Non-Euclidean Art Project

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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:


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.

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:

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:

Yet more how-to and creative ideas at MathCraft.

Written Component


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?


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.


  • 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


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