science

Small hexagrammic hexecontahedron

Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron rather than just read about it. In short: In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron.

Key takeaways

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

Reference excerpt

In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.

Geometry Its faces are hexagonal stars with two short and four long edges. Denoting the golden ratio by ϕ {\displaystyle \phi } and putting ξ = 1 4 + 1 4 1 + 4 ϕ ≈ 0.933 380 199 59 {\displaystyle \xi ={\frac {1}{4}}+{\frac {1}{4}}{\sqrt {1+4\phi }}\approx 0.933\,380\,199\,59} , the stars have five equal angles of arccos ⁡ ( ξ ) ≈ 21.031 988 967 51 ∘ {\displaystyle \arccos(\xi )\approx 21.031\,988\,967\,51^{\circ }} and one of 360 ∘ − arccos ⁡ ( ϕ − 2 ξ − ϕ − 1 ) ≈ 254.840 055 162 43 ∘ {\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 254.840\,055\,162\,43^{\circ }} . Each face has four long and two short edges. The ratio between the edge lengths is

1 / 2 − 1 / 2 × ( 1 − ξ ) / ( ϕ 3 − ξ ) ≈ 0.428 986 992 12 {\displaystyle 1/2-1/2\times {\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.428\,986\,992\,12} . The dihedral angle equals arccos ⁡ ( ξ / ( 1 + ξ ) ) ≈ 61.133 452 273 64 ∘ {\displaystyle \arccos(\xi /(1+\xi ))\approx 61.133\,452\,273\,64^{\circ }} . Part of each face is inside the solid, hence is not visible in solid models.

References Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links Weisstein, Eric W. "Small hexagrammic hexecontahedron". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Small hexagrammic hexecontahedron

Start with the simplest possible case. Write down what Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron

In research: Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron 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: Small hexagrammic hexecontahedron is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dual uniform polyhedra, Polyhedron stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron in 20 minutes

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

Frequently asked questions

What is Small hexagrammic hexecontahedron in simple terms?

In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron.

Why does Small hexagrammic hexecontahedron 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 Small hexagrammic hexecontahedron?

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 Small hexagrammic hexecontahedron.