# Non-Euclidean Art Project

**Objective:**
Create an artwork based on spherical geometry, polyhedra, or hyperbolic geometry.

Use techniques from this course, or research some other method to create an artwork involving non-Euclidean geometry. A spherical geometry project would likely involve working on the surface of a sphere. Polyhedra are models of the sphere, but with flat faces, and can have appealing symmetry. Hyperbolic geometry projects could use the Poincare disk model, or construct a three dimensional model of hyperbolic space.

## Art Component

The art project will involve some mathematical planning and understanding, and some artistic skill. Generally, a project with more complicated mathematics will require less artistic talents, and vice-versa, but an excellent project will feature both.

You are allowed to create any artwork that involves non-Euclidean geometry in an integral fashion, but there are a few clear ways to accomplish the goals of this project:

### Tessellation

Design a tessellation by recognizable figures on a sphere, polyhedron, or in hyperbolic space. Techniques from the Tessellation Art Project are still applicable, but require careful thinking about symmetry to realize on a non-flat surface. Tesselating a polyhedron has the advantage that you can execute the design flat and then fold or build the 3D model.

This sort of project generally has a high mathematical content, unless you choose one of the simpler polyhedra such as a cube, tetrahedron, or octahedron. An entirely geometric tessellation is acceptable, but would require a higher degree of craftsmanship.

- Escher's Sphere with Fish, Sphere with Angels and Devils, Sphere with Eight Grotesques, and Sphere with Reptiles are all examples of spherical tessellations.
- Japanese Temari balls are geometric spherical tessellations made with string. Some examples by Carolyn Yackel.
- Jill Britton has examples of tessellated polyhedra at http://britton.disted.camosun.bc.ca/jbpolytess.htm
- Gijs Korthals Altes has many paper 'nets' that fold into polyhedra, which you can use as a starting point.

### Polyhedron Construction

Choose a polyhedron and construct it out of some interesting materials or using challenging techniques. Your construction should be sturdy and symmetric.

Some techniques:

- Modular origami
- Soda straws with pipe cleaners, string, or rubber bands.
- Tensegrity structures

For still more ideas, browse flickrhivemind polyhderon tag, George Hart's Pavilion of Polyhedreality, or just search the web for polyhedron.

### Hyperbolic Model

Make a model of hyperbolic space. Here are a few ways to create hyperbolic models:

- From polygons, as in the Hyperbolic Paper Exploration.
- By folding paper to make 'hypars', as described in Make magazine
- Using crochet, as at the Hyperbolic Coral Reef Project

Yet more how-to and creative ideas at MathCraft.

## Written Component

### Full

Write a paper about the geometry you selected for your artwork, either spherical or hyperbolic.

The paper should be a minimum of 3 pages, typed, double-spaced, 1 inch margins. Figures are excluded from the page count.

- Explain how this geometry is different from Euclidean geometry. How are the axioms different? What polygons exist? Which ones do not exist?
- What do we know about the theory of tessellations in this geometry? How many regular tessellations are there? Do you think all triangles tessellate? Do all quadrilaterals tessellate?
- What do the isometries of this geometry look like?

### Partial

Create some basic documentation about your project (on a separate piece of paper), as if this was the information placard next to your work in a museum.

Include:

- Your name
- The title of the artwork
- The materials you used to make the artwork (media, type of paper, …)
- A short discussion of how you constructed it
- Anything else about the work that needs explaining, such as the theme, meaning, source of inspiration.

## More Information

- Examples of past student work: Non-Euclidean Art Project Gallery
- Non-Euclidean Art Project Grading Rubric

**Handin:**

The finished work. The written component. Any preliminary work you have done (as evidence of effort).