In geometry, a polyhedron ( pl. : polyhedra or polyhedrons; from Greek πολύ (poly-) ‘many’ and ἕδρον (-hedron) ‘base, seat’) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. Sometimes named solidmetria or the science of solid shapes/geometry or geometry that extends into three spatial dimensions; 3-D geometry.
Polyhedra have been a major theme in perspective, both in the spatial objects/themes depicted and in the perspective methods employed to image the scene. For example, graphical perspective methods like the legitimate construction, based on a metric grid, can allow extension to depiction of regular polyhedra.
Themes in Perspective
The development, theory, and practice of perspective have been linked to major subjects/themes including optics, geometry, architecture, surveying, and astronomy.
In addition to these traditional disciplines, which served as major topics, several subordinate themes emerged. Some were closely connected with geometry, namely, the two main methods of constructing an artificial linear perspective image (legitimate construction and distance point method), polyhedra, irregular objects, letters, scenography, building interiors, quadratura (ceiling paintings), columns, ancient ruins, idealised buildings, towns, landscapes, gardens, nature, and the human form.
More recently, new themes and categories of perspective have been developed, including photography, motion perspective (cinema), New Media perspective, and Virtual /Augmented Reality.
Platonic Solids
Platonic solids are convex three-dimensional shapes with identical faces and angles. They are also known as regular polyhedra.
Platonic solids
- Tetrahedron: A pyramid with four equilateral triangular faces
- Cube: A hexahedron with six square faces
- Octahedron: A shape with eight equilateral triangular faces
- Dodecahedron: A shape with twelve regular pentagonal faces
- Icosahedron: A shape with twenty equilateral triangular faces
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other.
Space – Tessellation of
Space tessellation, or tiling, involves dividing space (2-D or 3-D) into shapes (tiles) that fit perfectly without gaps or overlaps, like floor tiles or honeycomb structures, forming regular patterns or complex cellular divisions for analysis in fields like GIS, astronomy, and crystallography. These patterns can be simple (squares, hexagons) or complex polyhedra, used to cover surfaces or fill volumes, with applications in art, architecture, and data visualisation.
One interesting example is geodesic dome, which is a dome constructed from convex polyhedra that approximates a sphere or hemisphere.
Artists / Geometers / Scientists
Polyhedra have been a common theme for a variety of prominent creatives.
Maurits Cornelis Escher (1898 – 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired by mathematics. His work features mathematical objects and operations, including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations.
The Dymaxion map, invented by R. Buckminster Fuller, is a unique, uninterrupted polyhedral projection of Earth’s surface onto an icosahedron, revealing the world as “one island in one ocean” with minimal distortion of continent shapes and sizes, challenging traditional maps by removing obvious geographic divisions. It transforms from a flat map into a 3-D globe, emphasising global interconnectedness.
Harold Scott MacDonald “Donald” Coxeter (1907 – 2003) was a British-Canadian mathematician. He is regarded as one of the greatest geometers of the 20th century, who made important contributions to the theory of non-Euclidean geometry. He developed a type of non-Euclidean and mathematical perspective investigated by M.C. Escher.
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Perspective in Professional Practice
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