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Snub trihexagonal tiling

Snub trihexagonal tiling is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Snub trihexagonal tiling rather than just read about it. In short: In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex.

Snub trihexagonal tiling — main illustration
Snub trihexagonal tiling — illustration

Key takeaways

  • Snub trihexagonal tiling belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Snub trihexagonal tiling to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Snub trihexagonal tiling from memory before moving on to harder problems.

Reference excerpt

In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}. Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille). There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry. There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)

Circle packing The snub trihexagonal tiling leads to a circle packing, each vertex becoming the center of a circle of fixed diameter. Every circle is in contact with 5 other circles in the packing (kissing number). The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling.

Related polyhedra and tilings

Symmetry mutations This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

6-fold pentille tiling

In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower. Each of its pentagonal faces has four 120° and one 60° angle. It is the dual of the uniform snub trihexagonal tiling, and has rotational symmetries of orders 6-3-2 symmetry.

Variations The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.

Related k-uniform and dual k-uniform tilings There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labeling F for V34.6, C for V32.4.3.4, B for V33.42, H for V36:

Fractalization Replacing every V36 hexagon by a rhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4. Replacing every V36 hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 32.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4. Replacing every V36 hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.42.6, and one vertex of 3.4.6.4. In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since there is no chiral symmetry (the domains have 3:2 side lengths of 1 + 1 3 : 2 + 2 3 {\displaystyle 1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}} in the rhombitrihexagonal; 1 + 2 3 : 2 + 4 3 {\displaystyle 1+{\frac {2}{\sqrt {3}}}:2+{\frac {4}{\sqrt {3}}}} in the truncated hexagonal; and 1 + 3 : 2 + 2 3 {\displaystyle 1+{\sqrt {3}}:2+2{\sqrt {3}}} in the truncated trihexagonal).

Related tilings

See also

Tilings of regular polygons List of uniform tilings

References

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65) Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p. 39

Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern R, Dual p. 77-76, pattern 5 Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114

External links Weisstein, Eric W. "Uniform tessellation". MathWorld. Weisstein, Eric W. "Semiregular tessellation". MathWorld. Klitzing, Richard. "2D Euclidean tilings s3s6s - snathat - O11".

Illustrations

Snub trihexagonal tiling illustration
Snub trihexagonal tiling illustration
Snub trihexagonal tiling illustration
Snub trihexagonal tiling: There is one related 2-uniform tiling, which mixes the vertex configurations 3.3.3.3.6 of the snub trihexagonal tiling and 3.3.3.3.3.3 of the triangular tiling.
There is one related 2-uniform tiling, which mixes the vertex configurations 3.3.3.3.6 of the snub trihexagonal tiling and 3.3.3.3.3.3 of the triangular tiling.
Snub trihexagonal tiling illustration

Worked examples

Example 1 — a first encounter with Snub trihexagonal tiling

Start with the simplest possible case. Write down what Snub trihexagonal tiling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Snub trihexagonal tiling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Snub trihexagonal tiling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Snub trihexagonal tiling

In research
Snub trihexagonal tiling appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Snub trihexagonal tiling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Snub trihexagonal tiling is common in secondary-school and first-year university syllabi. It links to neighbouring topics Chiral figures, Euclidean tilings, Isogonal tilings, so understanding it makes those chapters shorter.
In everyday life
Look for Snub trihexagonal tiling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Snub trihexagonal tiling in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Snub trihexagonal tiling means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Snub trihexagonal tiling out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Snub trihexagonal tiling in simple terms?

In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex.

Why does Snub trihexagonal tiling matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Snub trihexagonal tiling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Snub trihexagonal tiling.

Tags

  • Chiral figures
  • Euclidean tilings
  • Isogonal tilings
  • Semiregular tilings
  • Snub tilings

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