Login
Login

List of isotoxal polyhedra and tilings

{{short description|none}}

In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge.Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, {{isbn|0-521-55432-2}}, p. 371 Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric.

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).

Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.

The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)

Convex isotoxal polyhedra

The dual of a convex polyhedron is also a convex polyhedron.{{Cite web|title=duality|url=http://maths.ac-noumea.nc/polyhedr/dual_.htm|access-date=2020-10-01|website=maths.ac-noumea.nc}}

There are nine convex isotoxal polyhedra based on the Platonic solids: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.

The vertex figures of the quasiregular forms are (squares or) rectangles; the vertex figures of the duals of the quasiregular forms are (equilateral triangles and equilateral triangles, or) equilateral triangles and squares, or equilateral triangles and regular pentagons.

class="wikitable"

!Form

!Regular

!Dual regular

!Quasiregular

!Quasiregular dual

Wythoff symbol

! q {{pipe}} 2 p

! p {{pipe}} 2 q

! 2 {{pipe}} p q

! 

Vertex configuration

!pq

!qp

!p.q.p.q

!

valign=top

|p=3
q=3

|75px
Tetrahedron
{3,3}
{{CDD|node_1|3|node|3|node}}
3 | 2 3

|75px
Tetrahedron
{3,3}
{{CDD|node|3|node|3|node_1}}
3 | 2 3

|75px
Tetratetrahedron
(Octahedron)
{{CDD|node|3|node_1|3|node}}
2 | 3 3

|75px
Cube
(Rhombic hexahedron)

valign=top

|p=4
q=3

|75px
Cube
{4,3}
{{CDD|node_1|4|node|3|node}}
3 | 2 4

|75px
Octahedron
{3,4}
{{CDD|node|4|node|3|node_1}}
4 | 2 3

|75px
Cuboctahedron
{{CDD|node|4|node_1|3|node}}
2 | 3 4

|75px
Rhombic dodecahedron

valign=top

|p=5
q=3

|75px
Dodecahedron
{5,3}
{{CDD|node_1|5|node|3|node}}
3 | 2 5

|75px
Icosahedron
{3,5}
{{CDD|node|5|node|3|node_1}}
5 | 2 3

|75px
Icosidodecahedron
{{CDD|node|5|node_1|3|node}}
2 | 3 5

|75px
Rhombic triacontahedron

Isotoxal star-polyhedra

The dual of a non-convex polyhedron is also a non-convex polyhedron. (By contraposition.)

There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:

class="wikitable"
FormQuasiregularQuasiregular dual

valign="top"

!p=3
q=3

|75px75px
Tetrahemihexahedron

|75px
Tetrahemihexacron

valign="top"

!rowspan="2"|p=4
q=3

|75px75px
Cubohemioctahedron

|75px
Hexahemioctacron

valign="top"

|75px75px
Octahemioctahedron

|75px
Octahemioctacron (visually indistinct from Hexahemioctacron) (*)

valign="top"

!rowspan="2"|p=5
q=3

|75px75px
Small icosihemidodecahedron

|75px
Small icosihemidodecacron (visually indistinct from Small dodecahemidodecacron) (*)

valign="top"

|75px75px
Small dodecahemidodecahedron

|75px
Small dodecahemidodecacron

(*) Faces, edges, and intersection points are the same; only, some other of these intersection points, not at infinity, are considered as vertices.

There are sixteen non-convex isotoxal polyhedra based on the Kepler–Poinsot polyhedra: the four (regular) Kepler–Poinsot polyhedra, the six (quasiregular) common cores of dual Kepler–Poinsot polyhedra (including four hemipolyhedra), and their six duals (including four (infinite) hemipolyhedron-duals):

class="wikitable"
Form

!Regular

!Dual regular

!Quasiregular

!Quasiregular dual

Wythoff symbol

! q {{pipe}} 2 p

! p {{pipe}} 2 q

! 2 {{pipe}} p q

! 

Vertex configuration

!pq

!qp

!p.q.p.q

!

valign=top

!rowspan="3"|p=5/2
q=3

|rowspan="3"|75px75px
Great stellated dodecahedron
{5/2,3}

{{CDD|node_1|5|rat|d2|node}}{{CDD|3|node}}
3 | 2 5/2

|rowspan="3" valign=top|75px75px
Great icosahedron
{3,5/2}

{{CDD|node}}{{CDD|5|rat|d2|node}}{{CDD|3|node_1}}
5/2 | 2 3

|75px75px
Great icosidodecahedron
 
{{CDD|node}}{{CDD|5|rat|d2|node_1|3|node}}
2 | 3 5/2

|75px
Great rhombic triacontahedron

valign=top

|75px75px
Great icosihemidodecahedron

|75px
Great icosihemidodecacron

valign="top"

|75px75px
Great dodecahemidodecahedron

|75px
Great dodecahemidodecacron

valign=top

!rowspan="3"|p=5/2
q=5

|rowspan="3"|75px75px
Small stellated dodecahedron
{5/2,5}

{{CDD|node_1|5|rat|d2|node}}{{CDD|5|node}}
5 | 2 5/2

|rowspan="3"|75px75px
Great dodecahedron
{5,5/2}

{{CDD|node}}{{CDD|5|rat|d2|node}}{{CDD|5

node_1}}
5/2 | 2 5

|75px75px
Dodecadodecahedron
 
{{CDD|node}}{{CDD|5|rat|d2|node_1|5|node}}
2 | 5 5/2

| 75px
Medial rhombic triacontahedron

valign="top"

|75px75px
Small icosihemidodecahedron

|75px
Small dodecahemicosacron

valign="top"

|75px75px
Great dodecahemidodecahedron

|75px
Great dodecahemicosacron

Finally, there are six other non-convex isotoxal polyhedra: the three quasiregular ditrigonal (3 | p q) star polyhedra, and their three duals:

class="wikitable" width=320
Quasiregular

!Quasiregular dual

3 {{pipe}} p q

! 

valign=top

|75px75px
Great ditrigonal icosidodecahedron
3/2 {{pipe}} 3 5
{{CDD|3|node|d3|rat|d2|node|5|node_1|3}}

|75px
Great triambic icosahedron

75px75px
Ditrigonal dodecadodecahedron
3 {{pipe}} 5/3 5
{{CDD|3|node}}{{CDD|5|rat|d3|node_1|5|node}}{{CDD|3}}

|75px
Medial triambic icosahedron

75px75px
Small ditrigonal icosidodecahedron
3 {{pipe}} 5/2 3
{{CDD|3|node}}{{CDD|5|rat|d2|node_1|3|node}}{{CDD|3}}

|75px
Small triambic icosahedron

Isotoxal tilings of the Euclidean plane

There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal. (The self-dual square tiling recreates itself in all four forms.)

class="wikitable" width=320

!Regular

!Dual regular

!Quasiregular

!Quasiregular dual

valign=top

|75px
Hexagonal tiling
{6,3}
{{CDD|node|6|node|3|node_1}}
6 | 2 3

|75px
Triangular tiling
{3,6}
{{CDD|node_1|6|node|3|node|3}}
3 | 2 3

|75px
Trihexagonal tiling
{{CDD|node|6|node_1|3|node}}
2 | 3 6

|75px
Rhombille tiling

valign=top

|75px
Square tiling
{4,4}
{{CDD|node|4|node|4|node_1}}
4 | 2 4

|75px
Square tiling
{4,4}
{{CDD|node_1|4|node|4|node}}
2 | 4 4

|75px
Square tiling
{4,4}
{{CDD|node|4|node_1|4|node}}
4 | 2 4

|75px
Square tiling
{4,4}

Isotoxal tilings of the hyperbolic plane

There are infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right (p q r) groups.

Here are six (p q 2) families, each with two regular forms, and one quasiregular form. All have rhombic duals of the quasiregular form, but only one is shown:

class="wikitable"
[p,q]

!{p,q}

!{q,p}

!r{p,q}

!Dual r{p,q}

Coxeter-Dynkin

!{{CDD|node_1|p|node|q|node}}

!{{CDD|node|p|node|q|node_1}}

!{{CDD|node|p|node_1|q|node}}

!{{CDD|node|p|node_f1|q|node}}

align=center valign=top

!valign=center|[7,3]

|64px
{7,3}

|64px
{3,7}

|64px
r{7,3}

|64px

align=center valign=top

!valign=center|[8,3]

|64px
{8,3}

|64px
{3,8}

|64px
r{8,3}

|64px

align=center valign=top

!valign=center|[5,4]

|64px
{5,4}

|64px
{4,5}

|64px
r{5,4}

|64px

align=center valign=top

!valign=center|[6,4]

|64px
{6,4}

|64px
{4,6}

|64px
r{6,4}

|64px

align=center valign=top

!valign=center|[8,4]

|64px
{8,4}

|64px
{4,8}

|64px
r{8,3}

|64px

align=center valign=top

!valign=center|[5,5]

|64px
{5,5}

|64px
{5,5}

|64px
r{5,5}

|64px

Here's 3 example (p q r) families, each with 3 quasiregular forms. The duals are not shown, but have isotoxal hexagonal and octagonal faces.

class="wikitable"

!Coxeter-Dynkin

!{{CDD|3|node_1|p|node|q|node|r}}

!{{CDD|3|node|p|node_1|q|node|r}}

!{{CDD|3|node|p|node|q|node_1|r}}

(4 3 3)

|64px
3 | 4 3

|64px
3 | 4 3

|64px
4 | 3 3

(4 4 3)

|64px
4 | 4 3

|64px
3 | 4 4

|64px
4 | 4 3

(4 4 4)

|64px
4 | 4 4

|64px
4 | 4 4

|64px
4 | 4 4

Isotoxal tilings of the sphere

All isotoxal polyhedra listed above can be made as isotoxal tilings of the sphere.

In addition as spherical tilings, there are two other families which are degenerate as polyhedra. Even ordered hosohedron can be semiregular, alternating two lunes, and thus isotoxal:

References

{{Reflist}}

  • {{cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (6.4 Isotoxal tilings, 309–321)
  • {{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 | issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003| bibcode=1954RSPTA.246..401C | s2cid=202575183 }}

{{Tessellation}}

Category:Polyhedra