mathematics

Paracompact uniform honeycombs

Paracompact uniform honeycombs 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 Paracompact uniform honeycombs rather than just read about it. In short: In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family.

Key takeaways

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Reference excerpt

In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in two dimensions.

Regular paracompact honeycombs Of the uniform paracompact H3 honeycombs, 11 are regular, meaning that their group of symmetries acts transitively on their flags. These have Schläfli symbol {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}, and are shown below. Four have finite Ideal polyhedral cells: {3,3,6}, {4,3,6}, {3,4,4}, and {5,3,6}.

Coxeter groups of paracompact uniform honeycombs

This is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains (rank 4 paracompact coxeter groups). The honeycombs are indexed here for cross-referencing duplicate forms, with brackets around the nonprimary constructions. The alternations are listed, but are either repeats or don't generate uniform solutions. Single-hole alternations represent a mirror removal operation. If an end-node is removed, another simplex (tetrahedral) family is generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored. These nonsimplectic (pyramidal) Coxeter groups are not enumerated on this page, except as special cases of half groups of the tetrahedral ones. Seven uniform honeycombs that arise here as alternations have been numbered 152 to 158, after the 151 Wythoffian forms not requiring alternation for their construction.

The complete list of nonsimplectic (non-tetrahedral) paracompact Coxeter groups was published by P. Tumarkin in 2003. The smallest paracompact form in H3 can be represented by or , or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4] : = . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is or , constructed as [4,4,1+,4] = [∞,4,4,∞] : = . Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1+,4)] = [((3,∞,3)),((3,∞,3))] or , [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or , [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or . = , = , = . Another nonsimplectic half groups is ↔ . A radical nonsimplectic subgroup is ↔ , which can be doubled into a triangular prism domain as ↔ .

Linear graphs

[6,3,3] family

[6,3,4] family There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or

[6,3,5] family

[6,3,6] family There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or

[3,6,3] family There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or

[4,4,3] family There are 15 forms, generated by ring permutations of the Coxeter group: [4,4,3] or

[4,4,4] family There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or .

Tridental graphs

[3,41,1] family There are 11 forms (of which only 4 are not shared with the [4,4,3] family), generated by ring permutations of the Coxeter group:

[4,41,1] family There are 7 forms, (all shared with [4,4,4] family), generated by ring permutations of the Coxeter group:

[6,31,1] family There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group: [6,31,1] or .

Cyclic graphs

[(4,4,3,3)] family There are 11 forms, 4 unique to this family, generated by ring permutations of the Coxeter group: , with ↔ .

[(4,4,4,3)] family There are 9 forms, generated by ring permutations of the Coxeter group: .

[(4,4,4,4)] family There are 5 forms, 1 unique, generated by ring permutations of the Coxeter group: . Repeat constructions are related as: ↔ , ↔ , and ↔ .

[(6,3,3,3)] family There are 9 forms, generated by ring permutations of the Coxeter group: .

[(6,3,4,3)] family There are 9 forms, generated by ring permutations of the Coxeter group:

[(6,3,5,3)] family There are 9 forms, generated by ring permutations of the Coxeter group:

[(6,3,6,3)] family There are 6 forms, generated by ring permutations of the Coxeter group: .

Loop-n-tail graphs

[3,3[3]] family There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [3,3[3]] or . 7 are half symmetry forms of [3,3,6]: ↔ .

[4,3[3]] family There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [4,3[3]] or . 7 are half symmetry forms of [4,3,6]: ↔ .

[5,3[3]] family There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [5,3[3]] or . 7 are half symmetry forms of [5,3,6]: ↔ .

[6,3[3]] family There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [6,3[3]] or . 7 are half symmetry forms of [6,3,6]: ↔ .

Multicyclic graphs

[3[ ]×[ ]] family There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group: . Two are duplicated as ↔ , two as ↔ , and three as ↔ .

[3[3,3]] family There are 4 forms, 0 unique, generated by ring permutations of the Coxeter group: . They are repeated in four families: ↔ (index 2 subgroup), ↔ (index 4 subgroup), ↔ (index 6 subgroup), and ↔ (index 24 subgroup).

Summary enumerations by family

Linear graphs

Tridental graphs

Cyclic graphs

Loop-n-tail graphs Symmetry in these graphs can be doubled by adding a mirror: [1[n,3[3]]] = [n,3,6]. Therefore ring-symmetry graphs are repeated in the linear graph families.

See also Uniform tilings in hyperbolic plane List of regular polytopes#Tessellations of hyperbolic 3-space Uniform honeycombs in hyperbolic space

Notes

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Illustrations

Worked examples

Example 1 — a first encounter with Paracompact uniform honeycombs

Start with the simplest possible case. Write down what Paracompact uniform honeycombs 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 Paracompact uniform honeycombs 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 Paracompact uniform honeycombs 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 Paracompact uniform honeycombs

In research: Paracompact uniform honeycombs 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 Paracompact uniform honeycombs 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: Paracompact uniform honeycombs is common in secondary-school and first-year university syllabi. It links to neighbouring topics Honeycombs (geometry), so understanding it makes those chapters shorter.
In everyday life: Look for Paracompact uniform honeycombs 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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Frequently asked questions

What is Paracompact uniform honeycombs in simple terms?

Why does Paracompact uniform honeycombs 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.

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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.

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