Mathieu Dutour Sikirić

Mathieu Dutour Sikirić

Uniform Partition of 3-space

A tiling is defined as a uniform partition of 3-space if it is composed of Archimedean polyhedra and its symmetry group is transitive on its vertices. This page provides visual representations, Delaney symbols, and technical data for these structures.

Historical Context & Corrections

Correct enumerations can be found in the works of Deza & Shtogrin (2000), Grünbaum (1996), and Johnson (1991). Historical listings by Andreini, Critchlow, and others often contained errors or omissions, which this resource aims to clarify.

Technical Details

  • Visualization: Pictures generated using 3dt by Olaf Delgado.
  • Encoding: Structures are encoded using Delaney symbols.
  • Space Groups: Referred to standard crystallographic nomenclature (see Wikipedia).
  • Scripts: Scripts for generating these tilings are available here.

The 28 Uniform Partitions

Structure 1 (DS-5, A-2, J-21)

Delaunay tessellation of the A3 lattice.

  • Composition: Tetrahedra & Octahedra (Ratio 1:2)
  • Space Group: Fm-3m
  • Delaney symbol
Structure 1

Structure 2 (DS-24, A-2', J-52)

One of only two vertex-transitive tessellations in the tetrahedron-octahedron continuum.

  • Composition: Tetrahedra & Octahedra (Ratio 1:2)
  • Space Group: P63/mmc
  • Delaney symbol
Structure 2

Structure 3 (DS-25, J-61)

Created by inserting a layer of Prism3 between layers of structure 1.

  • Composition: Tetrahedron, Octahedron, Prism3 (2:1:3)
  • Space Group: R-3m
  • Delaney symbol
Structure 3

Structure 4 (DS-26, J-62)

Created by inserting a layer of Prism3 between layers of structure 2.

  • Composition: Tetrahedron, Octahedron, Prism3 (2:1:3)
  • Space Group: P63/mmc
  • Delaney symbol
Structure 4

Structure 5 (DS-19, A-16, J-23)

  • Composition: Tetrahedron, Rhombicuboctahedron, Cube (2:1:1)
  • Space Group: Fm-3m
  • Delaney symbol
Structure 5

Structure 6 (DS-6, A-13, J-25)

  • Composition: Tetrahedron, Truncated Tetrahedron (1:1)
  • Space Group: Fd-3m
  • Delaney symbol
Structure 6

Structure 7 (DS-8, A-15, J-12)

  • Composition: Octahedron, Cuboctahedron (1:1)
  • Space Group: Pm-3m
  • Delaney symbol
Structure 7

Structure 8 (DS-7, A-14, J-13)

  • Composition: Octahedron, Truncated Cube (1:1)
  • Space Group: Pm-3m
  • Delaney symbol
Structure 8

Structure 9 (DS-18, A-17, J-14)

  • Composition: Cuboctahedron, Rhombicuboctahedron, Cube (1:1:3)
  • Space Group: Pm-3m
  • Delaney symbol
Structure 9

Structure 10 (DS-20, A-21, J-22)

  • Composition: Cuboctahedron, Truncated Tetrahedron, Truncated Octahedron (1:2:1)
  • Space Group: Fm-3m
  • Delaney symbol
Structure 10

Structure 11 (DS-3, A-2, J-21)

Lamination over plane tiling by triangles.

Structure 11

Structure 12 (DS-27, J-63)

Superimposed layers of twisted triangle prisms.

Structure 12

Structure 13 (DS-13, A-11', J-65)

Structure 13

Structure 14 (DS-14, A-11, J-44)

Lamination over tiling by squares and triangles.

Structure 14

Structure 15 (DS-28, J-64)

Similar to Structure 12 but with intermediate cube layers.

Structure 15

Structure 16 (DS-21, A-9, J-47)

  • Composition: Prism3, Prism4, Prism6 (2:3:1)
  • Space Group: P6/mmm
  • Delaney symbol
Structure 16

Structure 17 (DS-12, A-12, J-48)

Structure 17

Structure 18 (DS-11, A-8, J-43)

Lamination over Kagome.

  • Composition: Prism3, Prism6 (2:1)
  • Space Group: P6/mmm
  • Delaney symbol
Structure 18

Structure 19 (DS-15, A-7, J-46)

  • Composition: Prism3, Prism12 (2:1)
  • Space Group: P6/mmm
  • Delaney symbol
Structure 19

Structure 20 (DS-10, A-19, J-18)

  • Composition: Rhombicuboctahedron, Truncated Cube, Cube, Prism8 (1:1:3:3)
  • Space Group: Pm-3m
  • Delaney symbol
Structure 20

Structure 21 (DS-17, A-20, J-18)

  • Composition: Truncated Tetrahedron, Truncated Cube, Truncated Cuboctahedron (2:1:1)
  • Space Group: Fm-3m
  • Delaney symbol
Structure 21

Structure 22 (DS-1, A-1, J-11)

The standard cubic tiling.

Structure 22

Structure 23 (DS-22, A-10, J-49)

  • Composition: Cube, Prism6, Prism12 (3:2:1)
  • Space Group: P6/mmm
  • Delaney symbol
Structure 23

Structure 24 (DS-10, A-6, J-45)

Structure 24

Structure 25 (DS-16, A-18, J-17)

Delaunay polytopes of orbit (0, 1/4, 1/5) in the cube lattice.

  • Composition: Cube, Truncated Octahedron, Truncated Cuboctahedron (3:1:1)
  • Space Group: Pm-3m
  • Delaney symbol
Structure 25

Structure 26 (DS-4, A-18, J-17)

Structure 26

Structure 27 (DS-9, A-22, J-19)

Delaunay polytopes of orbit (1/3, 1/4, 1/5) in the cube lattice.

  • Composition: Prism8, Truncated Cuboctahedron (3:1)
  • Space Group: Im-3m
  • Delaney symbol
Structure 27

Structure 28 (DS-2, A-3, J-16)

Voronoi polytope of the A3 lattice (Kelvin polytope).

Structure 28