geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a

nonconvex uniform polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

, indexed as U₆₀. It is given a

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

sr.

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

for the vertices of an inverted snub dodecadodecahedron are all the

even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...

s of : (±2α, ±2, ±2β), : (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)), : (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)), : (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and : (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)), with an even number of plus signs, where : β = (α²/τ+τ)/(ατ−1/τ), where τ = (1+)/2 is the golden mean and α is the negative real

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

of τα⁴−α³+2α²−α−1/τ, or approximately −0.3352090. Taking the

odd permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...

s of the above coordinates with an odd number of plus signs gives another form, the

enantiomorph In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to ...

of the other one.

Related polyhedra

Medial inverted pentagonal hexecontahedron

The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

. It is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

Denote the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

\phi

, and let

\xi\approx -0.236\,993\,843\,45

be the largest (least negative) real zero of the polynomial

P=8x^4-12x^3+5x+1

. Then each face has three equal angles of

\arccos(\xi)\approx 103.709\,182\,219\,53^

, one of

\arccos(\phi^2\xi+\phi)\approx 3.990\,130\,423\,41^

and one of

360^-\arccos(\phi^\xi-\phi^)\approx 224.882\,322\,917\,99^

. Each face has one medium length edge, two short and two long ones. If the medium length is

2

, then the short edges have length :

1-\sqrt\approx 0.474\,126\,460\,54

, and the long edges have length :

1+\sqrt\approx 37.551\,879\,448\,54

. The

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

equals

\arccos(\xi/(\xi+1))\approx 108.095\,719\,352\,34^

. The other real zero of the polynomial

P

plays a similar role for the

medial pentagonal hexecontahedron In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces. Proportions Denote the golden ratio by \phi, and let \xi\a ...

References

* p. 124

External links