science

Semiregular polytope

Semiregular polytope 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 Semiregular polytope rather than just read about it. In short: In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L.

Key takeaways

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

Reference excerpt

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.

Gosset's list In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the k21 polytopes, where the rectified 5-cell is the special case of k = 0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work of Makarov (1988) for four dimensions, and Blind & Blind (1991) for higher dimensions.

Gosset's 4-polytopes (with his names in parentheses)

Rectified 5-cell (Tetroctahedric), Rectified 600-cell (Octicosahedric), Snub 24-cell (Tetricosahedric), , or Semiregular E-polytopes in higher dimensions

5-demicube (5-ic semi-regular), a 5-polytope, ↔ 221 polytope (6-ic semi-regular), a 6-polytope, or 321 polytope (7-ic semi-regular), a 7-polytope, 421 polytope (8-ic semi-regular), an 8-polytope,

Euclidean honeycombs

Semiregular polytopes can be extended to semiregular honeycombs. The semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb (3D), gyrated alternated cubic honeycomb (3D) and the 521 honeycomb (8D). Gosset honeycombs:

Tetrahedral-octahedral honeycomb or alternated cubic honeycomb (Simple tetroctahedric check), ↔ (Also quasiregular polytope) Gyrated alternated cubic honeycomb (Complex tetroctahedric check), Semiregular E-honeycomb:

521 honeycomb (9-ic check) (8D Euclidean honeycomb), Gosset (1900) additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:

Hypercubic honeycomb prism, named by Gosset as the (n – 1)-ic semi-check (analogous to a single rank or file of a chessboard) Alternated hexagonal slab honeycomb (tetroctahedric semi-check),

Hyperbolic honeycombs

There are also hyperbolic uniform honeycombs composed of only regular cells (Coxeter & Whitrow 1950), including:

Hyperbolic uniform honeycombs, 3D honeycombs: Alternated order-5 cubic honeycomb, ↔ (a quasiregular polytope) Tetrahedral-octahedral honeycomb, Tetrahedron-icosahedron honeycomb, Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells: Rectified order-6 tetrahedral honeycomb, Rectified square tiling honeycomb, Rectified order-4 square tiling honeycomb, ↔ Alternated order-6 cubic honeycomb, ↔ (quasiregular) Alternated hexagonal tiling honeycomb, ↔ Alternated order-4 hexagonal tiling honeycomb, ↔ Alternated order-5 hexagonal tiling honeycomb, ↔ Alternated order-6 hexagonal tiling honeycomb, ↔ Alternated square tiling honeycomb, ↔ (quasiregular) Cubic-square tiling honeycomb, Order-4 square tiling honeycomb, = (regular) Tetrahedral-triangular tiling honeycomb, 9D hyperbolic paracompact honeycomb: 621 honeycomb (10-ic check),

Illustrations

Worked examples

Example 1 — a first encounter with Semiregular polytope

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

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

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

Frequently asked questions

What is Semiregular polytope in simple terms?

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L.

Why does Semiregular polytope 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 Semiregular 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 Semiregular polytope.