science

Small retrosnub icosicosidodecahedron

Small retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron rather than just read about it. In short: In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.

Key takeaways

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

Reference excerpt

In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. It is given a Schläfli symbol sr{⁵/₃,³/₂}. The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries. George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).

Convex hull Its convex hull is a nonuniform truncated dodecahedron.

Cartesian coordinates Let ξ = − 3 2 − 1 2 1 + 4 ϕ ≈ − 2.86676039917 {\displaystyle \xi =-{\frac {3}{2}}-{\frac {1}{2}}{\sqrt {1+4\phi }}\approx -2.86676039917} be the smallest (most negative) zero of the polynomial P = x 2 + 3 x + ϕ − 2 {\displaystyle P=x^{2}+3x+\phi ^{-2}} , where ϕ {\displaystyle \phi } is the golden ratio. Equivalently, ξ = − 1 − ϕ + ϕ + ϕ + ⋯ = − 1 − β {\displaystyle \xi =-1-{\sqrt {\phi +{\sqrt {\phi +{\sqrt {\phi +\cdots }}}}}}\,=-1-\beta } where β ≈ 1.86676039917 {\displaystyle \beta \approx 1.86676039917} (OEIS: A275828) is a root of β 2 − β − ϕ = 0. {\displaystyle \beta ^{2}-\beta -\phi =0.} Let the point p {\displaystyle p} be given by

p = ( ϕ − 1 ξ + ϕ − 3 ξ ϕ − 2 ξ + ϕ − 2 ) {\displaystyle p={\begin{pmatrix}\phi ^{-1}\xi +\phi ^{-3}\\\xi \\\phi ^{-2}\xi +\phi ^{-2}\end{pmatrix}}} . Let the matrix M {\displaystyle M} be given by

M = ( 1 / 2 − ϕ / 2 1 / ( 2 ϕ ) ϕ / 2 1 / ( 2 ϕ ) − 1 / 2 1 / ( 2 ϕ ) 1 / 2 ϕ / 2 ) {\displaystyle M={\begin{pmatrix}1/2&-\phi /2&1/(2\phi )\\\phi /2&1/(2\phi )&-1/2\\1/(2\phi )&1/2&\phi /2\end{pmatrix}}} .

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Small retrosnub icosicosidodecahedron

Start with the simplest possible case. Write down what Small retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron

In research: Small retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron in 20 minutes

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

Frequently asked questions

What is Small retrosnub icosicosidodecahedron in simple terms?

Why does Small retrosnub icosicosidodecahedron 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 retrosnub icosicosidodecahedron?

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 retrosnub icosicosidodecahedron.