All the spherical imagery in Chapter 10 of the second edition of Divided Spheres was generated with DisplaySphere, a program specially developed by Chris Kitrick to render and capture spherical geometry utilizing graphics hardware typically available on today’s computers. It is specifically designed to minimize the amount of geometric data required to build spherical frameworks and maximize the visual effects that can be rendered in real time to deepen one’s understanding of the geometry.

It has many features that are useful for visualizing and understanding spherical structures.

DisplaySphere Vertices, edges, and faces can all be visually manipulated in size, shape, and color along with lighting, fog, transparency and stereo rendering. Visualizations can be captured in single or multiple (film) images. The state of the program is also captured alongside image captures allowing users to return to the exact state of the geometry and all the user effects at a later time and continue making changes. Just click on the feature for a sample: face styles, flat/smooth shading, fog, stereo pairs, inscribed polyhedra/transparency, and zooming, to name a few.

DisplaySphere imports industry standard OFF (Object File Format) geometry files, which allows for vertices, normals, and faces to be explicitly defined. Edges are automatically determined. Users can easily control visualization of vertices, edges, or faces. Vertices can be rendered as highly tessellated spheres of any size, while edges can be rendered as cylindrical or box shapes. Object faces can be offset, scaled, extruded and further enhanced with automatically generated holes. There are also multiple ways to apply color to any of these polyhedral components, explicitly or automatically based on the uniqueness of the underlying geometry. With these options users can highlight any number of characteristics of the inputted geometry. Lighting, fog, transparency and stereo rendering further broaden one’s ability to understand the inner workings of polyhedral forms. The latest release of DisplaySphere includes all the geometry and images from chapter 10 of the second edition of Ed Popko’s Divided Spheres as sample material. Users can now directly interact with the geometry and move beyond the printed material to develop a deeper understanding of the ideas and methods presented.

DisplaySphere The software is freely available through GitHub and includes full documentation in PDF format along with sample geometry including the ones used in Chapter 10.


From the documentation:

DisplaySphere is a Windows application specifically enhanced for viewing spherical geometry. It uses the OpenGL API for rendering geometry and is optimized for four platonic polyhedra: the icosahedron, octahedron, cube, and tetrahedron. For these polyhedra only the definition of one face, or a symmetrical subset of the entire polyhedron, is required since the rest of the polyhedron can be constructed from rotational duplicates of the single face. Though optimized for spherical geometry the program will display non‐spherical geometry with all the same controls available.

The application will allow basic viewing control of one or more geometries that can be inputted in multiple formats and allow topological components to be viewed separately or simultaneously; such as faces, edges, and vertices. Faces can scaled, offset, extruded, and even Edges and vertices are constructed from face information with multiple variations available for form, size, and color. In addition, there is support for image capture both in single and multiple frames. The size of the render, and the colors used are user selectable. More
advanced rendering features allow for orthographic and stereo projection.

Utilization of hardware‐based rendering, using the OpenGL API, allows efficient real time rendering of the geometry. The application also uses multi‐sampling per pixel to achieve a relatively high level of image quality.