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Order-3 apeirogonal tiling

{{Short description|Periodic tiling of the hyperbolic disk}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|Ui3_0}}

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.

Images

Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.

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The edges of the tiling, shown in blue, form an order-3 Cayley tree.

Uniform colorings

Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:

class=wikitable

!Regular

!colspan=3|Truncations

align=center

|120px
{∞,3}
{{CDD|node_1|infin|node|3|node}}

|120px
t0,1{∞,∞}
{{CDD|node_1|infin|node_1|infin|node}}

|120px
t1,2{∞,∞}
{{CDD|node|infin|node_1|infin|node_1}}

|120px
t{∞[3]}
{{CDD|node_1|split1-ii|branch_11|labelinfin}}

Colspan=4|Hyperbolic triangle groups
align=center

|120px
[∞,3]

|colspan=2|120px
[∞,∞]

|120px
[(∞,∞,∞)]

= Symmetry=

The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.

A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞∞) with gyration points removed, becomes (*∞∞).

class="wikitable collapsible collapsed"

!colspan=7| Subgroups of [(∞,∞,∞)] (*∞∞∞)

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!Index

!1

!colspan=3|2

!colspan=2|4

align=center

!Diagram

|120px

|120px

|120px

|120px

|120px

|120px

align=center

!Coxeter

|[(∞,∞,∞)]
{{CDD|node_c1|split1-ii|branch_c3-2|labelinfin}}

|[(1+,∞,∞,∞)]
{{CDD|labelh|node|split1-ii|branch_c3-2|labelinfin}} = {{CDD|labelinfin|branch_c3-2|iaib-cross|branch_c3-2|labelinfin}}

|[(∞,1+,∞,∞)]
{{CDD|node_c1|split1-ii|branch_h0c2|labelinfin}} = {{CDD|labelinfin|branch_c1-2|iaib-cross|branch_c1-2|labelinfin}}

|[(∞,∞,1+,∞)]
{{CDD|node_c1|split1-ii|branch_c3h0|labelinfin}} = {{CDD|labelinfin|branch_c1-3|iaib-cross|branch_c1-3|labelinfin}}

|[(1+,∞,1+,∞,∞)]
{{CDD|labelh|node|split1-ii|branch_h0c2|labelinfin}}

|[(∞+,∞+,∞)]
{{CDD|node_h2|split1-ii|branch_h2h2|labelinfin}}

align=center

!Orbifold

|*∞∞∞

|colspan=3|*∞∞∞∞

|∞*∞∞∞

|∞∞∞×

align=center

!Diagram

|

|120px

|120px

|120px

|120px

|120px

align=center

!Coxeter

|

|[(∞,∞+,∞)]
{{CDD|node_c1|split1-ii|branch_h2h2|labelinfin}}

|[(∞,∞,∞+)]
{{CDD|node_h2|split1-ii|branch_c3h2|labelinfin}}

|[(∞+,∞,∞)]
{{CDD|node_h2|split1-ii|branch_h2c2|labelinfin}}

|[(∞,1+,∞,1+,∞)]
{{CDD|node_c1|split1-ii|branch_h0h0|labelinfin}}

|[(1+,∞,∞,1+,∞)]
{{CDD|labelh|node|split1-ii|branch_c3h2|labelinfin}} = {{CDD|labelinfin|branch_c3h2|2a2b-cross|branch_c3h2|labelinfin}}

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!Orbifold

|

|colspan=3|∞*∞

|colspan=2|∞*∞∞∞

align=center

!colspan=7|Direct subgroups

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!Index

!2

!colspan=3|4

!colspan=2|8

align=center

!Diagram

|120px

|120px

|120px

|120px

|colspan=2|120px

align=center

!Coxeter

|[(∞,∞,∞)]+
{{CDD|node_h2|split1-ii|branch_h2h2|labelinfin}}

|[(∞,∞+,∞)]+
{{CDD|labelh|node|split1-ii|branch_h2h2|labelinfin}} = {{CDD|labelinfin|branch_h2h2|iaib-cross|branch_h2h2|labelinfin}}

|[(∞,∞,∞+)]+
{{CDD|node_h2|split1-ii|branch_h0h2|labelinfin}} = {{CDD|labelinfin|branch_h2h2|iaib-cross|branch_h2h2|labelinfin}}

|[(∞+,∞,∞)]+
{{CDD|node_h2|split1-ii|branch_h2h0|labelinfin}} = {{CDD|labelinfin|branch_h2h2|iaib-cross|branch_h2h2|labelinfin}}

|colspan=2|[(∞,1+,∞,1+,∞)]+
{{CDD|labelh|node|split1-ii|branch_h0h0|labelinfin}} = {{CDD|node_h4|split1-ii|branch_h4h4|labelinfin}}

align=center

!Orbifold

|∞∞∞

|colspan=3|∞∞∞∞

|colspan=2|∞∞∞∞∞∞

align=center

!colspan=7|Radical subgroups

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!Index

!colspan=3|∞

!colspan=3|∞

align=center

!Diagram

|120px

|120px

|120px

|120px

|120px

|120px

align=center

!Coxeter

|[(∞,∞*,∞)]

|[(∞,∞,∞*)]

|[(∞*,∞,∞)]

|[(∞,∞*,∞)]+

|[(∞,∞,∞*)]+

|[(∞*,∞,∞)]+

align=center

!Orbifold

|colspan=3|∞*∞∞

|colspan=3|∞∞

Related polyhedra and tilings

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.

{{Order-3 tiling table}}

{{Order i-3 tiling table}}

{{Order i-i tiling table}}

{{Order i-i-i tiling table}}

Functions on tilings

Functions on this tiling that have symmetry properties tied to it are called modular functions; the modular forms are a special case. This is visually evident in the visualizations of Klein's j-invariant on the Poincaré disk, as well as the Eisenstein series.

See also

{{Commons category|Order-3 apeirogonal tiling}}

References

{{reflist}}

{{refbegin}}

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

{{refend}}

{{Tessellation}}

Category:Apeirogonal tilings

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isohedral tilings

Category:Order-3 tilings

Category:Regular tilings