Art Gallery (artwork)
Art Gallery Foam core board mounted photos, 24 in 2018

A 3D spherical panoramic camera image printed on hexagons and pentagons mounted on foam core board assembled into a truncated icosahedron. I call this idea a Personal Panoramic Photographic Polyhedral Pavilion. I used both black and white foam core board to result in a soccer ball like exterior pattern.

Background and Inspiration

I created this work as a demonstration tool for a talk I gave at the 2018 Joint Mathematics Meetings in San Diego, California. I had purchased a Ricoh Theta S camera in 2016 and was exploring some ways of visualizing the 3D panoramic photos. One approach to hardcopy display of these images is to simply use the equirectangular projection as shown below. Another is to map the images onto the surface of a small polyhedron, such as a Platonic or Archimedean solid. In this case I used a truncated icosahedron which has a pattern similar to the classic soccer ball. I called this work Art Gallery after M.C. Escher's 1956 work Print Gallery where a self-referential gallery containing his prints plays a central role.

The equirectangular projection photo from the exhibition Mathematics as Muse that was mapped to the inside of the truncated icosahedron to create the work Art Gallery.

Panoramic Imagery

The panorama was invented by the painter Robert Barker in 1787 after painting Edinburgh from Calton Hill. Watercolor painting was 25 feet in diameter. He originally called his invention the French name la nature à coup d’œil meaning nature at a glance. CaltonHill.jpg (Barker, 1793) Cross-section-of-the-rotund_0.jpg Cyclorama in Leicester Square, London. Gettysburg Slazburg Ohio Met Muybridge (1878) Panorama of San Francisco from Nob Hill. Thirteen individual $18\times22$ plates, April 1878. Printed 17 feet wide.

Spherical Panoramic Cameras

A Ricoh Theta S spherical panoramic camera.

Conventional cameras view a small solid angle, limiting the both the field of view and projective distortion. However, multiple individual pictures are need to have full spherical coverage. Cameras that can directly take spherical panoramic photos, such as the Ricoh Theta S, have become available as relatively inexpensive consumer products. Unlike a traditional camera, this camera has two hemispherical lenses, allowing it to see simultaneously in every direction around the camera. These cameras produce an equirectangular projection, where each latitude row has the same number of pixels, which results in severe distortion at the poles.

The Ricoh Theta S, released in 2015, uses two coupled cameras each having an ultrawide-angle fixed-focus lens. Two 12 megapixel CMOS sensors acquire images on opposite hemispheres which are electronically stitched together to create an equirectangular projection image containing 5376 by 2688 pixels.

Two opposite hemispheres acquired using a Ricoh Theta S. Only a round region of each sensor is illuminated, about 6.75 megapixels of the available 12 megapixels. The two pictures are then combined in software.
The Theta S, and other spherical panoramic cameras, are omnidirectional, thus have a much wider field of view (full image) than a conventional camera (central color region). However, that is at the expense of spatial resolution by roughly a factor of five in each direction. The typical human visual field (black outlined region) is just over 180 degrees horizontally and about 30 degrees up, and 70 degrees down.

The Mapping Problem

A challenge is to produce a hardcopy of the spherical panoramic imagery. This problem is related to mapping the spherical surface of the Earth to a flat map. If the region is small enough, this can be done with little distortion. However, Euler proved in 1775 that no distortion free flat 2D map of surface of the sphere exists.

Besides the equirectangular projection, another commonly used solution is the Mercator projection which was developed in 1569. The Mercator projection is useful in navigation because ship traveling with a constant compass bearing will follow a straight line on the map.

Fuller's Dymaxion Map
There are many other types of projections. Jack (Jarke J.) van Wijk has developed several exotic projections he calls Myriahedral Projections. I was fortunate to attend a talk he gave at the 2013 Bridges conference.

Buckminster Fuller solved this problem by mapping the surface of the Earth to a cuboctahedron and published in Life Magazine, later patented in 1946 (US Patent 2,393,676. He called this the Dymaxion World. A modified version, the Dymaxion Map, is based on a regular icosahedron was developed by Fuller in 1954. One of Fuller's goals was to help people see the interconnectedness of our planet, what is referred to as Spaceship Earth.

Fuller's Dymaxion Map unfolded

Dick Termes

On August 16, 2015, I had the opportunity to meet and visit with Dick Termes at his gallery and studio near Spearfish, South Dakota. Termes developed the concept of 6-point perspective around 1968-9, where the points of convergence lie on the surface of a sphere. He calls this type of painting on the sphere a Termesphere. Four of the vanishing points lie equally spaced around the equator with the other two at the north and south poles respectively.

A photo of the Termesphere Gallery from 2015.
An example of Total Environmental Photography from Termes.

Termes had also been interested in spherical panoramic photography with a technique he calls Total Environmental Photography (US Patent 4,214,821. Here a Platonic solid is used as a mounting base for positioning cameras at fixed relative angles to capture a spherical panorama. Note this was prior to the development of consumer grade digital cameras.

Creating a Personal Panoramic Photographic Polyhedral Pavilions

A spherical panoramic photo of Niagara Falls showing a typical point in cyan.
A spherical panoramic photo of Niagara Falls
A spherical panoramic photo of Niagara Falls

The truncated icosahedron has 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices.

A spherical panoramic photo of Niagara Falls showing a typical point in cyan.
A spherical panoramic photo of Niagara Falls showing a typical point in cyan.

Concept: Panoramic Photographic Polyhedral Pavilions} Acquire a spherical panoramic picture. Identify a roughly spherical polyhedron large enough to get eye at center. Map an image to the surface of the spherical polyhedron. Example: target polyhedron is the truncated icosahedron. Create polyhedron. Place head inside the polyhedron. View!