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History

Skeletal polyhedra have appeared in the literature at least since the Italian Renaissance. In 1509 Leonardo da Vinci drew illustrations of several to illustrate Luca Pacioli's book De Divina Proportione ("On Divine Proportion"). Their mathematical significance was first studied in the mid-twentieth century by Branko Grünbaum, who intially referred to them as polyhedra with hollow faces.^[1]

They led to new classes of regular polyhedra which he listed in 1979.^[2] Andreas Dress noticed an omission and enumerated a more complete list in 1985.^[3] The regular skeletal polyhedra have since become known as the Grünbaum-Dress polyhedra.^[4][5]

In 1994 Grünbaum extended the idea to less regular polyhedra^[1] and in 2016 Egon Schulte and Abigail Williams explored uniform skeletal polyhedra.^[6]

See also

• Saddle polyhedron: If the faces are filled in, with for example minimal surfaces, the figure is instead known as a saddle polyhedron.^[7]

References

Angular defect

To continue Polyhedron#Convex polyhedra

The sum of the exterior angles of any convex polygon equals 2π radians (360 degrees). This is [sometimes? - see Talk:Exterior angle theorem] known as the exterior angle theorem. The exterior angle at a corner can be seen as the difference or defect between a straight line and the interior angle. The sum of the exterior angles is just the sum of these angle defects.<Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton (2008), Page 220.>

Convex polyhedra have an analogous angle defect. At each vertex, the face angles of a convex polyhedron always sum to less than 2π. The sum of these angle defects for every vertex is always equal to 4π radians (720 degrees).<Coxeter, Regular Polytopes. Page 23.><Richeson (2008), Page 223-225.>

Angle excess

To continue Polyhedron#Topological characteristics

[See Richeson, as above and ff.]

Uniform honeycombs

From Uniform polytope

Related to the subject of finite uniform polytopes are uniform honeycombs in Euclidean and hyperbolic spaces. Euclidean uniform honeycombs are generated by affine Coxeter groups and hyperbolic honeycombs are generated by the hyperbolic Coxeter groups. Two affine Coxeter groups can be multiplied together.

There are two classes of hyperbolic Coxeter groups, compact and paracompact. Uniform honeycombs generated by compact groups have finite facets and vertex figures, and exist in 2 through 4 dimensions. Paracompact groups have affine or hyperbolic subgraphs, and infinite facets or vertex figures, and exist in 2 through 10 dimensions.

Uniform honeycombs

From Honeycomb (geometry)

A uniform honeycomb is a honeycomb in Euclidean 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples,^[1] also called the Archimedean honeycombs.

A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Two are quasiregular (made from two types of regular cells):

More information Type, Regular cubic honeycomb ...

Type	Regular cubic honeycomb	Quasiregular honeycombs
Cells	Cubic	Octahedra and tetrahedra
Slab layer
Slab construction	= {}×{4,4}	= s{2,6,3}

Close

The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers.

References