science

Icositruncated dodecadodecahedron

Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron rather than just read about it. In short: In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45. Convex hull Its convex hull is a nonuniform truncated icosidodecahedron.

Key takeaways

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

Reference excerpt

In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45.

Convex hull Its convex hull is a nonuniform truncated icosidodecahedron.

Cartesian coordinates Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of

( ± [ 2 − 1 φ ] , ± 1 , ± [ 2 + φ ] ) , ( ± 1 , ± 1 φ 2 , ± [ 3 φ − 1 ] ) , ( ± 2 , ± 2 φ , ± 2 φ ) , ( ± 3 , ± 1 φ 2 , ± φ 2 ) , ( ± φ 2 , ± 1 , ± [ 3 φ − 2 ] ) , {\displaystyle {\begin{array}{crrlc}{\Bigl (}&\pm {\bigl [}2-{\frac {1}{\varphi }}{\bigr ]},&\pm \,1,&\pm {\bigl [}2+\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,1,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm {\bigl [}3\varphi -1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,2,&\pm \,{\frac {2}{\varphi }},&\pm \,2\varphi &{\Bigr )},\\{\Bigl (}&\pm \,3,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm \,\varphi ^{2}&{\Bigr )},\\{\Bigl (}&\pm \,\varphi ^{2},&\pm \,1,&\pm {\bigl [}3\varphi -2{\bigr ]}&{\Bigr )},\end{array}}}

where φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} is the golden ratio.

Related polyhedra

Tridyakis icosahedron

The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.

See also Catalan solid Duals to convex uniform polyhedra Uniform polyhedra List of uniform polyhedra

References Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 Photo on page 96, Dorman Luke construction and stellation pattern on page 97.

External links Weisstein, Eric W. "Icositruncated dodecadodecahedron". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Icositruncated dodecadodecahedron

Start with the simplest possible case. Write down what Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron

In research: Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron 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: Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron in 20 minutes

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

Frequently asked questions

What is Icositruncated dodecadodecahedron in simple terms?

In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45. Convex hull Its convex hull is a nonuniform truncated icosidodecahedron.

Why does Icositruncated dodecadodecahedron 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 Icositruncated dodecadodecahedron?

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 Icositruncated dodecadodecahedron.