science

Small ditrigonal dodecacronic hexecontahedron

Small ditrigonal dodecacronic 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 ditrigonal dodecacronic hexecontahedron rather than just read about it. In short: In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron.

Key takeaways

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

Reference excerpt

In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. 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 12 + 1 4 5 ) ≈ 12.661 078 804 43 ∘ {\displaystyle \arccos({\frac {5}{12}}+{\frac {1}{4}}{\sqrt {5}})\approx 12.661\,078\,804\,43^{\circ }} , one of arccos ⁡ ( − 5 12 − 1 60 5 ) ≈ 116.996 396 851 70 ∘ {\displaystyle \arccos(-{\frac {5}{12}}-{\frac {1}{60}}{\sqrt {5}})\approx 116.996\,396\,851\,70^{\circ }} and one of 360 ∘ − arccos ⁡ ( − 1 12 − 19 60 5 ) ≈ 217.681 445 539 45 ∘ {\displaystyle 360^{\circ }-\arccos(-{\frac {1}{12}}-{\frac {19}{60}}{\sqrt {5}})\approx 217.681\,445\,539\,45^{\circ }} . Its dihedral angles equal arccos ⁡ ( − 44 − 3 5 61 ) ≈ 146.230 659 755 53 ∘ {\displaystyle \arccos({\frac {-44-3{\sqrt {5}}}{61}})\approx 146.230\,659\,755\,53^{\circ }} . The ratio between the lengths of the long and short edges is 31 + 5 5 38 ≈ 1.110 008 944 41 {\displaystyle {\frac {31+5{\sqrt {5}}}{38}}\approx 1.110\,008\,944\,41} .

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

External links Weisstein, Eric W. "Small ditrigonal dodecacronic hexecontahedron". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Small ditrigonal dodecacronic hexecontahedron

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

In research: Small ditrigonal dodecacronic 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 ditrigonal dodecacronic 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 ditrigonal dodecacronic 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 ditrigonal dodecacronic 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 ditrigonal dodecacronic hexecontahedron in 20 minutes

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

Frequently asked questions

What is Small ditrigonal dodecacronic hexecontahedron in simple terms?

In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron.

Why does Small ditrigonal dodecacronic 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 ditrigonal dodecacronic 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 ditrigonal dodecacronic hexecontahedron.