science

Small hexagonal hexecontahedron

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

Key takeaways

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

Reference excerpt

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

Geometry Treating it as a simple non-convex solid (without intersecting surfaces), it has 180 faces (all triangles), 270 edges, and 92 vertices (twelve with degree 10, twenty with degree 12, and sixty with degree 3), giving an Euler characteristic of 92 − 270 + 180 = +2.

Faces The faces are irregular hexagons. Denoting the golden ratio by ϕ {\displaystyle \phi } and putting ξ = 1 4 − 1 4 1 + 4 ϕ ≈ − 0.433 380 199 59 {\displaystyle \xi ={\frac {1}{4}}-{\frac {1}{4}}{\sqrt {1+4\phi }}\approx -0.433\,380\,199\,59} , the hexagons have five equal angles of arccos ⁡ ( ξ ) ≈ 115.682 268 170 75 ∘ {\displaystyle \arccos(\xi )\approx 115.682\,268\,170\,75^{\circ }} and one of arccos ⁡ ( ϕ − 2 ξ − ϕ − 1 ) ≈ 141.588 659 146 23 ∘ {\displaystyle \arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 141.588\,659\,146\,23^{\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 ) ≈ 0.777 024 337 46 {\displaystyle 1/2+1/2\times {\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.777\,024\,337\,46} . The dihedral angle equals arccos ⁡ ( ξ / ( 1 + ξ ) ) ≈ 139.893 813 264 51 ∘ {\displaystyle \arccos(\xi /(1+\xi ))\approx 139.893\,813\,264\,51^{\circ }} .

Construction Disregarding self-intersecting surfaces, the small hexagonal hexecontahedron can be constructed as a Kleetope of a pentakis dodecahedron. It is therefore a second order Kleetope of the regular dodecahedron. In other words, by adding a shallow pentagonal pyramid to each face of a regular dodecahedron, we get a pentakis dodecahedron. By adding an even shallower triangular pyramid to each face of the pentakis dodecahedron, we get a small hexagonal hexecontahedron. The 60 vertices of degree 3 correspond to the apex vertex of each triangular pyramid of the Kleetope, or to each face of the pentakis dodecahedron. The 20 vertices of degree 12 and 12 vertices of degree 10 correspond to the vertices of the pentakis dodecahedron, and also respectively to the 20 hexagons and 12 pentagons of the truncated icosahedron, the dual solid to the pentakis dodecahedron.

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

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

Illustrations

Worked examples

Example 1 — a first encounter with Small hexagonal hexecontahedron

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

In research: Small hexagonal 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 hexagonal 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 hexagonal 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 hexagonal 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 hexagonal hexecontahedron in 20 minutes

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

Frequently asked questions

What is Small hexagonal hexecontahedron in simple terms?

In geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron.

Why does Small hexagonal 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 hexagonal 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 hexagonal hexecontahedron.