Display Help

• The polyhedron itself. Drag to rotate it.
• The page title.
• The polyhedron’s name. May be missing.
• The polyhedron’s discoverer. May be missing. A #value is a reference to a page or diagram within the publications listed here.
• The polyhedron’s unique identifier. This will be the vertex configuration optionally followed by sub-identifiers in brackets. An optional initial letter identifies a set of configurations with the same dihedral angles. An optional digit identifies a particular incidence symbol. Optionally, after a comma may be either an angle or the number of sides or an offset and the number of sides.
• Details about the polyhedron. See below for more information.
• Properties of the polyhedron. See below for more information.
• Various controls that modify the display of the polyhedron. See below for more information.

❶

❷

❸

❹

❺

❻

❼

❽

The details ❻ for the polyhedron give the following:

• Vertices:

  Shows the number of vertices per translation unit and the vertex configuration. The configuration is a list of the faces around any vertex. Repeated faces are given by superscripts: 3.4.4.4 is displayed as 3.4⁴ and 3.4.3.4 is displayed as (3.4)².

• Valency:

  The number of faces (or edges) meeting at a vertex.

• Faces:

  A value in brackets gives the number of face-transitive sets the polyhedron has, unless the polyhedron is isohedral. An example is the rhombicuboctahedron: Faces (3) : 26 = 2 × (4{3}, 6{4}₁, 3{4}₂) It has 3 sets of faces (one triangle and two squares), 26 faces, 2×4 triangular faces, 2×6 square faces of one kind and 2×3 square faces of a second kind. Colours correspond to those within the display. For an infinite polyhedron, the values given are for a single translation unit.

• Edges:

  A value in brackets gives the number of edge-transitive sets the polyhedron has, unless the polyhedron is isotoxal. An example is the snub cuboctahedron: Edges (3) : 60 = 12 × (α₁, 2α₂, 2β) It has 3 sets of edges (α₁, α₂ and β), 60 edges, 12 of edge α₁, 12×2 of edge α₂ and 12×2 of edge β. For an infinite polyhedron, the values given are for a single translation unit.

• Vertex definition:

  This consists of four lines. The first line is a list of faces around a vertex, with subscripts to distinguish the different faces. The next three lines define the edges between these faces. The last of these gives the dihedral angles between the faces, with subscripts to distiguish the different edges with the same dihedral angles. The middle two lines comprise the incidence symbol made up of the vertex symbol and the adjacency symbol. For a full description of these see this page. The vertex symbol defines the symmetries of a vertex and the adjacency symbol defines their connections. No two polyhedra on these pages will have the same vertex definition.

• Dihedral angles:

  This is a list of dihedral angles and their values in degrees or radians.

• Genus:

  This is the genus of the polyhedron.

• Volume:

  When displayed, gives the proportion of space filled. For slabs gives the proportion of the slab filled.

• ∑ vertex angles:

  This is the sum of the angles of the faces meeting at a vertex. It is conjectured that this value is never greater than 720°.

• Sphere/Cylindrical/Plane/Layer/Space group:

  The space group of the polyhedron.

• Honeycomb:

  If given, the polyhedron can be found by selecting a subset of polyhedron from the honeycomb. See the honeycomb introduction page for more information about honeycombs.

• Structure:

  When displayed, shows how a polyhedron is structured. See below.

Various further properties ❼ of the polyhedron are displayed:

• Regular:

  Y if the polyhedron is regular, otherwise N.

• Isogonal:

  Y if the polyhedron is isogonal, otherwise N.

• Isohedral:

  Y if the polyhedron is isohedral, otherwise N.

• Isotoxal:

  Y if the polyhedron is isotoxal, otherwise N.

• Isopleural:

  Y if the polyhedron is isopleural, otherwise N.

• Dipleural:

  Y if the polyhedron is dipleural, otherwise N.

• Equitransitive:

  Y if the polyhedron is equitransitive, otherwise N.

• Flexible:

  Y if the polyhedron is flexible, otherwise N.

• Chiral:

  Y if the polyhedron is chiral, otherwise N. When chiral, the enantiomorph menu option is available.

• Complement:

  The complement can be self complementary or reverse complementary. Alternatively, a Y indicates that the polyhedron has a complement that can be displayed. No attempt is made to display the complement of spherical, cylindical or slab polyhedra. Use the options menu option to display the complement or press the C key.

The slider controls ❽ can all be controlled from the keyboard. One key will move the slider up and when used with the Shift key will move it down. For all keys go to the shortcuts page.

• Polyhedron:

  Select a polyhedron for this page. The ← and → keys can be used to step through the polyhedra.

• Offset:

  Used by a single tube polyhedron. Adjusts offset between rows of current polyhedron. Can be adjusted using the O key.

• Sides:

  Used only by tube polyhedron and prisms. Adjusts the sides of the current polyhedron. Can be adjusted using the S key.

• Angle:

  Used only by flexible polyhedra. Adjusts a dihedral angle of the current polyhedron. Can be adjusted using the N key or the [ and ] keys.

• X repeat:
  Y repeat:
  Z repeat:
  

  Adjusts the amount of repetion along an axis of the current polyhedron. Can be adjusted using the X, Y or Z keys. Alternatively the number keys can be used to set all three values at once. Finally the D key will set them to their default values.

• Perspective:

  Adjusts the amount of perspecitve of all polyhedra. Can be adjusted using the P key.

• Outline width:

  Adjusts the width of the outline of all polyhedra. Can be adjusted using the W key.

Structure

Not all polyhedra have structure. When they do, structure shows how a polyhedron can be constructed by combining simpler polyhedra. Colour is used to highlight the currently displayed structure. These structures can be selected and cycled through them using either a menu option or the R key.

Each structure consists of blue polyhedra connected by yellow or yellow and red polyhedra. A descriptor such as v(p) ⟺ w(q) means that polyhedron p connects to w of polyhedron q and polyhedron q connects to v of polyhedron p. A descriptor such as 2(t) ⟺ v(p) / w(q) means that two tilings t are connected upwards by v of polyhedron p and downwards by w polyhedral holes q.

Some polyhedra are given by a name:

• Py₄: Square pyramid
• Cp₃: Triangular cupola
• Bc₄: Square bicupola

Most polyhedra are described by their vertex configuration (see vertices above). A subscript is used to distinguish between polyhedra sharing a vertex configuration. A polyhedron (p)_n with a numeric subscript indicates a tube with n sides. Otherwise n can be one of the following:

• C: Corrugated tiling
• Cr: Crinkled tiling
• F: Folded tiling
• Rh: Rhombic tube

Sometimes a polyhedron will have a rhombic face Rh, for example 4².Rh is a rhombic prism. When structural polyhedra with rhombic faces occur, they will be joined by such faces, hence the full polyhedron will not contain these faces.