Wolfram Cloud Document

This Demonstration shows Hamiltonian tours on various polyhedra.

RibbonPolyhedron (link)

by: Eric Weisstein

Display the set of polygons obtained by extending polyhedron edges perpendicularly inward by a given distance

TwistyPuzzles.com

All of the classic polyhedra have been turned into twisty puzzles.

Preston Alden’s 49×49×49 cube

Announced Saturday, Aug 10, 2024: a new world record in twisty puzzles. It’s 30 kg, 34 cm tall, consists of 13,827 pieces, 14406 stickers and needed four years to construct.

So ... how many positions does this have? First, a few numbers:

This is a very fast growing number:

The numbers are more manageable when expressed with exponents:

SignedPermutations (link)

by: Ed Pegg Jr and Jan Mangaldan

Get all signed permutations of a list

The vertices of an icosahedron:

All the Platonic and Archimedean solids are easy to represent:

Let’s get the mesh regions back into polyhedra:

PolyhedralGraphFaces (link)

by: Ed Pegg Jr

Get the faces corresponding to a polyhedral graph

Find the faces corresponding to a set of polyhedral graph edges:

Show the faces:

All polyhedral graphs with 1 to 10 vertices, grouped by number of vertices, in “g6” format:

PolyhedronCanonicalForm (link)

by George W. Hart

Other code at Canonical Polyhedra (data) and Canonical Polyhedra (demo).

Show

Generate the canonical form of a convex polyhedron. The center of each edge is tangent to the unit sphere:

We can look at the canonical forms for the seven polyhedral graphs with six vertices:

The volume of the second is curiously close to Pi:

The polylist earlier was all the polyhedral graphs with a given number of vertices.
The dual graphs give all the polyhedral graphs with a given number of faces:

Canonical forms of the seven hexahedra:

Canonical forms of the 34 heptahedra:

One tetrahedron, two pentahedra, seven hexahedra, 34 heptahedra, 257 octahedra, 2606 nonahedra, and 32300 decahedra.

Schlegel Diagrams / Tutte Embeddings (link)

A Schlegel diagram is a planar 2D representation of a polyhedron with the following properties:
1. One outer face is taken as a regular polygon.
2. The inner faces are convex polygons.

EulerCharacteristic (F+V-E = 2)

The Euler Characteristic of a polyhedron is 2

Conway’s Toroid with 36 Equilateral Triangular Faces (link)

The smallest known single-hole toroidal polyhedron made up of only equilateral triangles was found by Conway and consists of 36 triangles. Some adjacent triangles are coplanar.

PolyhedronFaceReflect (link)

Reflect a polyhedron over a given face

A sample toroid:

Helix of Tetrahedra (link)

Tetrahedra are arranged in triples along a straight line. The edges of the tetrahedra trace three helices. The vertices of the tetrahedra lie on a cylinder, as well as on two helices, one that turns left and one that turns right.

Box Packing (Hoffman’s Packing Puzzle)

Colored Szilassi Polyhedron (link)

This Demonstration shows a colored Szilassi polyhedron and its net.

Nets of Polyhedra (link)

by: Stephen Wolfram

Generate nets of polyhedra, suitable for constructing the polyhedra out of paper.

RandomPolyhedralNet (link)

Create an unfolding net for a given polyhedron

Icosians

The icosians are a set of Hamiltonian quaternions based on the 120 vertices of the 600-cell.

With them, we can make the Hexakis Icosahedron:

The above can be used as an equivalent of the Icosahedral group.

Constructing Polyhedra Using the Icosahedral Group (link)

Great Rhombic Triacontahedron Sculpture (link)

Cut out all but an S-shaped piece from each of the 30 faces of a great rhombic triacontahedron (GRT) to produce this geometrical sculpture.

Octahedra Constructed from Eight Identical Triangles (link)

The Tetartoid (link)

Playing with Stellations of the Icosahedron (link)

by: Michael Rogers

Play with the facets and cells of stellations of the icosahedron. Cut off a segment (along an axis through a vertex, edge, or face) and see inside. Different color schemes help identify the symmetry, facets, and cells of icosahedral stellations.

Triacontahedron Stellations (link)

Op Art on a Sphere (link)

This Demonstration shows op art graphics on a sphere. You can choose patterns with underlying tetrahedral, octahedral or icosahedral symmetry.

Prince Rupert’s Cube (link)

Three Interpenetrating Golden Bricks (link)

The bricks are fit tightly together as a Borromean ring. When they are compressed, the vertices of the resulting golden rectangles coincide with the vertices of an icosahedron. When the golden rectangles are extended until equal to the square of the golden ratio, the edges coincide with the edges of a dodecahedron.

Heppes’s Two-Tip Tetrahedron (link)

A face of a polyhedron is stable if and only if the orthogonal projection of the center of mass of the polyhedron onto the plane of the face lies inside the face or on an edge. In other words, when the polyhedron is placed on that face, the center of mass is over the face. This tetrahedron has two unstable faces. If you place it on one of the unstable faces, it will topple to the other unstable face and then topple to one of the stable faces.

Reshetov’s Unistable Polyhedra with 14, 15, 16, and 17 Faces (link)

Space-Filling Polyhedra (link)

This illustrates four of the various polyhedra that can fill space. Drag the graphic to see the resulting polyhedra from different vantage points.

There are many more. (On Space Groups and Dirichlet-Voronoi Stereohedra)

The Statue of Regiomontanus (link)

If you want a statue to look as big as possible, where should you stand? If you are too close, the statue will be foreshortened. If far away, the statue will appear small. In the fifteenth century, Johannes Regiomontanus solved this question. The statue appears largest when an imaginary circle that passes through the top and bottom of the statue also passes horizontally through the viewer’s eyes.

What major error did Regiomontanus find?

Densest Tetrahedral Packing (link)

Conway’s Billiard Ball Loop (link)

This Demonstration shows a loop of a billiard ball in a regular tetrahedron discovered by J. H. Conway. Each vertex is a vertex of a triangle on a face with side length one-tenth the length of an edge of the tetrahedron. There are three such loops.

Steinhaus’ Billiard Ball Loop (link)

Curves and Surfaces of Constant Width (link)

by: Ian Calvert

Arbitrary Curves of Constant Width (link)

by: Ed Pegg Jr

Both the circle and the Reuleaux triangle are examples of curves of constant width. Such curves, if fitted into a square, can rotate in constant contact with all four sides. Any triangle can serve as a template for a curve of constant width by putting three pairs of arcs of circles around it, centered at each of the three vertices, as shown by this Demonstration.

Barbier’s theorem proves that a curve with constant width 1 has a perimeter of π.

Meissner Tetrahedra (link)

https://arxiv.org/abs/2107.05769

Peabodies of Constant Width

Here’s one example of recent papers on new surfaces of constant width. But with the new methods are needed polyhedra that will work well with the methods.

Biggest Little Polyhedron (link)

Generalized Waterman Polyhedra (link)

Some Polyhedra with Identical Triangular Faces (link)

Cluster of 30 Tetrahedra

Thirty tetrahedra are arranged in a cluster such that one edge of each tetrahedron coincides with the diagonal of the square face of a small rhombicosidodecahedron. Two vertices of each tetrahedron meet vertices of an icosahedron embedded in the tetrahedron. The vertices of the icosahedron are defined by the intersecting points of golden lines within the regular triangles of the tetrahedron.

Cayley Graphs