In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, one
octagon
In geometry, an octagon () is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
, and one
apeirogon
In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an in ...
on each
vertex. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of tr.
Related polyhedra and tilings
Symmetry
The dual of this tiling represents the fundamental domains of
ˆž,4 (*∞42) symmetry. There are 15 small index subgroups constructed from
ˆž,4by 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
subgroup index-8 group,
+,∞,1+,4,1+(∞2∞2) is the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of
ˆž,4
A larger subgroup is constructed as
ˆž,4* index 8, as
ˆž,4+ (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞
4), and another
ˆž*,4 index ∞ as
ˆž+,4 (∞*2) with gyration points removed as (*2
∞). And their direct subgroups
ˆž,4*sup>+,
ˆž*,4sup>+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2
∞).
See also
*
Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619).
Notation of Euclidean tilings
Eucl ...
*
List of uniform planar tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their du ...
References
*
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
*
External links
*
*
Hyperbolic and Spherical Tiling Gallery
{{Tessellation
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings