Richard Buckminster Fuller was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more than 30 books and coining or popularizing such terms as "Spaceship Earth", "Dymaxion", "ephemeralization", "synergetics", and "tensegrity".
In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
A map is a symbolic depiction of relationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or another durable medium, or may be displayed on a programmable medium such as a computer screen. Some maps change interactively. Although maps are commonly used to depict geography, they may represent any space, real or fictional. The subject being mapped may be two-dimensional, such as Earth's surface; three-dimensional, such as Earth's interior; or may even be from an abstract space of any dimension.
In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.
In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography.
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.
The Chamberlin trimetric projection is a map projection where three points are fixed on the globe and the points on the sphere are mapped onto a plane by triangulation. It was developed in 1946 by Wellman Chamberlin for the National Geographic Society. Chamberlin was chief cartographer for the Society from 1964 to 1971. The projection's principal feature is that it compromises between distortions of area, direction, and distance. A Chamberlin trimetric map therefore gives an excellent overall sense of the region being mapped. Many National Geographic Society maps of single continents use this projection.
Bernard Joseph Stanislaus Cahill, American cartographer and architect, was the inventor of the octahedral "Butterfly Map". An early proponent of the San Francisco Civic Center, he also designed hotels, factories and mausoleums like the Columbarium of San Francisco.
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth is preserved in the image of the projection; that is, the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map with a conformal projection cross at a 39° angle.
Synergetics is the empirical study of systems in transformation, with an emphasis on whole system behaviors unpredicted by the behavior of any components in isolation. R. Buckminster Fuller (1895–1983) named and pioneered the field. His two-volume work Synergetics: Explorations in the Geometry of Thinking, in collaboration with E. J. Applewhite, distills a lifetime of research into book form.
The Geoscope was a proposal by Buckminster Fuller around 1960 to create a 200-foot-diameter (61 m) globe that would be covered in colored lights so that it could function as a large spherical display. It was envisioned that the Geoscope would be connected to computers which would allow it to display both historical and current data, and enable people to visualize large scale patterns around the world. Several projects by his students to build a "miniature Earth", starting with a 20-foot version at Cornell University in 1952, were precursors of the Geoscope proposal. Before proposing the Geoscope, Fuller had invented the Dymaxion map, a novel map projection for the whole Earth.
The Waterman "Butterfly" World Map is a map projection created by Steve Waterman. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a polyhedral globe with the shape of a truncated octahedron, evoking the butterfly map principle first developed by Bernard J.S. Cahill (1866–1944) in 1909. Cahill and Waterman maps can be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans.
The Cahill–Keyes projection is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier 1909 projection by Bernard Cahill. The projection was designed to achieve a number of desirable characteristics, namely symmetry of component maps (octants), scalability allowing the map to continue to work well even at high resolution, uniformity of geocells, metric-based joining edges, minimized distortion compared to a globe, and an easily understood orientation to enhance general usability and teachability.
AuthaGraph is an approximately equal-area world map projection invented by Japanese architect Hajime Narukawa in 1999. The map is made by equally dividing a spherical surface into 96 triangles, transferring it to a tetrahedron while maintaining area proportions, and unfolding it in the form of a rectangle: it is a polyhedral map projection. The map substantially preserves sizes and shapes of all continents and oceans while it reduces distortions of their shapes, as inspired by the Dymaxion map. The projection does not have some of the major distortions of the Mercator projection, like the expansion of countries in far northern latitudes, and allows for Antarctica to be displayed accurately and in whole. Triangular world maps are also possible using the same method. The name is derived from "authalic" and "graph".
The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon elliptic functions. It is conformal everywhere except for the four singularities at the vertices of the polyhedron. Because of the nature of polyhedra, this map projection can be tessellated infinitely in the plane. It was developed by Laurence Patrick Lee in 1965.
In map projections, an interruption is any place where the globe has been split. All map projections are interrupted at at least one point. Typical world maps are interrupted along an entire meridian. In that typical case, the interruption forms an east/west boundary, even though the globe has no boundaries.
Snyder equal-area projection is a polyhedral map projection used in the ISEA discrete global grids. It is named for John P. Snyder, who developed the projection in the 1990s.
A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.
The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity. Consequently, its vertices can be repositioned by folding at the edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.
