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Uniform honeycomb

Uniform honeycomb is a mathematics 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 Uniform honeycomb rather than just read about it. In short: In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex.

Uniform honeycomb — main illustration
Uniform honeycomb — illustration

Key takeaways

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

Reference excerpt

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as n-honeycomb or an n-dimensional honeycomb. An n-dimensional uniform honeycomb can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, and n-dimensional hyperbolic space. A 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation. Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter–Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson. Wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example, 4.4.4.4 represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an n-dimensional uniform tessellation vertex figures are defined by an (n−1)-polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.

Examples of uniform honeycombs

See also Uniform tiling List of uniform tilings Uniform tilings in hyperbolic plane Honeycomb (geometry) Wythoff construction Convex uniform honeycomb List of regular polytopes

References George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49–56. Norman Johnson Uniform Polytopes, Manuscript (1991) Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1. N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.

External links Weisstein, Eric W. "Uniform tessellation". MathWorld. Tessellations of the Plane Klitzing, Richard. "2D Euclidean tesselations".

Illustrations

Uniform honeycomb illustration
Uniform honeycomb illustration
Uniform honeycomb illustration
Uniform honeycomb illustration
Uniform honeycomb illustration

Worked examples

Example 1 — a first encounter with Uniform honeycomb

Start with the simplest possible case. Write down what Uniform honeycomb claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Uniform honeycomb 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 Uniform honeycomb 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 Uniform honeycomb

In research
Uniform honeycomb appears in mathematics 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 Uniform honeycomb 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
Uniform honeycomb is common in secondary-school and first-year university syllabi. It links to neighbouring topics Honeycombs (geometry), Uniform tilings, so understanding it makes those chapters shorter.
In everyday life
Look for Uniform honeycomb 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 Uniform honeycomb in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Uniform honeycomb 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 Uniform honeycomb out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Uniform honeycomb in simple terms?

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex.

Why does Uniform honeycomb matter?

Because it connects several mathematics 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 Uniform honeycomb?

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 Uniform honeycomb.

Tags

  • Honeycombs (geometry)
  • Uniform tilings

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