphere, and show that the sphere is the most optimal shape among them. Furthermore, our calculation and analysis use the surface area to volume ratio. The lower the ratio the more optimal the shape is. The full surface area to volume ratio calculations of the equilateral pyramid and the cylinder are provided below, while the ratios of other shapes are referenced.

**5.0 References**

- Stan Alama, ‘On Isoperimetric Problems’, retrieved May 4th, 2023 from https://ms.mcmaster.ca/alamas/main/Isoperimetric_Undergrad_handout.pdf
- Surface-area-to-volume ratio. (2023, April 23). In
*Wikipedia*. https://en.wikipedia.org/wiki/Surface-area-to-volume_ratio - Thomas C. Hales, ‘The Honeycomb Conjecture’ (June 8, 1999), retrieved May 4th, 2023 from https://pdodds.w3.uvm.edu/files/papers/others/2000/hales2000a.pdf
- Jordana Cepelewicz, ‘Mathematicians Complete Quest to Build ‘Spherical Cubes’’ (February 10, 2023), retrieved May 4th, 2023 from https://www.quantamagazine.org/mathematicians-complete-quest-to-build-spherical-cubes-20230210/
- Weaire–Phelan structure. (2022, September 5). In
*Wikipedia*. https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure - Thaddeus Cesari, ‘Honeycomb Mirrors Make NASA’s Webb the Most Powerful Space Telescope’ (December 7, 2018), retrieved May 4th, 2023 from https://www.nasa.gov/image-feature/goddard/2018/honeycomb-mirrors-makes-nasa-s-webb-the-most-powerful-space-telescope
- Marc Chamberland, ‘The Miraculous Space Efficiency of Honeycomb’ (July 22, 2015), retrieved May 4th, 2023 from https://slate.com/technology/2015/07/hexagons-are-the-most-scientifically-efficient-packing-shape-as-bee-honeycomb-proves.html
- Matt Strimas-Mackey, ‘Fishnets and Honeycomb: Square vs. Hexagonal Spatial Grids’ (March 20, 2020), retrieved May 4th, 2023 from https://strimas.com/post/hexagonal-grids/
- Brilliant.org, ‘Math of Soap Bubbles and Honeycombs’, retrieved May 4th, 2023 from https://brilliant.org/wiki/math-of-soap-bubbles-and-honeycombs/
- Regular dodecahedron. (2023, April 9). In
*Wikipedia*. https://en.wikipedia.org/wiki/Regular_dodecahedron