mathematics

Pentagonal polytope

Pentagonal polytope 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 Pentagonal polytope rather than just read about it. In short: In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H.

Key takeaways

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

Reference excerpt

In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral).

Family members The family starts as 1-polytopes and ends with n = 5 as infinite tessellations of 4-dimensional hyperbolic space. There are two types of pentagonal polytopes; they may be termed the dodecahedral and icosahedral types, by their three-dimensional members. The two types are duals of each other.

Dodecahedral The complete family of dodecahedral pentagonal polytopes are:

Line segment, { } Pentagon, {5} Dodecahedron, {5, 3} (12 pentagonal faces) 120-cell, {5, 3, 3} (120 dodecahedral cells) Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ 120-cell facets) The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of one less dimension. Their vertex figures are the simplices of one less dimension.

Icosahedral The complete family of icosahedral pentagonal polytopes are:

Line segment, { } Pentagon, {5} Icosahedron, {3, 5} (20 triangular faces) 600-cell, {3, 3, 5} (600 tetrahedron cells) Order-5 5-cell honeycomb, {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ 5-cell facets) The facets of each icosahedral pentagonal polytope are the simplices of one less dimension. Their vertex figures are icosahedral pentagonal polytopes of one less dimension.

Related star polytopes and honeycombs The pentagonal polytopes can be stellated to form new star regular polytopes:

In two dimensions, we obtain the pentagram {5/2}, In three dimensions, this forms the four Kepler–Poinsot polyhedra, {3,5/2}, {5/2,3}, {5,5/2}, and {5/2,5}. In four dimensions, this forms the ten Schläfli–Hess polychora: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5,3,5/2}, {5/2,3,5}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {3,3,5/2}, and {5/2,3,3}. In four-dimensional hyperbolic space there are four regular star-honeycombs: {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}. In some cases, the star pentagonal polytopes are themselves counted among the pentagonal polytopes. Like other polytopes, regular stars can be combined with their duals to form compounds;

In two dimensions, a decagrammic star figure {10/2} is formed, In three dimensions, we obtain the compound of dodecahedron and icosahedron, In four dimensions, we obtain the compound of 120-cell and 600-cell. Star polytopes can also be combined.

Notes

References Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] Archived 2016-07-11 at the Wayback Machine (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36] Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q, r} in four dimensions, pp. 292–293)

Illustrations

Worked examples

Example 1 — a first encounter with Pentagonal polytope

Start with the simplest possible case. Write down what Pentagonal polytope 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 Pentagonal polytope 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 Pentagonal polytope 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 Pentagonal polytope

In research: Pentagonal polytope 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 Pentagonal polytope 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: Pentagonal polytope is common in secondary-school and first-year university syllabi. It links to neighbouring topics Multi-dimensional geometry, Regular polytopes, so understanding it makes those chapters shorter.
In everyday life: Look for Pentagonal polytope 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 Pentagonal polytope in 20 minutes

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

Frequently asked questions

What is Pentagonal polytope in simple terms?

In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H.

Why does Pentagonal polytope 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 Pentagonal polytope?

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 Pentagonal polytope.