science

Medial rhombic triacontahedron

Medial rhombic triacontahedron 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 Medial rhombic triacontahedron rather than just read about it. In short: In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron.

Key takeaways

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

Reference excerpt

In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron. Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is φ ≈ 1.618 {\displaystyle \varphi \approx 1.618} , the golden ratio. It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron. The diagonals in the rhombs of the convex solid have a ratio of 1 to φ {\displaystyle \varphi } . The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to φ 3 ≈ 4.236 {\displaystyle \varphi ^{3}\approx 4.236} . So the ratio of rhomb diagonals in the medial solid is 1 to φ 2 ≈ 2.618 {\displaystyle \varphi ^{2}\approx 2.618} . This solid is to the compound of small stellated dodecahedron and great dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs. The faces have two angles of arccos ⁡ ( 1 3 5 ) ≈ 41.810 314 895 78 ∘ {\displaystyle \arccos({\frac {1}{3}}{\sqrt {5}})\approx 41.810\,314\,895\,78^{\circ }} , and two of arccos ⁡ ( − 1 3 5 ) ≈ 138.189 685 104 22 ∘ {\displaystyle \arccos(-{\frac {1}{3}}{\sqrt {5}})\approx 138.189\,685\,104\,22^{\circ }} . Its dihedral angles equal arccos ⁡ ( − 1 2 ) = 120 ∘ {\displaystyle \arccos(-{\frac {1}{2}})=120^{\circ }} . Part of each rhomb lies inside the solid, hence is invisible in solid models.

Related hyperbolic tiling It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:

Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

See also Great rhombic triacontahedron Great triambic icosahedron

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

External links Weisstein, Eric W. "Medial Rhombic Triacontahedron". MathWorld. David I. McCooey: animation and measurements Uniform polyhedra and duals

Illustrations

Worked examples

Example 1 — a first encounter with Medial rhombic triacontahedron

Start with the simplest possible case. Write down what Medial rhombic triacontahedron 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 Medial rhombic triacontahedron 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 Medial rhombic triacontahedron 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 Medial rhombic triacontahedron

In research: Medial rhombic triacontahedron 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 Medial rhombic triacontahedron 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: Medial rhombic triacontahedron 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 Medial rhombic triacontahedron 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 Medial rhombic triacontahedron in 20 minutes

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

Frequently asked questions

What is Medial rhombic triacontahedron in simple terms?

Why does Medial rhombic triacontahedron 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 Medial rhombic triacontahedron?

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 Medial rhombic triacontahedron.