Infinite-order triangular tiling
Concept in geometry
title: "Infinite-order triangular tiling" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["hyperbolic-tilings", "infinite-order-tilings", "isogonal-tilings", "isohedral-tilings", "regular-tilings", "triangular-tilings"] description: "Concept in geometry" topic_path: "general/hyperbolic-tilings" source: "https://en.wikipedia.org/wiki/Infinite-order_triangular_tiling" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Concept in geometry ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a4/H3_33inf_UHS_plane_at_infinity.png" caption="{3,3,∞}]] honeycomb has {3,∞} vertex figures."] ::
In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.
Symmetry
A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction. ::data[format=table] | [[File:Infinite-order triangular tiling.svg|150px]]Alternated colored tiling | [[File:Iii symmetry mirrors.png|150px]] *∞∞∞ symmetry | [[File:Apolleangasket symmetry.png|150px]]Apollonian gasket with *∞∞∞ symmetry | |---|---|---| ::
Related polyhedra and tiling
This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.
Other infinite-order triangular tilings
A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here: :[[File:Ideal-triangle hyperbolic tiling.svg|240px]]
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
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