science

Small dodecicosidodecahedron

Small dodecicosidodecahedron 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 dodecicosidodecahedron rather than just read about it. In short: In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices.

Key takeaways

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

Reference excerpt

In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.

Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common).

Dual

The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.

Proportions Faces have two angles of arccos ⁡ ( 5 8 + 1 8 5 ) ≈ 25.242 832 961 52 ∘ {\displaystyle \arccos({\frac {5}{8}}+{\frac {1}{8}}{\sqrt {5}})\approx 25.242\,832\,961\,52^{\circ }} , one of arccos ⁡ ( − 1 8 + 9 40 5 ) ≈ 67.783 011 547 44 ∘ {\displaystyle \arccos(-{\frac {1}{8}}+{\frac {9}{40}}{\sqrt {5}})\approx 67.783\,011\,547\,44^{\circ }} and one of 360 ∘ − arccos ⁡ ( − 1 4 − 1 10 5 ) ≈ 241.731 322 529 52 ∘ {\displaystyle 360^{\circ }-\arccos(-{\frac {1}{4}}-{\frac {1}{10}}{\sqrt {5}})\approx 241.731\,322\,529\,52^{\circ }} . Its dihedral angles equal arccos ⁡ ( − 19 − 8 5 41 ) ≈ 154.121 363 125 78 ∘ {\displaystyle \arccos({\frac {-19-8{\sqrt {5}}}{41}})\approx 154.121\,363\,125\,78^{\circ }} . The ratio between the lengths of the long and short edges is 7 + 5 6 ≈ 1.539 344 662 92 {\displaystyle {\frac {7+{\sqrt {5}}}{6}}\approx 1.539\,344\,662\,92} .

References

Coxeter, H. S. M. (May 13, 1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246 (916): 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087. Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links Weisstein, Eric W. "Uniform Polyhedron". MathWorld. Weisstein, Eric W. "Small dodecicosidodecahedron". MathWorld. Weisstein, Eric W. "Small dodecacronic hexecontahedron". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Small dodecicosidodecahedron

Start with the simplest possible case. Write down what Small dodecicosidodecahedron 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 dodecicosidodecahedron 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 dodecicosidodecahedron 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 dodecicosidodecahedron

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

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

Frequently asked questions

What is Small dodecicosidodecahedron in simple terms?

Why does Small dodecicosidodecahedron 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 dodecicosidodecahedron?

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 dodecicosidodecahedron.