1. Paper Mache genus k surface or other polyhedra

Paper mache is a versatile medium for exploring complex geometric shapes. The project could involve creating alot of different geometric forms, like polyhedra, toroids, or even fractal structures. The mathematical aspect focuses on the geometry of the shapes being constructed, including concepts such as vertices, edges, faces, Euler’s characteristic, and the relationship between geometric properties and physical form.

By Daryl J. Ashton By George Hart 

I would begin by designing the geometric shape on paper or using software tools (e.g., GeoGebra or Blender) to visualize the structure. Then I would create a framework or mold for the shape using wire or thin cardboard. Then I would layer papier-mâché over the framework, and carve out the intricate details or smooth surfaces. 

2. Constructing a Tensegrity Structure

Tensegrity structures represent a physical manifestation of mathematical principles in equilibrium, vector forces, and spatial geometry. Constructing a tensegrity structure involves understanding the balance of forces and the geometric arrangement necessary to maintain stability. This project can explore concepts of symmetry, tension, compression, and equilibrium in a tangible form.

To make this, I would start by using a CAD software to model the structure and its forces. The construction phase involves selecting materials (e.g., elastic cords for tension and rods for compression elements), assembling the components based on the design, and adjusting the tension to achieve balance and stability. This process not only brings mathematical concepts to life but also challenges us to think critically about physical forces and structural design. The first thing that comes to mind would be to also simulate this in comsol to simulate and understand all the physical forces at play. 

3. Make a hexaflexagon

Hexaflexgons are hexagons that can be made and folded from a strip of paper. It would still have 6 sides, but it can be folded to show different faces. The first hexaflexagon was made by a British mathematician Arthur Stone who was playing around with strips of paper. In order to make a simple hexaflexagon, a person must fold a paper into 10 equilateral triangles, and then creating a hexagon by attaching the last triangle to the first triangle.

I would make this be using a strip of paper with cool designs to be able to display different faces when it is finished. I would fold into 10 different triangles, and then glued the 1st and 10th triangle together forming a hexagonal shape.