Truncated icosidodecahedron

Key takeaways

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

Reference excerpt

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces. It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15-zonohedron.

Names

The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of an icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. Alternate interchangeable names are:

Truncated icosidodecahedron (Johannes Kepler) Rhombitruncated icosidodecahedron (Magnus Wenninger) Great rhombicosidodecahedron (Robert Williams, Peter Cromwell) Omnitruncated dodecahedron or icosahedron (Norman Johnson) The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection). There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.

Area and volume The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:

A = 30 ( 1 + 3 + 5 + 2 5 ) a 2 ≈ 174.292 0303 a 2 . V = ( 95 + 50 5 ) a 3 ≈ 206.803 399 a 3 . {\displaystyle {\begin{aligned}A&=30\left(1+{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 174.292\,0303a^{2}.\\V&=\left(95+50{\sqrt {5}}\right)a^{3}&&\approx 206.803\,399a^{3}.\end{aligned}}}

If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

Cartesian coordinates Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:

(±⁠1/φ⁠, ±⁠1/φ⁠, ±(3 + φ)), (±⁠2/φ⁠, ±φ, ±(1 + 2φ)), (±⁠1/φ⁠, ±φ2, ±(−1 + 3φ)), (±(2φ − 1), ±2, ±(2 + φ)) and (±φ, ±3, ±2φ), where φ = ⁠1 + √5/2⁠ is the golden ratio. Therefore, the circumradius is the common distance of these points from the origin. For unit edge length, giving

R = ⁠√24φ+19/2⁠ = ⁠√31+12√5/2⁠ ≈ 3.802.

Dissection The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is φ. The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons. An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.

Orthogonal projections The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.

Spherical tilings and Schlegel diagrams The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Schlegel diagrams are similar, with a perspective projection and straight edges.

Geometric variations Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.

Truncated icosidodecahedral graph

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Truncated icosidodecahedron

Start with the simplest possible case. Write down what Truncated icosidodecahedron 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 Truncated icosidodecahedron 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 Truncated icosidodecahedron 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 Truncated icosidodecahedron

In research: Truncated icosidodecahedron 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 Truncated icosidodecahedron 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: Truncated icosidodecahedron is common in secondary-school and first-year university syllabi. It links to neighbouring topics Archimedean solids, Individual graphs, Planar graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Truncated icosidodecahedron 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.

How to study Truncated icosidodecahedron in 20 minutes

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

Frequently asked questions

What is Truncated icosidodecahedron in simple terms?

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon fa…

Why does Truncated icosidodecahedron 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 Truncated icosidodecahedron?

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 Truncated icosidodecahedron.