mathematics

Uniform k 21 polytope

Uniform k 21 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 Uniform k 21 polytope rather than just read about it. In short: In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.

Key takeaways

Uniform k 21 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 Uniform k 21 polytope to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Uniform k 21 polytope from memory before moving on to harder problems.

Reference excerpt

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence. Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure.

Family members The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family. They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry. The complete family of Gosset semiregular polytopes are:

triangular prism: −121 (2 triangles and 3 square faces) rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells) demipenteract: 121, 5-ic semiregular figure (16 5-cell and 10 16-cell facets) 2 21 polytope: 221, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets) 3 21 polytope: 321, 7-ic semiregular figure (576 6-simplex and 126 6-orthoplex facets) 4 21 polytope: 421, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets) 5 21 honeycomb: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets) 6 21 honeycomb: 621, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets) Each polytope is constructed from (n − 1)-simplex and (n − 1)-orthoplex facets. The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schläfli symbol of {31,n−1,1} rather than the regular {3n−2,4}. This construction is an implication of two "facet types". Half the facets around each orthoplex ridge are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex. Each has a vertex figure as the previous form. For example, the rectified 5-cell has a vertex figure as a triangular prism.

Elements

See also Uniform 2k1 polytope family Uniform 1k2 polytope family

References T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 A. Boole Stott (1910). "Geometrical deduction of semiregular from regular polytopes and space fillings" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. XI (1). Amsterdam: Johannes Müller. Archived from the original (PDF) on 29 April 2025. P. H. Schoute (1911). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Section I. XI (3). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 January 2025. P. H. Schoute (1913). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Sections II, III, IV. XI (5). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 February 2025. H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985 H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988 G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150–154 John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 411–413: The Gosset Series: n21)

External links PolyGloss v0.05: Gosset figures (Gossetoicosatope) Regular, SemiRegular, Regular faced and Archimedean polytopes Archived 2011-07-19 at the Wayback Machine

Illustrations

Worked examples

Example 1 — a first encounter with Uniform k 21 polytope

Start with the simplest possible case. Write down what Uniform k 21 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 Uniform k 21 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 Uniform k 21 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 Uniform k 21 polytope

In research: Uniform k 21 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 Uniform k 21 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: Uniform k 21 polytope is common in secondary-school and first-year university syllabi. It links to neighbouring topics Multi-dimensional geometry, Uniform polytopes, so understanding it makes those chapters shorter.
In everyday life: Look for Uniform k 21 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 Uniform k 21 polytope in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Uniform k 21 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 Uniform k 21 polytope out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Uniform k 21 polytope in simple terms?

Why does Uniform k 21 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 Uniform k 21 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 Uniform k 21 polytope.